S-PLUS (and R) - Department of Statistics

S-PLUS (and R) Manual to AccompanyAgresti’s Categorical Data Analysis (2002)2 nd editionLaura A. Thompson, 2006 ©

Table of ContentsIntroduction and Changes from First Edition......................................1A. Obtaining the R Software for Windows.............................................................................1B. Libraries in S-PLUS and Packages in R...........................................................................1C. Setting contrast types using Options()..............................................................................3D. Credit for functions...............................................................................................................3E. Editing functions ...................................................................................................................3F. A note about using Splus Menus .......................................................................................4G. Notice of errors.....................................................................................................................4H. Introductions to the S Language........................................................................................4I. References..............................................................................................................................4J. Acknowledgements...............................................................................................................5Chapter 1: Distributions and Inference for CategoricalData: ...........................................................................................................................................6A. Summary of Chapter 1, Agresti.........................................................................................6B. Categorical Distributions in S-PLUS and R.....................................................................6C. Proportion of Vegetarians (Statistical Inference for Binomial Parameters) ...............8D. The Mid-P-Value ...............................................................................................................11E. Pearson’s Chi-Squared Statistic .....................................................................................11F. Likelihood Ratio Chi-Squared Statistic...........................................................................12G. Maximum Likelihood Estimation.....................................................................................12Chapter 2: Describing Contingency Tables.......................................16A. Summary of Chapter 2, Agresti........................................................................................16B. Comparing two proportions...............................................................................................18C. Partial Association in Stratified 2 x 2 Tables..................................................................19D. Conditional Odds Ratios ...................................................................................................23E. Summary Measures of Assocation: Ordinal Trends.....................................................24Chapter 3: Inference for Contingency Tables .................................28A. Summary of Chapter 3, Agresti........................................................................................28B. Confidence Intervals for Association Parameters .........................................................29C. Testing Independence in Two-way Contingency Tables .............................................35D. Following Up Chi-Squared Tests.....................................................................................37E. Two-Way Tables with Ordered Classification................................................................39F. Small Sample Tests of Independence ............................................................................41G. Small-Sample Confidence Intervals For 2x2 Tables....................................................44Chapter 4: Generalized Linear Models ..................................................50i

A. Summary of Chapter 4, Agresti........................................................................................50B. Generalized Linear Models for Binary Data ...................................................................51C. Generalized Linear Models for Count Data ...................................................................56D. Overdispersion in Poisson Generalized Linear Models ...............................................61E. Negative Binomial GLIMs..................................................................................................63F. Residuals for GLIMs...........................................................................................................65G. Quasi-Likelihood and GLIMs............................................................................................67H. Generalized Additive Models (GAMs).............................................................................68Chapter 5 : Logistic Regression....................................................................72A. Summary of Chapter 5, Agresti.......................................................................................72B. Logistic Regression for Horseshoe Crab Data .............................................................73C. Goodness-of-fit for Logistic Regression for Ungrouped Data....................................77D. Logit Models with Categorical Predictors......................................................................78E. Multiple Logistic Regression............................................................................................82F. Extended Example (Problem 5.17).................................................................................88Chapter 6 – Building and Applying Logistic RegressionModels ....................................................................................................................................92A. Summary of Chapter 6, Agresti........................................................................................92B. Model Selection ..................................................................................................................93C. Using Causal Hypotheses to Guide Model Fitting ........................................................94D. Logistic Regression Diagnostics......................................................................................96E. Inference about Conditional Associations in 2 x 2 x K Tables ..................................102F. Estimation/Testing of Common Odds Ratio .................................................................105G. Using Models to Improve Inferential Power.................................................................106H. Sample Size and Power Considerations......................................................................107I. Probit and Complementary Log-Log Models................................................................109J. Conditional Logistic Regression and Exact Distributions ...........................................111K. Bias-reduced Logistic Regression .................................................................................116Chapter 7 –Logit Models for Multinomial Responses............117A. Summary of Chapter 7, Agresti......................................................................................117B. Nominal Responses: Baseline-Category Logit Models..............................................118C. Cumulative Logit Models.................................................................................................121D. Cumulative Link Models..................................................................................................125E. Adjacent-Categories Logit Models.................................................................................127F. Continuation-Ratio Logit Models....................................................................................128G. Mean Response Models .................................................................................................134H. Generalized Cochran-Mantel Haenszel Statistic for Ordinal Categories ...............139Chapter 8 –Loglinear Models for Contingency Tables .........141A. Summary of Chapter 8, Agresti......................................................................................141ii

B. Loglinear Models for Three-way Tables .......................................................................142C. Inference for Loglinear Models ......................................................................................145D. Loglinear Models for Higher Dimensions ....................................................................147E. Loglinear-Logit Model Connection.................................................................................150F. Contingency Table Standardization...............................................................................151Chapter 9 –Building and Extending Loglinear Models .........152A. Summary of Chapter 9, Agresti......................................................................................152B. Model Selection and Comparison.................................................................................153C. Diagnostics for Checking Models .................................................................................155D. Modeling Ordinal Assocations ......................................................................................156E. Assocation Models ...........................................................................................................158F. Association Models, Correlation Models, and Correspondence Analysis ...............164G. Poisson Regression for Rates .......................................................................................170H. Modeling Survival Times.................................................................................................172I. Empty Cells and Sparseness...........................................................................................174Chapter 10 – Models for Matched Pairs .............................................176A. Summary of Chapter 10, Agresti ...................................................................................176B. Comparing Dependent Proportions...............................................................................177C. Conditional Logistic Regression for Binary Matched Pairs........................................178D. Marginal Models for Square Contingency Tables......................................................181E. Symmetry, Quasi-symmetry, and Quasi-independence ...........................................186F. Square Tables with Ordered Categories .....................................................................189G. Measuring Agreement Between Observers ...............................................................192H. Kappa Measure of Agreement......................................................................................195I. Bradley-Terry Model for Paired Preferences................................................................196J. Bradley-Terry Model with Order Effect.........................................................................199K. Marginal and Quasi-symmetry Models for Matched Sets.........................................200Chapter 11 –Analyzing Repeated Categorical ResponseData.........................................................................................................................................203A. Summary of Chapter 11, Agresti ...................................................................................203B. Comparing Marginal Distributions: Multiple Responses ............................................203C. Marginal Modeling: Maximum Likelihood Approach...................................................205D. Marginal Modeling: Maximum Likelihood Approach. Modeling a RepeatedMultinomial Response ..........................................................................................................211E. Marginal Modeling: GEE Approach. Modeling a Repeated Multinomial Response215F. Marginal Modeling: GEE Approach. Modeling a Repeated Multinomial OrdinalResponse................................................................................................................................219G. Markov Chains: Transitional Modeling .........................................................................221iii

Chapter 12 – Random Effects: Generalized Linear MixedModels for Categorical Responses..........................................................226A. Summary of Chapter 12, Agresti ...................................................................................226B. Logit GLIMM for Binary Matched Pairs.........................................................................227C. Examples of Random Effects Models for Binary Data...............................................230D. Random Effects Models for Multinomial Data .............................................................243E. Multivariate Random Effects Models for Binary Data.................................................245iv

1Introduction and Changes from First EditionThis manual accompanies Agresti’s Categorical Data Analysis (2002). It provides assistance indoing the statistical methods illustrated there, using S-PLUS and the R language. Although I have usedthe Windows versions of these two softwares, I suspect there are few changes in order to use the code inother ports. I have included examples of almost all of the major (and some minor) analyses introduced byAgresti. The manual chapters parallel those from Agresti so that, for example, in Chapter 2 I discussusing the software to conduct analyses from Agresti’s Chapter 2. In most cases I use the data providedin the text. There are only one or two occasions where I use data from the problems sections in Agresti.Discussion of results of analyses is brief since the discussion appears in the text. That said, one of theimprovements in the current manual over the previous version of the manual (Thompson, 1999) is that itis more self-contained. In addition, I include a summary of the corresponding chapter from Agresti at thebeginning of each chapter in the manual. However, it will still be helpful to refer to the text to understandcompletely the analyses in this manual.In the manual, I frequently employ functions that come from user-contributed libraries (packages)of S-PLUS (or R). In the text, I note when a particular library or package is used. These libraries are notautomatically included with the software, but can be easily downloaded from the internet, especially forthe newest version of R for Windows. I mention in the next section how to get these libraries and how toinstall them for use. Many times, the library or package has its own help manual or help section. I willdemonstrate how to access these from inside the software.I used S-PLUS 6.1 for Windows and R versions 1.8.1, 1.9, and 2.0 for Windows for the analyses.However, S-PLUS for Windows versions as far back as 3.0 will do many of the analyses (but not all).This is not so easily said for R, as user-contributed packages frequently apply to the newer versions of R(e.g., at least 1.3.0). Many of the analyses can be applied to either S-PLUS or R. Some need smallchanges in order to apply to both softwares; these changes I have provided where necessary. In general,when I refer to both of the softwares, I will use the “generic” name, S. Also, if I do not indicate that I amusing either S-PLUS or R for a particular command, that means it will work in both softwares.To separate input to R or S-PLUS and output from other text in this manual, I have put normaltext in Arial font and commands and output in courier font. The input commands are in bold font,whereas the output is not. Also, all assignments will use the “

2yags2 (V. Carey) - (used for chapter 11)Most of these libraries can be obtained in .zip form from URL http://lib.stat.cmu.edu/DOS/S/Swin orhttp://lib.stat.cmu.edu/DOS/S. Currently, the URL http://www.stats.ox.ac.uk/pub/MASS4/Winlibs/ containsmany ports to S-PLUS 6.0 for Windows. To install a library, first download it to any folder on yourcomputer. Next, “unzip” the file using an “unzipping” program. This will extract all the files into a newfolder in the directory into which you downloaded the zip file. Move the entire folder to the librarydirectory under your S-PLUS directory (e.g., c:/program files/Insightful/splus61/library).To load a library, you can either pull down the File menu in S-PLUS and select Load Library or type oneof the following in a script or command windowlibrary(“libraryname”,first=T)library(libraryname)# loads libraryname into first database positionTo use the library’s help manual from inside S-PLUS or R type in a script or command windowhelp(library=“libraryname”)The R packages used in this manual that do not come with the software are (note: I still need toadd libraries I will use in chapters 11-15)MASS – (VR bundle, Venables and Ripley)rmutil (J. Lindsey) – (used with gnlm) http://alpha.luc.ac.be/~lucp0753/rcode.htmlgnlm (J. Lindsey) – http://alpha.luc.ac.be/~lucp0753/rcode.htmlrepeated (J. Lindsey) – http://alpha.luc.ac.be/~lucp0753/rcode.htmlSuppDists (B. Wheeler) – (used in chapter 1)combinant (V. Carey) – (used in chapters 1, 3)methods – (used in chapter 3)Bhat (E. Luebeck)– (used throughout)mgcv (S. Wood) – (used for fitting GAM models)modreg (B. Ripley) – (used for fitting GAM models)gee and geepack (J. Yan) – (used in chapter 11)yags (V. Carey) – (used in chapter 11)gllm – (used for generalized log linear models)GlmmGibbs (Myles and Clayton) – (used for generalized linear mixed models, chap. 12)glmmML (G. Broström) – (used for generalized linear mixed models, chapter 12)CoCoAn (S. Dray) – (used for correspondence analysis)e1071 (A. Weingessel) – (used for latent class analysis)vcd (M. Friendly)– (used in chapters 2, 3, 5, 9 and 10)brlr (D. Firth) – (used in chapter 6)BradleyTerry (D. Firth) – (used in chapter 10)ordinal (P. Lindsey) – (used in chapter 7) http://euridice.tue.nl/~plindsey/rlibs.htmldesign (F. Harrell) – (used throughout) http://hesweb1.med.virginia.edu/biostatHmisc (F. Harrell) – (used throughout)VGAM (T. Yee) – (used in chapters 7 and 9)http://www.stat.auckland.ac.nz/~yee/VGAM/download.shtmlmph (J. Lang) – (used in chapters 7, 10)http://www.stat.uiowa.edu/~jblang/mph.fitting/mph.fit.documentation.htm#availabilityexactLoglinTest (B. Caffo) – (used in chapters 9, 10)http://www.biostat.jhsph.edu/~bcaffo/downloads.htmR packages can be installed from Windows using the install.packages function. This function can becalled from the console or from the pull-down “Packages” menu. A package is loaded in the same way

3as for S-PLUS. As well, the command help(package=pkg) can used to invoke the help menu forpackage pkg.To detach a library or package, named library.name or pkg, respectively, one can issue the followingcommands, among others.detach(“library.name”)detach(“package:pkg”)# S-PLUS# RC. Setting contrast types using Options()The options function can be used to set the type of contrasts used in estimating models. The defaultfor S-PLUS 6.1 is Helmert contrasts for factors and polynomial contrasts for ordered factors. The defaultfor R 1.6 is treatment contrasts for factors and polynomial contrasts for ordered factors. I use treatmentcontrasts for factors for most of the analysis so that they match Agresti’s estimates. However, there aretimes when I use sum-to-zero contrasts (contr.sum). The type of contrasts I use is usually indicated by acall to options prior to fitting the model, if necessary. If the call to options has been omitted, pleaseassume I am using treatment contrasts for factors.One can find out exactly what the contrasts are in a glm-type fit by using the functions model.matrixand contrasts. Issuing the comand contrasts(model.matrix(fit)) gives the contrasts.D. Credit for functionsThe author of a function is named if the function was not written by me. Whenever I use functions that donot come with S, I will mention the S-PLUS library or R package that includes them. I also give a URL, ifapplicable.E. Editing functionsIn several places in the text, I mention creating functions or editing existing functions that come witheither S-PLUS or R. There are many ways to edit or create a function. Some of them are specific to yourplatform. However, one procedure that works on all platforms from the command line is a call to fix. Forexample, to edit the method function update.default, and call the new version update.crosstabs, typethe following at the command line (R or S-PLUS)update.crosstabs

4F. A note about using S-PLUS MenusMany of the more common methods I will illustrate can be accomplished via the S-PLUS menus. If youwant to know what code corresponds to a particular menu command, issue the menu command and callup the History window (using the Window menu). All commands you have issued from menus will bethere in (gui) code form which can then be used in a command window or script.G. Notice of errorsAll code has been tested, but there are undoubtedly still errors. Please notify me of any errors in themanual or of easier ways to perform tasks. My email address is [email protected]. Introductions to the S LanguageThis manual assumes some working knowledge of the S language. There is not space to also describeit. Fortunately, there are many tutorials available for learning S. Some of them are listed in your User’sGuides that come with S-PLUS. Others are listed on the CRAN website for R (see Section A above).I. ReferencesAgresti, A. (2002). Categorical Data Analysis 2 nd edition. Wiley.Bishop, C. (1995). Neural Networks for Pattern Recognition. Cambridge University PressChambers, J. (1998). Programming with Data. Springer-Verlag.Chambers, J. and Hastie, T. (1992). Statistical Models in S. Chapman & Hall.Ewens, W. and Grant. G. (2001) Statistical Methods in Bioinformatics. Springer-Verlag.Gentleman, R. and Ilhaka, R. (2000). “Lexical Scope and Statistical Computing.” Journal ofComputational and Graphical Statistics, 9, 491-508.Green, P. and Silverman, B. (1994). Nonparametric Regression and Generalized Linear Models,Chapman & Hall.Fletcher, R. (1987). Practical Methods of Optimization. Wiley.Harrell, F. (1998). Predicting Outcomes: Applied Survival Analysis and Logistic Regression. Unpublishedmanuscript. Now available as Regression Modeling Strategies : With Applications to LinearModels, Logistic Regression, and Survival Analysis. Springer (2001).Liao, L. and Rosen, O. (2001). “Fast and Stable Algorithms for Computing and Sampling From theNoncentral Hypergeometric Distribution." The American Statistician, 55, 366-369.Lloyd, C. (1999). Statistical Analysis of Categorical Data, Wiley.McCullagh, P. and Nelder, J. (1989). Generalized Linear Models, 2 nd ed., Chapman & Hall.Ripley B. (2002). On-line complements to Modern Applied Statistics with SPLUS(http://www.stats.ox.ac.uk/pub/MASS4).Ross, S. (1997). Introduction to Probability Models. Addison-Wesley.

5Selvin, S. (1998). Modern Applied Biostatistical Methods Using SPLUS. Oxford University Press.Sprent, P. (1997). Applied Nonparametric Statistical Methods. Chapman & Hall.Thompson, L. (1999) S-PLUS Manual to Accompany Agresti’s (1990) Catagorical Data Analysis.(http://math.cl.uh.edu/~thompsonla/5537/Splusdiscrete.PDF).Venables, W. and Ripley, B. (2000). S programming. Springer-Verlag.Venables, W. and Ripley, B. (2002). Modern Applied Statistics with S. Springer-Verlag.Wickens, T. (1989). Multiway Contingency Tables Analysis for the Social Sciences. LEA.J. AcknowledgementsThanks to Gregory Rodd and Frederico Zanqueta Poleto for reviewing the manuscript and finding manyerrors that I overlooked. Remaining errors are the sole responsibility of the author.

6Chapter 1: Distributions and Inference forCategorical DataA. Summary of Chapter 1, AgrestiIn Chapter 1, Agresti introduces categorical data, including its types, distributions and statisticalinference. Categorical variables measured on a nominal scale take values that do not have a naturalordering, whereas categorical variables measured on an ordinal scale take values that do have anordering, but not necessarily a numerical ordering. Categorical variables are sometimes called discretevariables because they only take on a discrete or countable number of values.Agresti discusses three main distributions for categorical variables: binomial, multinomial, andPoisson. The binomial distribution describes the distribution of the sum of a fixed number of independentBernoulli trials (i.e., binary outcomes), where the probability of success is fixed across trials. Themultinomial distribution extends the binomial distribution to handle trials with possibly more than twooutcomes. The Poisson distribution describes the distribution of the number of events occurring in aspecified length of time or space, where the intervals of time or space are independent with respect to theoccurrence of an event. Formal assumptions related to the Poisson distribution (which derives from thehomogeneous Poisson process) can be found in any text on stochastic processes (see, for example,Ross, 1997). A nice explanation, however, can also be found in Ewens and Grant (2001).When observations modeled as coming from a binomial or Poisson distribution are much morevariable than that predicted by the respective theoretical distributions, the use of these distributions canlead to poor predictions because the excess variability is not considered. The prescence of this excessvariability is called overdispersion. There are options for dealing with overdispersion when it is suspectedin a data set that otherwise could be reasonably modeled using a conventional distribution like thebinomial or Poisson. One option is to incorporate random effects into the model (see Agresti, Chapter12). Another option is to use a distribution with a greater variance than that dictated by the binomial orPoisson (e.g., the beta-binomial distribution is a mixture of binomials with different probabilities ofsuccess).Given a particular probability distribution to describe the data, the likelihood function is theprobability of the data as a function of the parameters. The maximum likelihood estimate (MLE) is thevalue of the parameter that maximizes this function. Thus, the MLE is the value for the set of parametersthat give the observed data the highest probability of occurrence. Inference for categorical data dealswith the MLE and tests derived from maximum likelihood. Tests of the null hypothesis of zero-valuedparameters can be carried out via the Wald Test, Likelihood Ratio Test, and the Score Test. These testsare based on different aspects of the likelihood function, but are asymptotically equivalent. However, insmaller samples they yield different answers and are not equally reliable.Agresti also discusses confidence intervals derived from “inverting” the above three tests. Asmentioned on p. 13 of Agresti, “a 95% confidence interval for β is the set of β0for which the test of H 0 :β = β0has a P-value exceeding 0.05.” That is, it describes the set of β0values for which we would“keep” the null hypothesis (assuming a significance level of 0.05).The chapter concludes with illustration of statistical inference for binomial and multinomialparameters.B. Discrete Probability Distributions in S-PLUS and RS-PLUS and R come with support for many built-in probability distributions, and support for manyspecialized distributions can be downloaded from statlib or from the CRAN website (see IntroductionSection). Each distribution has functions for obtaining cumulative probabilities, density values, quantiles,and realizations of random variates. For example, in both S-PLUS and Rdbinom(x, size, prob) computes the density of the indicated binomial distribution at xpbinom(x, size, prob) computes the cumulative density of the indicated binomial distribution

8C. Proportion of Vegetarians (Statistical Inference for Binomial Parameters)An example of the use of the S-PLUS distribution functions comes from computing the asymptoticand exact confidence intervals on the proportion of vegetarians (p. 16). In a questionnaire given to anintroductory statistics class (n = 25), zero students said they were vegetarians. Assuming that thenumber responding yes is distributed binomially with success probability π , what is a 95% confidenceinterval for π , given that a point estimate, ˆ π , is 0? Agresti cites three approximate methods and twoexact methods. The approximate methods are given first.1. Approximate Confidence Intervals on π1) Inverting the Wald Test (AKA Normal Approximation)ˆ πwhich is computed quite easily in S-PLUS “by hand”.zα /2ˆ π (1 − ˆ π )n± (1.1)phat

9library(Hmisc, T)binconf(x=0, n=25, alpha=.05, method="wilson")PointEst Lower Upper0 1.202937e-017 0.13319233) A 95% likelihood-ratio (LR) confidence interval is the set of π0for which the likelihood ratio test has ap-value exceeding the significance level, α . The expression for the LR statistic is simple enough so thatwe can find the upper confidence bound analytically. However, we could have obtained it using uniroot(both R and S-PLUS) or using nlmin (S-PLUS only). These functions can be used to solve an equation,i.e., find its zeroes. uniroot is only for univariate functions. nlmin can also be used to solve a systemof equations. A function doing the same in R would be function nlm or function nls (for nonlinear leastsquares) in library nls.Using uniroot we set up the LR statistic as a function, giving as first argument the parameter for whichwe want the confidence interval. The remaining arguments are the observed successes, the total, and thesignificance level. uniroot takes an argument called interval that specifies where we want to searchfor the root. There cannot be a singularity at either of these values. Thus, we cannot use 0 and 1.LR

10[1] T$conv.type:[1] "x convergence"2. Exact Confidence Intervals on πThere are several functions available for computing exact confidence intervals in R and S-PLUS. In R,the function binom.test computes a Clopper-Pearson interval. But, the same function in S-PLUSdoesn’t give confidence intervals, at least not automatically.binom.test(x=0, n=25, conf.level=.95)# RExact binomial testdata: 0 and 25number of successes = 0, number of trials = 25, p-value = 5.96e-08alternative hypothesis: true probability of success is not equal to 0.595 percent confidence interval:0.0000000 0.1371852sample estimates:probability of success0The function binconf in the Hmisc library also gives Clopper-Pearson intervals via the use of the“exact” method.library(Hmisc, T)binconf(x = 0, n = 25, alpha = .05, method = "exact") # SPLUSPointEst Lower Upper0 0 0.1371852In addition, several individuals have written their own functions for computing Clopper-Pearson as well asother types of approximate intervals besides the normal approximation interval. A search of “exactbinomial confidence interval” on the “S-news” search page (http://lib.stat.cmu.edu/cgi-bin/iform?SNEWS)gave several user-made functions.The improved confidence interval described in Blaker (2000, cited in Agresti) can be computed in S-PLUSusing the following function, which is a slight modification of the function appearing in the Appendix of theoriginal reference.acceptbin

11}lower

12The S-PLUS function chisq.test is used for situations where one has a 2x2 cross-tabulation(preferably computed using the function table) or two factor vectors, and wants to test for independenceof the row and column variables.F. Likelihood Ratio Chi-Squared StatisticThe likelihood ratio chi-squared statistic is easily computed “by hand” for multinomial frequencies. Usingvectorized calculations, G 2 is for Mendel’s dataobs

13the φ1parameter using the scale argument so that when p[1] is subtracted from p[3], we should alwaysget a positive value. I have also started φ1at a value much larger than λ1.obj.function

14[1] 5.191676e-007$iterations:[1] 22We get the same result. One could supply the Hessian using the function deriv3. The Mass librarysupplies a function called vcov.nlminb to extract the inverse of observed information (or a finitedifference approximation if the Hessian is not supplied) from an nlminb object.In R, we can use the function nlm, which is similar in that it minimizes an objective function of severalvariables, allows boundary constraints, and uses a quasi-Newton optimizer. However, it does not allowbox constraints on the parameters.obj.res

15}attr(obj.res(la1, la2, pi1, y1, y2, y3, y4), "gradient")optim(par=c(11, 62, 2*(11+4)), fn=obj.function, gr=obj.gr, lower=c(0,0,0),method="L-BFGS-B", control=list(parscale=c(1,1,10)), y=c(11, 62, 4, 7))$par[1] 14.35235 57.89246 19.57130$value[1] -216.6862$countsfunction gradient25 25$convergence[1] 0$message[1] "CONVERGENCE: REL_REDUCTION_OF_F

16Chapter 2: Describing Contingency TablesA. Summary of Chapter 2, AgrestiChapter two in Agresti introduces two-way I × J contingency tables. If both the row and columnof a table denote random variables, then the probabilities { π ij} define the joint distribution of the twovariables. The marginal distributions are denoted by { πi +} for the row variable and { π + j} for the columnvariable. For a fixed value i of the row variable, the column variable has the conditional distribution{ π ,..., π } . The conditional distribution is especially important if the row variable is fixed by design1| i J|iand not free to vary for each observation.With diagnostic tests for a disease, the sensitivity of the test is the conditional probability that thediagnostic test is positive given that subject has the disease. The specificity of the test is the conditionalprobability that the test is negative given that the subject does not have the disease. In a 2x2 table withrows indicating disease status (yes, no) and columns indicating test result (positive, negative), thesensitivity is π1|1, and the specificity is π2|2.Row and column variables are independent if the conditional distribution of the column variablegiven the row variable is the same as the marginal distribution of the column variable (and vice versa).That is, πji |= π + jfor i = 1,…, I, and πij |= πi +j = 1,…, J. Equivalently, if all joint probabilities equal theproduct of their marginal probabilities: πij= πi +π + j, for all i and j. Thus, when the two variables areindependent, knowledge of the value of the row variable does not change the distribution of the columnvariable, and vice versa.When the row variable is an explanatory variable and the column is a response variable, thenthere is no joint distribution, and independence is referred to as homogeneity of the conditionaldistributions of the column variable given a value for the row variable.Y differ depending on how sampling was done. IfThe distributions of the cell counts { } ijobservations are to be collected over a certain period of time and cross-classified into one of the I × Jcategories, then a Poisson sampling model might be used where cell counts are treated as independentPoisson random variables with parameters { µ ij}. If the total sample size of observations is fixed inadvance (e.g., in a cross-sectional study), then a multinomial sampling model might be used where cell. If the row totalscounts are treated as multinomial random variables with index n and probabilities { π ij}are fixed in advance, perhaps as fixed-size random samples drawn from specific populations that are tobe compared, as in prospective studies, then a product multinomial sampling model may apply where foreach i, the counts { Yji}| have a multinomial distribution with index niand probabilities πji |j = 1,…, J . Ifboth row and column totals are fixed by design, then a hypergeometric sampling distribution applies forthe cell counts.However, there are times when certain sampling models are assumed, but sampling was actuallydone differently. For example, when the row variable is an explanatory variable, product multinomialsampling model may be used even though the row totals were not actually fixed in advance. Also, thePoisson model is used even when the total sample size is fixed in advance (see Chapter 8 of Agresti).Section 2.2 discusses comparing two proportions from two samples, including the difference ofproportions, relative risk, and odds ratio. The relative risk compares the proportions of the two groups ina ratio. If the rows of a 2x2 table represent groups and the columns represent a binary response, thenthe relative risk of a positive response is the ratio π 1|1π 1|2. A relative risk of 1.0 corresponds toindependence. A relative risk of C means that π 1|1= Cπ1|2. The odds ratio is the ratio of odds of apositive response by group

17π1|1 (1 −π1|1 ) π11π22θ = =(2.1)π1|2 (1 −π1|2 ) π12π21θ = , the row and column variables are independent. Values of θ farther from 1.0 in a givenθ = , the oddsWhen 1direction represent stronger association. For example, as on p. 45 of Agresti, when 0.25of “success” in group 1 (row 1) are 0.25 times the odds in group 2 (row 2), or equivalently, the odds ofsuccess in group 2 are 4 times the odds in group 1. The odds ratio can be used with a joint distribution ofthe row and column variables too. Indeed, it can be used with prospective (rows totals fixed),retrospective (column totals fixed), and cross-sectional designs. Finally, if the rows and columns areinterchanged, the value of the odds ratio does not change. The sample odds ratio uses the observedsample counts, nij.Odds ratios can be computed for retrospective sampling designs (case-control studies). Relativerisk cannot be computed because the outcome variable is fixed by design. However, if the probability ofthe outcome is close to zero for both groups, then the odds ratio can be used to approximate relative riskusing the formula on p. 47 of Agresti.In observational studies, confounding variables can be controlled with stratification orconditioning. The association between two variables X and Y given that another measured variable Ztakes the value z is called a conditional association. The 2 x 2 table resulting from cross-classifying allobservations with Z = z by their X and Y values is called a partial table. If Z is ignored, the X-Y table iscalled a marginal table. Simpson’s Paradox is the result that a marginal association can have a differentdirection than a conditional association. For example, in the death penalty example on p. 49-51, ignoringvictim’s race, the death penalty (Y) is more likely for whites than for blacks (X). However, conditioning onvictim’s race (either black or white), the death penalty is more likely for blacks than for whites. Theparadox in this case can be explained by the strong association between victim’s race (ignored in themarginal association) and defendant’s race and that between victim’s race and the death penalty. Thedeath penalty was more likely when the victims were white (regardless of defendant race). Also, whiteswere more likely to kill whites than any other victim/defendant race combination in the sample. So, thereare a lot of whites receiving the death penalty in the sample. On the other hand, blacks were more likelyto kill blacks. Thus, there are fewer blacks receiving the death penalty in the sample. But, if we look atonly white victims, there are relatively more blacks receiving the death penalty than whites. The same istrue for black victims. An unmodeled association between victim’s and defendant’s race hides thisconclusion.Does Simpson’s Paradox imply that we should distrust all contingency table analyses? After all,there are undoubtedly unmeasured variables that could be potential conditioning variables in allcontingency tables. Could these variables change the direction of marginal associations? Page 51 inAgresti paraphrases J. Cornfield’s result “that with a very strong XY association [marginal association], avery strong association must exist between the confounding variable Z and both X and Y in order for theeffect to disappear or change …”.For I x J x K tables (where X has I levels, Y has J levels, and Z has K levels), if X and Y areindependent in partial table k, then X and Y are conditionally independent given that Z takes on value k. IfX and Y are independent at all levels of Z, then X and Y are conditionally independent given Z. Or, X andY only depend on each other through Z. Once variability in Z is removed, by fixing it, X and Y are nolonger related statistically. Conditional independence does not imply marginal independence. For 2 x 2 xK tables, X and Y are conditionally independent given Z if the odds ratio between X and Y equals 1 ateach category of Z. For the general case of I x J x K tables, independence is equivalent to all( I −1)( J −1)local odds ratios equaling 1.0.An analogy to no three-way interaction in ANOVA is homogeneous association. A 2 x 2 x K tablehas homogeneous XY association if the conditional odds ratios comparing two categories of X to twocategories of Y are the same at each level of Z. When interaction exists, the conditional odds ratio forany pair of variables (say X and Y) changes across categories of the third (say Z), wherein the thirdvariable is called an effect modifier because the effect of X on Y (the response) changes depending onthe level of Z. For the general case of I x J x K tables, homogeneous XY association means that anyconditional odds ratio formed using two categories of X and two categories of Y is the same at eachcategory of Z.

18The chapter concludes with discussion of summary measures of association for nominal andordinal data. The measures described include Kendall and Stuart’s measure of proportional reduction invariance from the marginal distribution of the response to the conditional distributions given the value ofan explanatory vector, Theil’s uncertainty coefficient, the proportional reduction in entropy (or uncertainty)in response given explanatory variables, and measures for ordinal association such as concordance andGamma.B. Comparing two proportionsThe Aspirin and Heart Attacks example is used to illustrate several ways to compare proportions betweentwo groups. The two groups are aspirin-takers and placebo-takers. Fatal and non-fatal heart attacks arenot distinguished. The data are in Table 2.1 on p. 37 of Agresti. Setting up the data is easy:x

19temp$estimate[1]-temp$estimate[2][1] -0.007706024Other useful quantities are easily computed. Here, I calculate the relative risk and odds ratio using theobject temp above, as well as the original data vectors:Relative risk:temp$estimate[2]/temp$estimate[1][1] 1.817802Odds ratio (simple enough not to need temp):x[2]*(n[1]-x[1])/(x[1]*(n[2]-x[2]))[1] 1.832054C. Partial Association in Stratified 2 x 2 TablesThe Death Penalty example is given on p. 48 of Agresti to illustrate partial tables and conditional oddsratios. The effect of the defendant’s race (X) on the death penalty verdict (Y) is studied, treating thevictim’s race (Z) as a control variate. The 2 x 2 x 2 table can be created using the function crosstabs inS-PLUS (xtabs in R). The function print.crosstabs can also be called directly with output fromcrosstabs to control what is actually printed after the call.vic.race

20victim=blackdefendant|death|yes |no |RowTotl|-------+-------+-------+-------+white | 0 | 16 |16 ||0 |1 |0.1 ||0 |0.1 | ||0 |0.024 | |-------+-------+-------+-------+black | 4 |139 |143 ||0.028 |0.97 |0.9 ||1 |0.9 | ||0.0059 |0.21 | |-------+-------+-------+-------+ColTotl|4 |155 |159 ||0.025 |0.97 | |-------+-------+-------+-------+Test for independence of all factorsChi^2 = 419.5589 d.f.= 4 (p=0)Yates' correction not usedSome expected values are less than 5, don't trust stated p-valueThe differences in R are enough so that the above is worth repeating in R. Below is the same result(including set up) from R’s xtabs. Note that the fac.design has been replaced by expand.grid.The output is also much more terse by default.vic.race

21Getting the marginals can be done by re-calling the function without victim, or one can use the updatefunction to update the call by subtracting victim from the formula. update is a generic function forwhich methods are written for different classes of objects. There is no method for class crosstabs, so acall to update via the set of commands:temp

22|0.079 |0.64 | |-------+-------+-------+-------+black | 15 |176 |191 ||0.079 |0.92 |0.28 ||0.22 |0.29 | ||0.022 |0.26 | |-------+-------+-------+-------+ColTotl|68 |606 |674 ||0.1 |0.9 | |-------+-------+-------+-------+Test for independence of all factorsChi^2 = 1.468519 d.f.= 1 (p=0.2255796)Yates' correction not usedwhich is the bottom portion of Table 2.6 in Agresti.In R, we also create an update method for xtabs, which is a function that we will call update.xtabs.This will be the same as update.default, with the following substitutions:Change the linecall

23prop. defendant = white0.0 0.2 0.4 0.6 0.8 1.0whiteblackwhiteblackyesdeathnoIn the body of the plot are the segments for white and black victims. The sizes of the circles connectedby segments represent the relative count at that particular cell of the table. The dotted segmentrepresents the marginal table (not conditioning on race of victim). It shows that the proportion of whitedefendants is higher for the death penalty group than for those who did not get the death penalty.From conditional associations, however, we see that a white victim is much more likely to have a whitedefendant than a black defendant (due to the height of the “white” segment), and a black victim is muchmore likely to have a black defendant. Those who receive the death penalty are more frequently thosewho had white victims than those who had black victims (see the circles for the “yes” column).D. Conditional Odds RatiosAs the objects returned by xtabs in R and crosstabs in S-PLUS are already arrays, we can just useapply on the 2D slices representing the conditional tables.apply(temp,3,function(x) x[1,1]*x[2,2]/(x[2,1]*x[1,2]))white black0.4306105 0.0000000

24The R package vcd has a function oddsratio, which computes conditional odds ratios for 2x2x… tables,along with asymptotic confidence intervals. If we use it on the death penalty table to get odds ratioswithin victim’s race, we getsummary(oddsratio(temp, log=F, stratum=3))Odds Ratio lwr uprwhite 0.4208843 0.20498745 0.8641678black 0.9393939 0.04838904 18.2367947Note that the odds ratio for black victims is noticeably larger than that computed using apply. Thefunction oddsratio adds 0.5 to each count prior to computing the odds ratios. The estimate near 1.0implies that for black victims, the odds of getting the death penalty are equal for white and blackdefendants.A plot method is also available for oddsratio (plot.oddsratio), which may be quite useful with a largetable.E. Summary Measures of Assocation: Ordinal TrendsAn example to illustrate ordinal measures of association comes from the income and job satisfaction datain Table 2.8 (p. 57, Agresti). The respondents are black males in the U.S. Four ordinal classificationsconstitute the response categories for both variables (see the table). We might postulate that as incomeincreases, reported job satisfaction increases as well, and vice versa. Measures analogous to correlationmeasure the degree of monotonic relationship between the two variables. Agresti uses a concordancemeasure to quantify the association.A pair of individuals in the sample is concordant if the individual who is ranked higher (among the twoindividuals) on one variable, say X, also ranks higher on the second variable, say Y. The pair isdiscordant if the individual ranking higher on X ranks lower on Y. The pair is tied if the individuals havethe same classification on X and/or Y. If the number of concordant pairs, say C, exceeds the number ofdiscordant pairs, say D, then a positive monotonic relationship is supported. A statistic that is based onthis difference is Goodman and Kruskal’s Gamma. It is the estimated difference between the probabilityof concordance and the probability of discordance, given that a pair is untied. Gamma, γ , is estimatedas( C − D)ˆ γ =(2.2)( C + D)Gamma measures monotonic association between two variables, with range −1≤≤ 1. Positive andnegative associations have the corresponding sign changes in ˆ γ . Perfect monotonicity in the sample( ˆ γ = 1 ) occurs when there are either no discordant pairs (D = 0: ˆ γ = 1 ) or there are no concordantpairs (C = 0: ˆ γ =− 1 ).One can create a cross-classified table in S-PLUS out of Table 2.8 using the following commands (usextabs in R instead of crosstabs):income

25Call:crosstabs(formula = count ~ income + jobsat, data = table.2.8)96 cases in table+----------+|N ||N/RowTotal||N/ColTotal||N/Total |+----------+income |jobsat|VD |LD |MS |VS |RowTotl|-------+-------+-------+-------+-------+-------+40000 | 0 | 1 | 9 |11 |21 ||0 |0.048 |0.43 |0.52 |0.22 ||0 |0.077 |0.21 |0.31 | ||0 |0.01 |0.094 |0.11 | |-------+-------+-------+-------+-------+-------+ColTotl|4 |13 |43 |36 |96 ||0.042 |0.14 |0.45 |0.38 | |-------+-------+-------+-------+-------+-------+Test for independence of all factorsChi^2 = 5.965515 d.f.= 9 (p=0.7433647)Yates' correction not usedSome expected values are less than 5, don't trust stated p-valueHere is a function for computing Goodman and Kruskal’s gamma. There is a different version of thisfunction in Chapter 3 of this manual (called Gamma2.f). This version uses the computations from problem3.27 in Agresti, and also computes the standard error of Gamma. It is faster than Gamma.f.Gamma.f

26We can use this on table 2.8 by just inputting the result of the crosstabs call above:Gamma.f(temp)$gamma:[1] 0.2211009$C:[1] 1331$D:[1] 849Selvin (1998) computes the number of concordant and discordant pairs using the outer function alongwith ifelse statements (Selvin, p. 339). However, the procedure is memory intensive. The functionabove takes between 0.33 and 0.60 CPU seconds on a Pentium 4, 1.4 GHz, with 512 MB RAM.Other measures of association can be computed immediately from the chi-square value output fromchisq.test (e.g., phi, Cramer’s V, Cramer’s C). See Selvin p. 336ff for more details. The R packagevcd has a function assoc.stats that computes these association measures along with the Pearson andLR chi-square tests. For example, on the job satisfaction data (where we used xtabs instead ofcrosstabs),summary(assoc.stats(temp))jobsatincome VD LD MS VS40000 0 1 9 11X^2 df P(> X^2)Likelihood Ratio 6.7641 9 0.66167Pearson 5.9655 9 0.74336Phi-Coefficient : 0.249Contingency Coeff.: 0.242Cramer's V : 0.144A nice method for an object of class crosstabs is the “[“ method. This allows us to select smaller tablesin the following way. Suppose I want to print Table 2.8 with the last category of the income variableeliminated. This istemp[1:3,1:4]Call:crosstabs(formula = count ~ income + jobsat, data = table.2.8)75 cases in table+----------+|N ||N/RowTotal||N/ColTotal||N/Total |+----------+income |jobsat|VD |LD |MS |VS |RowTotl|-------+-------+-------+-------+-------+-------+

|0.25 |0.23 |0.23 |0.17 | ||0.01 |0.031 |0.1 |0.062 | |-------+-------+-------+-------+-------+-------+15000-2| 2 | 3 |10 | 7 |22 ||0.091 |0.14 |0.45 |0.32 |0.29 ||0.5 |0.23 |0.23 |0.19 | ||0.021 |0.031 |0.1 |0.073 | |-------+-------+-------+-------+-------+-------+25000-4| 1 | 6 |14 |12 |33 ||0.03 |0.18 |0.42 |0.36 |0.44 ||0.25 |0.46 |0.33 |0.33 | ||0.01 |0.062 |0.15 |0.12 | |-------+-------+-------+-------+-------+-------+ColTotl|4 |12 |34 |25 |75 ||0.053 |0.16 |0.45 |0.33 | |-------+-------+-------+-------+-------+-------+Test for independence of all factorsChi^2 = 1.432754 d.f.= 6 (p=0.9638357)Yates' correction not usedSome expected values are less than 5, don't trust stated p-value27

28Chapter 3: Inference for Contingency TablesA. Summary of Chapter 3, AgrestiThis chapter discusses interval estimation and testing for two-way contingency tables, bothunordered and ordered. Confidence intervals for association parameters like the odds ratio, difference inproportions, and relative risk for 2x2 tables can be computed using large-sample normality (Wolf’sprocedure). Score and profile likelihood confidence intervals are better alternatives, especially for smallersample sizes.For an I x J contingency table, the hypothesis that the row and column variables are independentmay be tested using a chi-square test (either likelihood ratio statistic or Pearson’s chi-squared statistic).Although the two statistics are asymptotically equivalent and asymptotically chi-squared, Pearson’s chisquaredstatistic tends to be better for smaller counts. If the row and column variables are ordered, thena trend may be used as the alternative hypothesis. Chi-squared tests can be followed up by residualanalysis and chi-squared tests on partitioned tables to indicate exactly where association lies. Rules forpartitioning likelihood ratio chi-squared tests appear on p. 84 of Agresti.If independence of the two classification factors of an I x J contingency table approximatelydescribes the table, then the MLEs of the cell probabilities under the hypothesis of independence canhave lower MSE than can the MLEs under the alternative (i.e., the sample proportions), except for whenthe sample size is very large. This is because the variance of the MLEs under independence is smallerbecause they are based on fewer parameters. However, they are biased. The bias can dominate thevariance when the sample size increases, thereby causing the independence MLEs to lose theiradvantage over the sample proportions. Agresti gives details on p. 85-86.If row and column variables are ordered, then a test of independence that has a trend alternativehas greater power to detect a true association (if the association is a trend of that type) than would ageneral-purpose chi-squared test. One reason for this is due to the fewer degrees of freedom for thetrend statistics versus the chi-squared statistics, as there are fewer parameters to estimate for the trendstatistics.There is a disadvantage in using trend tests with ordinal level variables because one has tochoose the scores for the levels of each of the variables. An inappropriate choice can downplay anassociation, making the test less sensitive to it. Agresti shows that the popular choice to use categorymidranks as scores can affect sensitivity of test statistics if the cell sizes are quite disparate. Asuggestion is to use equally-spaced scores unless there is an inherent numerical order in the levelsalready.When either the row or column variable has two categories, then the tests based on linear ormonotone trend that are discussed in Section 3.4 of Agresti reduce to certain well-known nonparametrictests. See p. 90 in Agresti. However, some care is needed when using midranks for the categories ofthe response variable (reducing to Wilcoxon Mann-WhitneyTest), as there may be very many ties (seeSprent, 1997).With a small total sample size, the exact tests of Section 3.5 are preferred over analogous largesampletests. Fisher’s Exact Test is for testing independence of row and column variables in a 2x2 table.The same ideas apply for an I x J table, though the computing time is increased. The test assumes fixedrow and column totals, but as these are sufficient statistics for the row and column probabilities, whichdetermine the null distribution, the test can be applied even if row and column totals are not fixed bydesign. If marginal totals were not fixed by design, an alternative is to use the “unconditional” test ofSection 3.5.5, which fixes row totals only and assumes independent binomial row samples. Anunconditional test may have more power than Fisher’s Exact Test because the null distribution is lessdiscrete, allowing more values for the p-value to assume. Thus, one is not forced to be as conservativeas with a conditional test.Small-sample “exact” confidence intervals for the odds ratio for a 2x2 table can be computedusing Cornfield’s tail method. However, if the data are highly discrete, then Agresti suggets anadjustment to the interval using the mid-p-value.

29B. Confidence Intervals for Association ParametersWald confidence intervals, score intervals, and profile likelihood intervals are illustrated for theAspirin and Myocardial Infarction example on p. 72 in Agresti. We briefly describe each type of interval.1. Wald Confidence IntervalsSuppose the statistic T n is used to estimate an unknown parameter θ = ET ( n). Wald confidenceintervals are based on asymptotic normality of the standardized statistic, ( T ) n −θse( T). Inverting this statisticgives the 100(1− α )% confidence interval on θ , Tn± zα/2σ ( Tn), where z α /2is the 100(1 − α /2) thquantile of the standard normal distribution, and σ ( ) is the standard deviation of T n . An estimate ofT nσ ( ) is given by the delta method (see Section 3.1.5 in Agresti), with sample statistics inserted whereparameters are required (e.g., replace cell probabilities with sample proportions). For the odds ratio (θ ),difference in proportions ( π1 − π2), and relative risk ( π1 π2) in a 2x2 table (assuming independentbinomial distributions for each of the two samples), the standard errors estimated using this method are:T n−1 −1 −1 −1( ) 1/211 12 21 22ˆ(log σ ˆ θ) = n + n + n + n(3.1)andˆ1(1 ˆ1) ˆ2(1 ˆ2)ˆ ˆ ˆ⎡π −π π −π⎤σπ (1− π2)=⎢−⎣ n1 n ⎥2 ⎦ˆ σ ( log ˆ π ˆ π ) = ⎡(1 − ˆ π ) ˆ π n + (1 − ˆ π ) ˆ π1/2−1 −1 −1 −11 2 1 1 1 2 2 21/2(3.2)⎣ n ⎤⎦ (3.3)A Wald confidence interval on either the log odds ratio or the log ratio is more stable than oneconstructed for the respective ratios, themselves. Take the antilog of the interval endpoints to get aconfidence interval on the ratio.We can apply these formulae to the Aspirin and Myocardial Infarction data easily using arithmeticoperations in S. However, sometimes a function is useful. The function Wald.ci below computesasymptotic confidence intervals for the proportion difference, odds ratio, and relative risk.To set up the data, we first do so as a data frame, then use the design.table function (S-PLUS) tochange it to an array.Drug

30We apply the function Wald.ci below to get the confidence intervals.Wald.ci

31So, the death rate for the placebo group ranges from 0.85 to 2.85 times that for the aspirin group (oddsratio confidence interval). As the interval for the odds ratio covers 1.0 and somewhat below 1.0, there isstill a small chance that aspirin has slight negative effects. The intervals on the relative risk anddifference of proportions give similar conclusions.Note that the relative risk estimate is very close to that of the odds ratio. This is not surprising becausethe sample proportion of deaths from myocardial infarction were small in both groups. The relationshipbetween the odds ratio and relative risk (see p. 47 of Agresti) indicates that the two will be similar in thiscircumstance.The same code works in R with minor changes. R does not have the design.table function (althoughwith the change of one line, you can source it, as well as factor.names into R. Contact me for the onelinechange). However, a glance at the function design.table in S-PLUS shows that the “workhorse” isthe tapply function. So, just change the last line as follows:Drug

32minimizer of L( 1, 2| y1, y2, n1, n2)that the gradients of the two functions L and c are related as follows:π π (see for example Fletcher, 1987). It can be shown that this implies∇ L( π , π | y , y , n, n ) = λ∇ c( π , π )(3.5)1 2 1 2 1 2 1 2implying that to find a local minimizer (note that we will negate L later to find a maximizer) we must solve(3.5) and also the equation c( π1, π2)= 0. Thus, we must solve a system of three equations in threeunknowns ( π1, π2, λ ) , then discard the solution for the constant λ (called the Lagrange multiplier).The function f below is the left-hand side of the system of equations we want to solve. This is a functionthat returns a three-dimensional S vector whose first two values represent the gradient of (the negativeof) the Lagrangian, with respect to ( π1, π2), with values for y i and n i given by the values in Table 3.1(Agresti). The third value in the vector is c( π1, π2).f

33The same idea can be carried out with R, using, for example, the optim function. Then, the followingcommands produce a score confidence interval. This time, the function g (here called gfun) is directlyused instead of just existing within solveNonlinear. Also, we control the step sizes (ndeps) for thecomputation of the finite difference gradient to be much smaller than the default (see also, the R packageBhat with function dfp to possibly get around this). And, we use a set of Deltas much narrower in orderto get better accuracy without needing much more computation time.gfun

34However, we still must type in the log likelihood as a function of θ , π1 +, and π + 1. One can find thenecessary transformations very easily using a computer algebra system like Mathematica (2001)..muchmore readily than using S!. These are then copied almost “word” for “word” into the following function,nlogf:library(Bhat)# neg. log-likelihood of "new" multinomial modelnlogf

35trying upper bound ------------------------starting at: 0.9856728initial guess: 0.5029411 0.03382424 1.153263begin Newton-Raphson search for profile lkh conf. bounds:eps value for stop criterium: 0.001nmax : 10CONVERGENCE: 6 iterationschisquare value is: 3.841459confidence bound of theta is 0.8617511log derivatives: 2.122385e-06 6.523067e-06label estimate log deriv log curv1 p1p 0.502946 2.12238e-06 339.3352 pp1 0.0336989 6.52307e-06 41.85023 theta 0.861751 6.44456 10.8561[1] 0.8617511 2.8960895Thus, the profile likelihood based confidence interval for the odds ratio is: (0.862, 2.896), fairly close tothe Wald interval.C. Testing Independence in Two-way Contingency TablesFor multinomial sampling with probabilities { π ij} in an I x J contingency table, the null hypothesis ofstatistical independence of the row and column variables is H : 0πij = πi +πj +for all i and j. Pearson’schi-squared statistic can be used to test this hypothesis. The expected frequencies in an I x Jcontingency table under H 0 are µij= nπij = nπi +πj +, with MLEs given by îjµ = n i +n j +n. Then, thechi-squared statistic, X 2 , is given on p. 78 of Agresti.The score test uses Pearson’s chi-squared statistics. But, as mentioned in Agresti, the likelihood ratiotest uses a different “chi-squared statistic” (they are both called chi-squared statistics because they bothhave asymptotic chi-squared distribution under the null hypothesis of independence). The likelihood ratiochi-squared statistic is given on p. 79 of Agresti. The degrees of freedom for each asymptotic chisquareddistribution is (I − 1)(J − 1). The limiting distributions apply when the total sample size is largeand the number of cells is fixed. Agresti p.79-80 gives further details.Agresti uses data on religious fundamentalism and degree of education to illustrate the two different chisquaredtests of independence. The data are in Table 3.2 (p.80).First, I set up the data in S-PLUS:religion.counts

36Pearson's chi-square test without Yates' continuity correctiondata: design.table(table.3.2)X-square = 69.1568, df = 4, p-value = 0To obtain the expected frequencies, we take advantage of the outer function, as well as the functionsrowSums and colSums:expected.freqs

37the rules of thumb required for the asymptotic chi-squared distribution to be appropriate, and the table istoo large to use some of the exact tests offered. To request p-value computation via Monte Carlosimulation, set the argument simulate.p.value (or sim) to TRUE. The argument B controls the numberof replicates in the simulation.chisq.test(table.3.2.array, sim=T, B=2000)Pearson's Chi-squared test with simulated p-value (based on 2000replicates)data: table.3.2.arrayX-squared = 69.1568, df = NA, p-value = < 2.2e-16Here, the p-value is similarly very low, rejecting the hypothesis of independence of the two variables.For the likelihood ratio test statistic, one can easily use the expected frequencies obtained above and theformula for the statistic on p. 79 of Agresti. For example,2*sum(table.3.2.array*log(table.3.2.array/expected.freqs)) # R: Use res$expected insteadof expected.freqs[1] 69.81162Alternatively, jumping ahead a bit, we can use the glm function for generalized linear models (seeChapter 4) and the Poisson distribution (which is the distribution of the counts in the table when the totalnumber of cases is not fixed – see p. 132 in Agresti) to get the likelihood ratio chi-squared statistic. Forexample, in either S-PLUS or R, we getfit.glm

38Note that the sum of the squared Pearson residuals equals the Pearson chi-squared statistic:sum(resid.pear^2)[1] 69.11429To get the standardized residuals, just modify resid.pear according to the formula on p. 81 of Agresti.ni

39update(fit, subset=(School=="Eclectic" | School=="Medical") & (Origin=="Biogenic" |Origin=="Environmental"))$deviance[1] 0.2941939Thus, we do not reject the hypothesis of independence for Subtable 1. Subtables 2 through 4 require abit more than just case selection via an update call. Subtable 2 compares the Eclectic and Medicalschools on whether the origin is a combination or not. So, we use the aggregate function to sumcolumns.First, I add a column called select that indicates the new categories. Then, I aggregate the counts intable.3.3 by School and select, and sum these counts. This gives the new table, which is supplied asthe data argument to glm. Finally, to restrict the analysis to just the Eclectic and Medical schools, I usethe subset argument.table.3.3$select

40Then, we repeat the income and jobsat values in table.2.8 count times. We can do this in at least twoways. One way is to use apply. The other is to just rep the row labels count times. Both methods arefairly comparable in efficiency, but method 2 requires a subsequent change of class of the columns tonumeric. Then, we apply cor, which returns an entire matrix, from which we select the 2,1 element.res

41psi

42consistent with the alternative hypothesis than the one corresponding to the observed table, plus theprobability corresponding to the observed table. One possible two-sided p-value is double this result,provided the answer does not exceed 1. Other possibilities are given in Agresti p. 93. Some of these canbe implemented using dhyper (see below).To demonstrate Fisher’s Exact Test, Agresti uses the data from Table 3.8 (p. 92). The functionfisher.test in S-PLUS gives a two-sided test; the version in R gives either a one- or two-sided test.The function can be used for I x J tables as well, with I, J ≤ 10, but the total sample size in the tablecannot exceed 200. (Interesting, the “Tea Tasting” example is given in the help page for fisher.test inR). In S-PLUS we get(test

4395 of Agresti, X 2 2 3 3= 6 and we need the supremum of PX ( ≥ 6) = 2 π (1 − π ) . In S, we can use generaloptimization functions for this. The function optim exists for both R and S-PLUS implementations (in S-PLUS it resides in the MASS library).library(MASS) # S-PLUS only(res

44num

45Liao and Rosen (2001) give an R function for probability calculations from a non-central hypergeometricdistribution. Their function can be used to obtain the confidence interval above. The syntax (usingAgresti’s notation) is hypergeometric( n1 +, n+ 1, n, θ ). This returns a set of functions such as thecumulative distribution function (CDF), the probability mass function, and moment functions. Theconfidence interval can be found by trial and error via the CDF, changing the odds ratio each time. Or,one can use optim. We can determine both endpoints with the same function by summing the squareddifferences of the two p-values from 0.025.f

46$countsfunction gradient58 46$convergence[1] 0res

47if(mode>ll){r1

48}MC(mean, list(prob=prob, shift=shift, ll=ll, uu=uu))MC(var, list(prob=prob, shift=shift, ll=ll, uu=uu))MC(sample.low.to.high, list(prob=prob, shift=shift, ll=ll, uu=uu) )MC(sample.high.to.low, list(prob=prob, shift=shift, ll=ll, uu=uu) )MC(r, list(prob=prob, shift=shift, ll=ll, uu=uu) )}return(mean, var, d, p, r);Thus, the small-sample confidence interval on the odds ratio, and its adjustment using the mid-p-valuecan be obtained using the same commands as above:f

The small-sample confidence interval on the difference of proportions, mentioned in Section 3.6.4 ofAgresti, can be computed using the methodology from Section 3.F.2 of this manual.49

50Chapter 4: Generalized Linear ModelsA. Summary of Chapter 4, AgrestiChapter 4 in Agresti deals with generalized linear models (GLIMs). GLIMs extend the generallinear model by allowing nonnormal response distributions and allowing a nonlinear mean function.There are three components of a GLIM, which are detailed on p. 116-117 of Agresti. Briefly, the randomcomponent consists of independent observations from a distribution in the natural exponential family.The pdf for this family is given in equation (4.1) of Agresti. Special discrete random variable cases arethe Poisson and binomial distributions. The systematic component relates a vector, η = ( η1,... η ) Tn, to aset of explanatory variables through a linear model: η = Xβ . η is called the “linear predictor”. The linkfunction links the random and systematic components. It describes the relationship between the mean ofthe response, µi, and the linear predictor, ηi= g( µi). When we specify a link function, g, we are sayingthat the systematic effects are additive on the scale given by the link function.A fourth component is sometimes specified explicitly. This is the variance function, which is afunction relating the variance to the mean (see Section 4.4 in Agresti). It is proportional to the variance ofthe response distribution, with proportionality constant the inverse of a parameter called the dispersionparameter. (If we use a particular random component, we automatically accept its variance function.However, there are methods where we can use a particular variance function that we believe describesthe random phenomenon, but then refrain from “choosing” a distribution for the random component.These methods use what are called “quasi-likelihood functions”.)Typical cases of GLIMs are the binomial logit model (Bernoulli response with log odds linkfunction) and the Poisson loglinear model (Poisson response with log link). Other cases are given inTable 4.1 in Agresti. For binomial and Poisson models, the dispersion parameter is fixed at 1.GLIMs are fit by solving the set of likelihood equations. This leads to maximum likelihoodestimates of the coefficients of the linear predictor. As the likelihood equations are usually nonlinear inthe coefficients β of the linear predictor, the solutions are found iteratively. Iterative Reweighted LeastSquares (IRLS) is the iterative method commonly used to fit GLIMs (it is used in S-PLUS). It uses Fisherscoring, which is based on the Newton-Raphson method, which achieves second-order convergence ofthe estimates. The difference between the two algorithms lies in the use of the observed informationmatrix for Newton-Raphson and the expected information matrix for Fisher scoring (see p. 145-146,Agresti). For canonical link models, these are the same. Fisher scoring will produce the estimatedasymptotic covariance matrix as a by-product, but it need not have second-order convergence. Plus, theobserved information matrix may be easier to calculate for complex models. The name IRLS comes fromthe iterative use of weighted least squares estimation, where the weights and responses (linearized formsof the link function evaluated at the observed data) change at each iteration. It is explained on p. 147 ofAgresti that IRLS and Fisher scoring are the same thing.Model deviance is the LR statistic for testing the null hypothesis that the model holds against thegeneral alternative of a saturated model. It is twice the difference between the saturated log likelihoodand the log likelihood maximized under the restrictions of the model being tested. In certain cases, thisquantity has an asymptotic chi-squared distribution. If the dispersion parameter is not fixed at 1, thentwice the difference between the saturated log likelihood and the restricted log likelihood is equal to thedeviance scaled by the dispersion parameter (hence called the scaled deviance). Model deviance for atwo-way contingency table is equivalent to the likelihood ratio chi-squared statistic. The deviance has anapproximate chi-squared distribution for large Poisson expected values and large binomial sample sizesper covariate combination. Thus, the model deviance for Bernoulli data (0/1, instead of counts out of atotal) is not asymptotically chi-squared.One can compare two nested models (i.e., one model is a subset of the other) using thedifference between their deviance values. The deviance for the larger model (more parameters toestimate) will be smaller. The comparison proceeds by first assuming that the larger model holds andtesting to see if the smaller model is not statistically significantly worse in deviance. The difference indeviance is then a LRT and has an asymptotic chi-squared null distribution. For binomial or Bernoulli

51data, the difference in deviance is the same, unlike their respective model deviances. Thus, the chisquaredapproximation holds for both. In general, the use of the chi-squared approximation is muchmore reliable for differences of deviances than model deviances themselves (see also, McCullagh andNelder, 1989). Model comparison is examined in detail in later chapters. Standardized Pearson anddeviance residuals are additional ways to assess the fit of a model.When we want to fit a GLIM such as the Poisson loglinear model to a data set, but the observedvariance of the data is greater than that allowed by the Poisson model, we may have a case for fitting anoverdispersed version of the model. If overdispersion is the result of subject heterogeneity, wheresubjects within each covariate combination still differ greatly (perhaps because we didn’t measureenough covariates), then a random effects version of the model (e.g., random effects Poisson regression,random effects logistic regression) may be appropriate. Another alternative is to fit a model with arandom component that allows for a greater variance than the ordinary Poisson or binomial. Someexamples are the negative binomial (for random count data) and the beta-binomial (for counts out of atotal).A third alternative is to use quasi-likelihood estimation. In quasi-likelihood estimation, we takeadvantage of the fact that the likelihood equations for GLIMS depend on the assumed responsedistribution only through its mean and variance (which may be a function of the mean). Distributions inthe natural exponential family are characterized by the relationship between the mean and the variance.Quasi-likelihood estimation is determined by this relationship. Thus, if we wanted to assume that thevariance of a random count was some specified function of its mean, but not equal to it, we could usequasi-likelihood estimation to estimate the coefficients of the linear predictor.Generalized Additive Models (GAMs) further generalize GLIMs by replacing the linear predictorwith smooth functions of the predictors (one for each predictor). A commonly used smooth function is thecubic spline. Each smooth function is assigned a degrees of freedom, which determines how rough thefunction will be. The GLIM is the special case of each smooth function being a linear function. GAMs arefit via penalized maximum likelihood estimation in S-PLUS (Chambers and Hastie, 1992). GAMs have anadvantage over procedures like lowess because they recognize the form of the response variable andonly give predictions within its bounds, which is not true of lowess (although, one can use a lowessfunction in a GAM). In general, the advantage of GAMs over GLIMS is that the form of the predictorsdoes not need to satisfy a particular functional relationship, like linear, logarithmic, etc. Finally, GAMsmay be used in an exploratory sense by determining a parametric function to use for a predictor based onits fitted smooth function.B. Generalized Linear Models for Binary DataSuppose the response is binary, taking one of two possible outcomes. Then, three special cases of theGLIM use an identity link (linear probability model), a logit link (logistic regression model), and an inversenormal CDF link (probit regression model). I briefly remind you what these models are, then fit some datato them using functions in S-PLUS and R.For a binary response, the regression modelπ ( x)= α + βxis called a linear probability model because it purports a linear relationship between the probability ofpositive response and the explanatory variables. Although it has a simple interpretation, the linearprobability model has a structural defect in that the predicted probability of positive response can exceed1 or be less than 0, due to the link function being the identity.The regression modelexp( α + βx)π ( x)=1 + exp( α + βx)

52is called a logistic regression model. This model corresponds to a binomial GLIM with log odds link (i.e.,g( π ( x)) = log( π( x) (1 −π( x )))).The regression modelπ ( x) = Φ ( α + βx)for a normal(0, 1) cdf, Φ , is called a probit regression model. It is a binomial GLIM with link function−Φ 1 ( π ( x )).The decision to use each of these link functions can depend on the expected rate of increase of π ( x ) asa function of x or can depend on a comparison of appropriate goodness-of-fit measures (note thatotherwise identical models with different link functions are not nested models).To illustrate a logit, probit, and linear probability model, Agresti uses Table 4.2 (Snoring and heartdisease, p. 121). The response variable is whether the subject had heart disease. The explanatoryvariable is the subject’s spouse’s report of the level of snoring (never, occasionally, nearly every night,every night). These levels are changed into scores (0, 2, 4, 5). The levels of snoring are treated asindependent binomial samples.To fit the three models using iterative reweighted least squares (IRLS), we can use the function glm, withfamily=”binomial” (for the logit and probit links, at least). However, glm doesn’t have a Newton-Raphson method or any other type of optimization method. Thus, for more general maximum likelihoodestimation, we might use the function optim (both S-PLUS and R). Conveniently, Venables and Ripley(2002) wrote a function for maximum likelihood estimation for a binomial logistic regression. We canmake minor changes to fit the linear probability model and the probit regression model.To set up the data, type,n

53}cat("\nResidual Deviance:", format(fit$value), "\n")cat("\nConvergence message:", fit$convergence, "\n")invisible(fit)The function is written for binomial logistic regression, but is easily modified for probit regression andlinear probability regression.Thus, to fit a linear probability model, we change the functions fmin and gmin within logitreg to readfmin

54Residual Deviance: 836.7943Convergence message: 0The warnings given with the linear probability model are somewhat expected, as the probability can beless than 0 or greater than 1. Also, note that we cannot use the default starting values.Approximate standard errors for the two parameter estimates can be obtained using the inverse of theobserved information matrix.sqrt(diag(solve(logit.fit$hessian)))[1] 0.11753115 0.03536285sqrt(diag(solve(lpm.fit$hessian)))[1] 0.002426329 0.001976457sqrt(diag(solve(probit.fit$hessian)))[1] 0.04981505 0.01670894We can obtain fitted probabilities for all three link functions.eta

55Predicted Probability0.00 0.05 0.10 0.15 0.20LogisticProbitLinear0 1 2 3 4 5Level of SnoringEstimation with IRLSIf you were wondering what estimates and standard errors IRLS would give, you can fit all three of theabove models using the glm function, which is described in more detail in the next section. For example,to fit the binomial with logit link using IRLS, typesnoring

56Notice also that, in the formula, we use the proportions for each category as the response, plus we addthe weights argument, which is the total number in each category.prop

57plot(y=plot.table.4.3$Sa,x=plot.table.4.3$W,xlab="Width (cm)",ylab="Number of Satellites", bty="L", axes=F, type="n")axis(2, at=1:15)axis(1, at=seq(20,34,2))text(y=plot.table.4.3$Sa,x=plot.table.4.3$W,labels=plot.table.4.3$x)It is probably possible to do the above using table instead of aggregate.Figure 4.4 plots the mean number of satellites for the mean width of eight width categories created fromthe W variable. In addition, a fitted smooth from the result of a generalized additive model (GAM) fit tothese data is superimposed. Generalized additive models (as extensions to GLIMs) are discussed brieflyin Section 4.8 in Agresti, and in the corresponding section of this manual.In the following code, first I cut the W variable into categories, then use aggregate again to get means ofSa and of W by the categories. Then, the points are plotted.table.4.3$W.fac

58To see exactly what the smoothed term in width in the gam call represents, we can plot (mean) width bythe smoothed term. The se=T argument adds plus and minus two pointwise standard deviations asdashed lines to the plot.plot.gam(res, se=T)s(plot.x, df = 5)-1.0 -0.5 0.0 0.524 26 28 30plot.xThe smoothed term does not deviate too much from linearity, and Figure 4.4 shows a linear trend relatingthe mean number of satellites to width. Agresti fits Poisson GLIMs with both log and identity links to thedata.We will use IRLS and glm to fit the Poisson models. For non-canonical links (e.g., identity), the estimatesmay differ slightly from Agresti’s. A Poisson loglinear model is fit usinglog.fit

59I’ve given the output from a call to summary (in S-PLUS), which summarizes model fitting by givingcoefficient estimates, their approximate SEs, residual deviance, and a summary of deviance residuals.The dispersion parameter by default is set to 1 (we will change that later). The correlation between thecoefficient estimates is quite high (almost perfectly negative). This can be reduced by using the meandeviation in width instead of width in the model. The null deviance is the deviance value (p. 119, Agresti)for a model with only an intercept. The residual deviance is the deviance value for a model with both anintercept and width. The reduction in deviance is a test of the width coefficient. That islog.fit$null.deviance-log.fit$deviance[1] 64.91309is the LR statistic for testing the significance of the Width variable in the model. Compared to a chisquareddistribution with 1 degree of freedom, the p-value of this test is quite low, rejecting the nullhypothesis of a zero-valued coefficient on width. We can get similar information from the Wald test givenby the t-value next to the coefficient estimate (z-value in R version). However, the LRT is usuallyconsidered more reliable (see Agresti, and also Lloyd 1999).The summary result for any glm.object in S-PLUS has the following attributes that can be extracted:attributes(summary(log.fit)) # S-PLUS$names:[1] "call" "terms" "coefficients" "dispersion" "df" "deviance.resid"[7] "cov.unscaled" "correlation" "deviance" "null.deviance" "iter" "nas"[13] "na.action"$class:[1] "summary.glm"The same summary call in R has a few additional components, notably AIC.attributes(summary(log.fit)) # R$names[1] "call" "terms" "family" "deviance"[5] "aic" "contrasts" "df.residual" "null.deviance"[9] "df.null" "iter" "deviance.resid" "aic"[13] "coefficients" "dispersion" "df" "cov.unscaled"[17] "cov.scaled"As an example, to extract the estimated coefficients, along with their standard errors, type:summary(log.fit)$coefficientsValue Std. Error t value(Intercept) -3.3047572 0.54222774 -6.094777W 0.1640451 0.01996492 8.216665Thus, the fitted model islog ˆ µ = − 3.305 + 0.164 x(4.2)The glm.object itself has the following components. It “inherits” all the attributes of lm.objectsattributes(log.fit) # S-PLUS (R has quite a few more)$names:[1] "coefficients" "residuals" "fitted.values" "effects" "R"[6] "rank" "assign" "df.residual" "weights" "family"[11] "linear.predictors" "deviance" "null.deviance" "call" "iter"[16] "y" "contrasts" "terms" "formula" "control"$class:[1] "glm" "lm"

60For example, the fitted response values (expected values) at each of the width values in the model canbe obtained by extracting the fitted.values.log.fit$fitted.valuesThe same answer can be obtained by the call fitted(log.fit). The functions, fitted, resid,coef are shortened versions to extract fitted values, residuals, and coefficients, respectively, from glmobjects.Using the predict method for glm, we can get the fitted response value for any width value we input.For example, the expected number of satellites at a width of 26.3 ispredict.glm(log.fit, type="response", newdata=data.frame(W=26.3))[1] 2.744581Agresti also fits a Poisson model with identity link to the Horseshoe Crab data. This is fit in S-PLUS usingid.fit

61guiModify( "CommentDate", Name = "GSD2$1",xPosition = "0.542857",yPosition = "3.52967", Angle = "90")# make arrows and textind

62If we examine Table 4.4 (p. 129 in Agresti) of the sample mean and variance of the number of satellitesper Width category, we see that the variance exceeds the mean in all cases. A Poisson model postulatesthat for each Width, the variance and mean of the number of satellites is the same. Thus, we haveevidence of overdispersion, where the “ordinary” Poisson model is not correct. Incidentally, the functiontapply (or by) can be used to get the sample means and variances:tapply(table.4.3$Sa, table.4.3$W.fac, function(x) c(mean=mean(x), variance=var(x)))# by(table.4.3$Sa, table.4.3$W.fac, function(x) c(mean=mean(x), variance=var(x)))The general density for the random component of a GLIM has an unspecified dispersion parameter that isrelated to the scale of the density. This density is given in equation (4.14) in Agresti (2002). Forexample, with a Gaussian random component, the dispersion parameter is the variance. For a binomialor Poisson random component, the dispersion is fixed at 1, so we have not needed to deal with it yet.However, if we believe that the mean response is like a Poisson mean, but the variance is proportional tothe Poisson mean (with proportionality constant the dispersion parameter), then we can leave thedispersion parameter unspecified and estimate it along with the mean parameter. In this case, we nolonger have a Poisson distribution as our random component. Plus, the estimate of the dispersionparameter must be used in standard error and deviance calculations. The general form of the scaleddeviance is given in equation (4.30) in Agresti.If the actual Poisson means (i.e., per Width value) are large enough (or, in the case of binomial data, thesample sizes at each covariate condition are large enough), then a simple way to detect overdispersion isto compare the residual deviance with the residual df, which will be equal in the asymptotic sense justmentioned. These two quantities are quite disparate for both link models. So, we decide to estimate thedispersion parameter instead of leaving it fixed at 1. We can do this in S-PLUS using a dispersion=0argument to summary.glm. In R, we must use the quasipoisson family first. (This makes sense after thediscussion of quasi-likelihood).summary.glm(log.fit, dispersion=0)$dispersion # S-PLUSPoisson3.181952# Rlog.fit.over

63The usual deviance must be divided by the dispersion estimate to get the scaled deviance used in F-testsfor model comparison (see Venables and Ripley, 2002). An anova method is available for glm objectsthat does these tests provided argument test=”F” is supplied. anova.glm will print the F-test that usesthe dispersion estimate from the larger model. Thus, if we wanted to compare the null model (with q = 1parameters) to log.fit (with p = 2) using the F-test (in Agresti’s notation)we would doF( D( y; µ0) − D( y; µ1))=( p−q)ˆ φ(4.4)null.fit

64mean and dispersion parameter. The pdf of the negative binomial is given in equation (4.12) of Agresti,with2EY ( ) = µ and var( Y)= µ + µ kwhere k is the (positive) dispersion parameter. The smaller the dispersion parameter, the larger thevariance as compared to the mean. But, with growing dispersion parameter, the variance converges tothe mean, and the random quantity converges to the Poisson distribution. Actually, negative binomial isthe distribution of a Poisson count with gamma-distributed rate parameter (see Section 13.4 in Agresti).There are several methods for going about fitting a negative binomial GLIM in S-PLUS and R. If thedispersion parameter is known, then the MASS library has a negative.binomial family function to usewith glm. For example, to fit a negative binomial GLIM to the Horseshoe Crab data, where the dispersionparameter (called theta in the function) is fixed to be 1.0, we can uselibrary(MASS)glm(Sa~W, family=negative.binomial(theta=1.0,link="identity"),data=table.4.3,start=predict(log.fit, type="link")) # for R use start=coef(log.fit)Call:glm(formula = Sa ~ W, family = negative.binomial(theta = 1, link = "identity"), data =table.4.3, start = predict(log.fit, type = "link"))Coefficients:(Intercept)W-11.62843 0.5537723Degrees of Freedom: 173 Total; 171 ResidualResidual Deviance: 202.8936A simpler version of negative.binomial (called neg.bin, with only the log link) is also available fromMASS, as well as a function called glm.nb. The function glm.nb allows one to estimate theta usingmaximum likelihood estimation. The output is similar to that of glm and has a summary method.library(MASS)nb.fit

65Theta: 0.932Std. Err.: 0.1682 x log-likelihood: -747.929The coefficient estimates are slightly different that those obtained by Agresti (p. 131).LRTs are also available for comparing nested negative binomial GLIMs (via the anova method).However, as theta must be held constant between the two models being compared, one must firstconvert the glm.nb object to a glm object, which will fix theta if the model is then update’d. Theconversion is done using the glm.convert function.To use custom GLIMs, one must create a family. The function make.family can be used to make a newglm family. You must specify the name of the family, the link functions allowed for it (e.g., logit, log,cloglog) and their derivatives, inverses, and initialization expressions (used to initialize the linearpredictor for Fisher scoring), and the variance and deviance functions.For example, in the function listing for negative.binomial, you can see where the links and varianceslists are defined and where the function make.family is used. The reason why the link functions are notactually typed out in function form is because all the links already appear in other glm families in S-PLUS.Their information is stored in the matrix glm.links. negative.binomial accesses the appropriate linkfunction using glm.links[,link].Beta-binomial regression models can be fit using gnlr in library gnlm for R.F. Residuals for GLIMsTo demonstrate how to obtain the residuals in Agresti’s Section 4.5.5, I use the Horseshoe Crab data fit.For example, the Pearson and deviance residuals are obtained fromresid(log.fit, type="deviance")pear.res

66Pearson Residuals-2 0 2 4 6Standardized Pearson Residuals-2 0 2 4 60 50 100 150observation0 50 100 150observationIssuing the command plot(log.fit) may also be useful:par(mfrow=c(1,2))old.par

67Std. deviance resid.0.0 0.5 1.0 1.5Scale-Location plot15149 56Cook's distance0.00 0.02 0.04 0.06Cook's distance plot115561490.5 1.0 1.5 2.00 50 100 150Predicted valuesObs. numberG. Quasi-Likelihood and GLIMsQuasi-likelihood estimation assumes only a mean-variance relationship rather than a completedistribution for the response variable. However, many times the relationship specified determines aparticular distribution. This is because distributions in the natural exponential family are characterized bythe relationship between the mean and the variance function. Thus, using the R function quasipoissonwith log link (and constant variance) gives the same estimates as using the Poisson family with log linkbecause the Poisson is a distribution in the natural exponential family. The reason the estimates are thesame is because the estimating equations used for quasi-likelihood estimation are the same as thelikelihood equations in the case of a specified distribution with that mean and variance relationship.In general then, quasi-likelihood estimates (QLEs) are not MLEs, but in some cases where we areassuming a particular distribution, but with a magnified variance (e.g., when the dispersion parametermultiplies the variance), the point estimates will be identical because the dispersion parameter drops outof the estimating equations.Agresti uses data from a teratology experiment to illustrate overdispersion and the use of quasi-likelihoodestimation. Rats in 58 litters were on iron-deficient diets and given one of four treatments (groups 1-4).The rats were made pregnant and killed after three weeks. The number of dead fetuses out of the totallitter size is the response variable. (The data set is available on Agresti’s CDA website). I copied the datainto a text file called teratology.ssc. It is then read into S-PLUS/R bytable.4.5

68The MLEs of the probabilities are then(pred

69where the s jare smooth functions of the predictors. When the s jare linear, the GAM becomes a GLIM.The functions s jare fit using scatterplot smoothers. One example of a scatterplot smoother is locallyweighted least squares regression (lowess). Another example is a smoothing spline. A common functionfor the s jis a restricted cubic spline. A cubic spline in a predictor variable is a piecewise cubic polynomialrepresentation of the predictor, where the pieces are determined by “knot” locations placed over therange of the predictor variable. The restriction occurs by forcing the resulting piecewise curve to havevanishing second derivatives at the boundaries. Then the number of degrees of freedom for each originalpredictor variable is equal to the number of knots minus 1. The more degrees of freedom allotted to apredictor variable, the more complex the relationship between the predictor and the response.A Fisher scoring-type outer loop (to update the working responses) with a backfitting inner loop (toupdate the smooth functions) can be used to find the estimates of the s j(see Chambers and Hastie,1992, p. 300ff). When we assume that the s jin a GAM are polynomial smoothing splines (like the cubicsplines above), then the algorithm corresponds to penalized maximum likelihood estimation, where thepenalty is for roughness of the s j(Green and Silverman, 1994).The function gam in S-PLUS can be used to fit GAMs. If the functions are restricted to be smoothingsplines (denoted by s in the gam formula) then gam uses a basis function algorithm, with a basis of cubicB-splines, to find the smooth functions (see Green and Silverman, 1994, p. 44ff). The number of knotscovering the range of the predictor values is chosen by default to be the number of unique data points (ifthese are less than 50), and otherwise “a suitable fine grid of knots is chosen”. The smoothing parameterused in the penalized least squares estimation to find the basis function representation of the smoothfunctions can be specified by the user or determined via generalized cross-validation (see Green andSilverman, 1994, p. 35). Equivalently, the number of degrees of freedom for a smooth function can bespecified.The function gam in S-PLUS “inherits” a lot of functionality from glm and lm. But, the output is moregraphical because the fitted smooth functions usually must be graphed to interpret the model (Chambersand Hastie, 1992).Horseshoe Crab Data – Bernoulli responseAgresti fits a GAM to the binary response of whether a female crab has at least one satellite, using alogit link and the width of carapace as a predictor. We fit a GAM using gam in S-PLUS and R, with 3degrees of freedom for width. Then, we plot the fitted smooth function.First, I get the binary response and the number of observations at each data point.table.4.3$Sa.bin0,1,0)plot.table.4.3

70Title = "Width, `x` (cm)") # S-PLUS only (GSD2 is the name of the graphsheet.Substitute the name of your graphsheet in place of it, if GSD2 is not it)Fit the GAM and plot the fit.res

71plot.gam(res, se=T)s(W, df = 3)-5 0 5 1022 24 26 28 30 32 34Wpar(old.par)# change back to old pty setting

72Chapter 5: Logistic RegressionA. Summary of Chapter 5, AgrestiThis chapter treats logistic regression in more detail than did Chapter 4. It begins with univariatelogistic regression. The interpretation of the coefficient, β , in the univariate logistic regression (equation5.1 in Agresti) is discussed initially. The direction of this coefficient indicates the direction of the effect ofthe variable x on the probability of a positive response. The magnitude of β is usually interpreted interms of the odds of a positive response. The odds change multiplicatively by exp( β ) for each one-unitincrease in x. The expression exp( β ) is actually an odds ratio: the odds (of positive response) atX = x + 1 to the odds at X = x . Prior to fitting a logistic regression model to data, one should checkthe assumption of a logistic relationship between the response and explanatory variables. A simple wayto do this is to use the linear relationship between the logit and the explanatory variable. The values ofthe explanatory variable can be plotted against the sample logits (p. 168, Agresti) at those values. Theplot should look roughly linear for a logistic model to be appropriate. If there are not enough responsedata at each unique x value (and categorizing x values is undesirable), then the technique of the lastsection in Chapter 4 can be used (i.e., GAM). There, we saw that a sigmoidal (or S-shaped) trendappeared in the plot of the response by predictor (Figure 4.7, Agresti).Logistic regression can be used with retrospective studies to estimate odds ratios. The fit of alogistic regression model to retrospective response data, given an explanatory variable whose values arenot known in advance, yields a coefficient estimate whose exponent is the same estimated odds ratio asif the response variable had been prospective.A logistic regression model is fit via maximum likelihood estimation. In practice, this can beachieved via IRLS, as mentioned in the previous chapter. That is, the MLE is the limit of a sequence ofweighted least squares estimates, where the weight matrix changes at each iteration (see Section 5.5 inAgresti).Inference for the maximum likelihood estimators is asymptotic. Confidence intervals can be Waldconfidence intervals, LR confidence intervals or score confidence intervals. The Wald, LR, and scoretests can be used to test hypotheses. The LRT is preferred, as it uses both the null maximized likelihoodvalue as well as the alternative maximized likelihood value (providing more information than the othertests), instead of just one of these values. When comparing two unsaturated fitted models, the differencebetween their individual LRT statistics in comparison with the saturated model has an approximate chisquarednull distribution, and this approximation is better than the chi-squared approximation by eachLRT statistic alone.Overall chi-squared goodness-of-fit tests for a logistic regression model can only be done forcategorical predictors, and for continuous predictors only if they are categorized. This is because thenumber of unique predictor combinations grows with increasing sample size when the predictors remaincontinuous.When a logit model includes categorical predictors (factors), there is a parameter for eachcategory of each predictor. However, one of those parameters per predictor is redundant. Setting theparameter for the last category equal to zero, and changing the definition of the remaining parameters tothat of deviation from this last parameter, will eliminate redundancy of parameters in the logit model. But,the interpretation of each parameter is modified. Another way to eliminate redundancy is to set “sum-tozeroconstraints”, where the sum of the parameters for a particular factor is constrained to equal zero.However, any constraints for the category parameters does not affect the meaning of estimates of theodds ratios between two categories or of the joint probabilities.Multiple logistic regression is the direct extension of univariate logistic regression. The multiplelogistic regression model is given in equations (5.8) and (5.9) in Agresti. In that representation, thequantity exp( β i) for the ith covariate represents the multiplicative effect on the odds of a 1-unit increasein that covariate, at fixed levels of the other covariates, provided there are no interactions between the ithcovariate and other covariates. With all categorical predictors, the model is called a logit model. A

73representation of a logit model with two predictors is given on p. 184 of Agresti. A logit model that doesnot contain an interaction between two categorical predictors assumes that the odds ratio(s) between oneof the predictors and the binary dependent variable is(are) the same at or given each level of the otherpredictor. (In the case of a binary predictor, there is just one odds ratio between that predictor and thedependent variable). Then, the predictor and the dependent variable are said to be conditionallyindependent. In terms of the model representation, this means that the parameters corresponding to thecategories of that predictor are all equal (see p. 184 in Agresti). A formal test of homogeneous oddsratios can be carried out using the chi-squared goodness-of-fit statistic G 2 .It is possible for additivity of predictors (i.e., no interactions) to hold on the logit scale, but notother link scales or vice versa.B. Logistic Regression for Horseshoe Crab DataOne interpretation of the horseshoe crab data of Table 4.3 in Agresti has a binary response witha positive response being that the female crab has at least one satellite. So, a logistic regression isplausible for describing the relationship between width of carapace and probability of at least onesatellite. If the widths are grouped into eight categories (Table 4.4, p. 129 Agresti), then a plot of themeans of the width categories by the proportion of female crabs within each category having satellites isin Figure 5.2 of Agresti, with a logistic regression fit superimposed.We plotted these proportions in Section I of Chapter 4, where we superimposed a GAM with logitlink and binary response. Now, we will superimpose a logistic regression function.Here is the logit fit, using glm.table.4.3$Sa.bin0,1,0) # change number of satellites to binaryresponse(crab.fit.logit

74# plot points and regression curve (note the ordering of the widths first)lines(y=prop, x=plot.x, pch=16, type="p")ind

75The Wald test is shown in the t value column under Coefficients (in R, this is a z value). To get theLRT, we need the log likelihood value at the estimate and at the null value 0. For this we can use thedeviance values.crab.fit.logit$null.deviance-crab.fit.logit$deviance[1] 31.30586A profile likelihood ratio confidence interval can be found easily in R using package Bhat. We first definethe negative log likelihood, then use plkhci in the same way as in Chapter 2.library(Bhat)# neg. log-likelihood of logistic model with width includednlogf

76I have since found that the MASS library has a function confint that computes profile-likelihoodconfidence intervals for the coefficients from glm objects. It is much simpler to use than plkhci for logitmodels. Below, I use it on the logit model fit to the crab datalibrary(MASS)confint(crab.fit.logit)Waiting for profiling to be done...2.5 % 97.5 %(Intercept) -17.8104562 -7.4577421W 0.3084012 0.7090312A plot of the predicted probabilities along with pointwise confidence intervals can be obtained usingoutput from the predict function, which gives the standard errors of the predictions.crab.predict

77The above plot is from R. Note that it extends on the left-hand side to only a width of 21.0 cm. However,Figure 5.3 in Agresti extends to a width of 20.0 cm. As 21.0 cm is the lowest width in the data set, inorder to predict the probability of a satellite at a width of 20.0 cm using the function predict we need touse the newdata argument. For example,predict(crab.fit.logit, type="response", se=T, newdata=data.frame(W=seq(20,32,1)))gives predictions and standard errors at widths from 20 to 32 cm, by cm.C. Goodness-of-fit for Logistic Regression for Ungrouped DataWith categorical predictors (grouped data), one may use a chi-squared goodness-of-fit statistic where theexpected number of positive and negative responses per predictor value (or predictor combination, withmore than one predictor) are obtained from the fitted model. When the predictors are continuous(ungrouped data), they must be categorized prior to using the test. In that case, one may assign themidpoint of the category to the observations in that category in order to compute the expected number ofpositives.In Agresti’s Table 5.2, the expected number of positives in each category (Fitted Yes) is obtained bysumming the predicted probabilities for each observation that falls within that category. Then, theobserved number of Yes’s are compared to these expected numbers in a chi-square test. RobertoBertolusso has sent me code to compute the values within Table 5.2, and also compute the goodness-offitstatistics. I present his code (somewhat modified) below. The original code appears in the code filesfor this document.First, we create a table with the successes and failures per width categorycont.table

78The Pearson chi-squared statistic is then easily computed(x.squared = sum((observed - fitted.all)^2/fitted.all))[1] 5.320099df

79To constrain the first category parameter to be zero, useoptions(contrasts=c("contr.treatment", "contr.poly"))Thus, to fit a logit model to the data in Table 5.3 on maternal alcohol consumption and child’s congenitalmalformations, we use glm with options set according to the constraint used.Alcohol

803 -5.060060 0.0063051704 -4.836282 0.0078740165 -3.610918 0.026315789The sample proportions tend to increase with alcohol consumption.A model that specifies independence between alcohol consumption and congenital malformations is fit by(Table.5.3.logit3

81(Intercept)scores -0.436482# chi-squared statisticsum(residuals(Table.5.3.LL, type="pearson")^2)[1] 2.050051# LR statisticTable.5.3.LL$null.deviance - Table.5.3.LL$deviance[1] 4.253277with fitted logits and proportionscbind(logit = predict(Table.5.3.LL), fitted.prop = predict(Table.5.3.LL, type ="response"))logit fitted.prop1 -5.960460 0.0025720922 -5.802180 0.0030118633 -5.485620 0.0041288464 -4.694219 0.0090650795 -3.744539 0.023100295Profile likelihood confidence intervals can be obtained using the plkhci function in the R package Bhat,which is illustrated in subsection B of this chapter, or more easily using the confint function from libraryMASS.library(MASS)confint(Table.5.3.LL)Waiting for profiling to be done...2.5 % 97.5 %(Intercept) -6.19303606 -5.7396909scores 0.01865425 0.5236161A logit model with an ordered categorical predictor can also be fit using orthogonal polynomial contrasts.However, by default, S-PLUS assumes the levels of the ordered factor are equally spaced. Forillustrative purposes, you could useAlcoholO

821 - pchisq(z2, df = 1)[1] 0.01037159which suggests strong evidence of a positive slope.E. Multiple Logistic Regression1. Multiple Logit Model – AIDS Symptoms DataAgresti introduces multiple logistic regression with a data set that has two categorical predictors. Thus,the model is a logit model. Table 5.5 in Agresti cross-classifies 338 veterans infected with the AIDS viruson the two predictors Race (black, white) and (immediate) AZT Use (yes, no), and the dependent variablewhether AIDS symptoms were present (yes, no). The model that is fit is the “main effects” modelAZT racelogit PY ( = 1) = α + β + β[ ]In S, we will represent the predictors as factors with two levels each. This ensures that we have thecorrect level specifications.table.5.5

83Analysis of Deviance TableModel 1: cbind(Yes, No) ~ AZTModel 2: cbind(Yes, No) ~ AZT + RaceResid. Df Resid. Dev Df Deviance P(>|Chi|)1 2 1.420612 1 1.38353 1 0.03708 0.84730Thus, the reduction in deviance (0.03708) is not significantly greater than chance, and we conclude thatRace probably does not belong in the model.As demonstrated above in Subsection D, one can easily change the type of constraint imposed on theparameters in order to estimate them uniquely. Use the options statement above in either R or S-PLUS.The estimated probabilities for each of the four predictor combinations is given by the function predict,and standard errors are given by setting the argument se to TRUE.res

84Now, add the sample proportions to the plot.attach(table.5.5)propAIDS

85table.4.3$C.fac

86lines(seq(18,34,1),res2) # add colors 2-4lines(seq(18,34,1),res3)lines(seq(18,34,1),res4)# add arrows and textarrows(x0=29, res1[25-17],x1=25, y1=res1[25-17], length=.09)text(x=29.1, y=res1[25-17], "Color 1", adj=c(0,0))arrows(x0=23, res2[26-17],x1=26, y1=res2[26-17], length=.09)text(x=21.1, y=res2[26-17], "Color 2", adj=c(0,0))arrows(x0=28.9, res3[24-17],x1=24, y1=res3[24-17], length=.09)text(x=29, y=res3[24-17], "Color 3", adj=c(0,0))arrows(x0=25.9, res4[23-17],x1=23, y1=res4[23-17], length=.09)text(x=26, y=res4[23-17], "Color 4", adj=c(0,0))Predicted Probability0.2 0.4 0.6 0.8 1.0Color 2Color 1Color 3Color 418 20 22 24 25 26 28 30 32 34Width, x(cm)As mentioned previously, there are some differences in the use of arrows and text across R and S-PLUS. Here is the code to plot Figure 5.5 in S-PLUS.plot(seq(18,34,1),res1,type="l",bty="L",ylab="Predicted Probability",axes=F)axis(2, at=seq(0,1,.2))axis(1, at=seq(18,34,2))guiModify( "XAxisTitle", Name = "GSD2$1$Axis2dX1$XAxisTitle", Title= "Width,`x`(cm)")lines(seq(18,34,1),res2)lines(seq(18,34,1),res3)lines(seq(18,34,1),res4)arrows(x1=29, y1=res1[25-17],x2=25, y2=res1[25-17],size=.25,open=T)text(x=29.1, y=res1[25-17], "Color 1", adj=0)

87arrows(x1=23, y1=res2[26-17],x2=26, y2=res2[26-17], size=.25,open=T)text(x=21.1, y=res2[26-17], "Color 2", adj=0)arrows(x1=28.9, y1=res3[24-17],x2=24, y2=res3[24-17], size=.25,open=T)text(x=29, y=res3[24-17], "Color 3", adj=0)arrows(x1=25.9, y1=res4[23-17],x2=23, y2=res4[23-17], size=.25,open=T)text(x=26, y=res4[23-17], "Color 4", adj=0)To test whether the width effect changes at each color, we can test the significance of a color by widthinteraction by fitting a new model with the addition of this interaction and comparing the model deviancewith that of the previous fit.crab.fit.logist.ia

88Judging by the p-value, we can go with the simpler (fewer parameters) quantitative color model.Other scores can be created by using logical operators. For example, the set of binary scores {1, 1, 1, 0}are created bytable.4.3$C.bin

89type -0.5980609duration -0.9190218 0.5496310type:duration 0.5481108 -0.9137683 -0.5964068The high negative correlation between duration and the intercept can probably be reduced bystandardizing duration:sorethroat.lg

90# R: legend(x=100, y=.6, legend=list("Tracheal","Laryngeal"), lty=1:2)prob0.0 0.2 0.4 0.6 0.8 1.0TLTT LL LLTLTLLT TLT T LL T L T LTrachealLaryngealT20 40 60 80 100 120durationWe can test for a type difference at a particular duration, say at 60 minutes, which is about 0.5 standarddeviations based on the mean and standard deviation of duration. The I( ) function is used so that(scale(duration)-0.5) is interpreted as is, meaning as a number, here.sorethroat.lgA

91One result we can obtain from setting different intercepts is a prediction of the duration at which prob(sorethroat upon awakening) = .25, .5, .75 for tracheal tubes/laryngeal mask. We can use the functiondose.p from the MASS library. The second argument to the function (i.e., c(1, 3) or c(2, 3) ) refers to thecoefficients specifying the common slope and separate intercept. For tracheal tubes, they are thetype.factrach coefficient and scale(duration) coefficient. The third argument refers to theprobability points (.25, .5, .75).For tracheal tubes, we have the following predictions:library(MASS)dose.p(sorethroat.lgB, c(1, 3), (1:3)/4)Dose SEp = 0.25: -0.61499711 0.3808615p = 0.50: -0.04219353 0.3567465p = 0.75: 0.53061006 0.4543964And, for laryngeal mask, we have:dose.p(sorethroat.lgB, c(2, 3), (1:3)/4)Dose SEp = 0.25: -1.4799537 0.5190731p = 0.50: -0.9071502 0.3720825p = 0.75: -0.3343466 0.3231864Of course, the predicted durations are in standard deviation units. The probability of sore throat ispredicted to be higher at lower durations (“doses”) when one is using the laryngeal mask.To print residual plots for this model, use the plot.glm function, which is called via plot with firstargument a glm object.par(mfrow=c(2,2))plot(sorethroat.lgB,ask=T)

92Chapter 6 – Building and Applying Logistic RegressionModelsA. Summary of Chapter 6, AgrestiThis chapter discusses logistic regression further, emphasizing practical issues related to modelchoice and model assessment. In Section 6.6, other link functions besides logit link are discussed forbinary data.We would like to fit a model that is rich enough to describe the data, but does not overfit the data.We also must be aware of issues such as multicollinearity among predictors. Multicollinearity may causerelated predictors to appear nonsignificant marginally. Stepwise procedures (forward selection, backwardelimination, or both) can be used to select predictors for an exploratory analysis. Forward selection addsterms sequentially until further terms do not improve the fit (based on a criterion such as Akaike’sInformation Criterion, AIC). Backward elimination starts with a model containing many terms andsequentially removes terms that do not add “significantly” to the fit. Once the final model is obtained, p-values must be interpreted cautiously because they are usually based on knowing the model form prior tolooking at the data (whereas, with stepwise selection we use the data to choose the model). Bootstrapadjustments may be needed for hypothesis tests and standard errors. In certain cases, conceptuallyimportant predictors should be included in a model, even though they may not be statistically significant.Model building can take advantage of causal diagrams that dictate conditional independencerelations among a set of variables. In this way, the causal diagram guides what models should be fit.Agresti gives an example from a British data set on extra-marital sex.Section 6.2 expounds on diagnostics for logistic regression analysis. These include residuals(Pearson and deviance), influence diagnostics or case-deletion diagnostics (e.g., Dfbetas, andconfidence interval diagnostic that measures the change in a joint confidence region on a set ofparameters after deleting each observation), and measures of predictive power (e.g. R-squared likemeasures, ROC curves).Section 6.3 deals with conditional inference from 2 x 2 x K tables. Testing conditional independenceof a binary response Y and a binary predictor X conditional on the level of a third variable Z can be doneusing a test of the appropriate parameter of a logit model or using the Cochran-Mantel-Haenszel (CMH)Test (The CMH test is a score statistic alternative for the LR test of the logit model parameters). The logitmodel tests differ depending on whether the association between the predictor and response is assumedthe same at each level of the third variable (i.e., no XZ interaction) or it is assumed to differ.The CMH test conditions on both response Y and predictor X totals within each level of Z. Then, thefirst cell count in each table has a hypergeometric distribution (independent of the other tables). TheCMH statistic compares the sum of the first cell counts across the K tables to its expected value ofconditional independence within strata. The asymptotic distribution of the statistic is chi-squared.Section 6.4 discusses the use of parsimonious models to improve inferential power and estimation.One example of this concept is illustrated by showing the improved power in testing for an association inthe presence of a logit model when a predictor has ordinal levels. In this case, the use of numericalscores in place of the nominal levels of the ordinal predictor results in fewer parameters in the model andultimately results in better power and smaller asymptotic variability of the cell probability estimates.Section 6.5 discusses power and sample size calculations for the two-sample binomial test, test ofnonzero coefficient in logistic regression and in multiple logistic regression, and chi-squared test incontingency tables. The formulas provide rough indications of power and/or sample size, based onassumptions about the distribution of predictors and of the model probabilities expected.When samples are small compared to the number of parameters in a logistic regression model,conditional inference may be used. With conditional inference, inference for a parameter conditions onsufficient statistics for remaining parameters, thereby eliminating them. What remains is a conditionallikelihood that only depends on the parameter of interest. In many respects, a conditional likelihood canbe used like an ordinary likelihood, giving conditional MLEs and asymptotic standard errors. Exactinference for parameters uses the conditional distributions of their sufficient statistics. Conditional

93inference is also used for sparse tables (with many zeroes) and tables that display “separation” where thesuccess cases all correspond to one level of the risk factor.B. Model Selection for Horseshoe Crab DataBackward elimination is done for the Horseshoe Crab data of Table 4.3 in order to select aparsimonious logistic regression model that predicts the probability of a female crab having a satellite.We begin by putting all the variables from Table 4.3 into a model. Agresti uses two dummy variables forthe variable spline condition, which we create by forming factors on the two variables.options(contrasts=c("contr.treatment", "contr.poly"))table.4.3$C.fac

94To perform stepwise selection of predictor variables for various types of fitted models (including glmobjects), one can use the function step with a lower and upper model specified. The criterion is AIC.Only the final model is printed unless trace=T is specified. An example of a call to the glm method isstep.glm(fit, scope=list(lower = formula(fit), upper = ~ .^2 ), scale=1, trace=T,direction="both")The above call specifies both forward and backward stepwise selection of terms (direction="both").The scope of the selection has a lower bound or starting model as “fit”. The upper bound modelincludes all two-way interactions. The component “anova” gives a summary of the trace path.To illustrate stepwise procedures, we perform backward elimination on a model fitted to the horseshoecrab data. This model includes up to a three-way interaction among Color, Width, and Spine Condition.We fit this model in S-PLUS or R usingcrab.fit.logist.stuffed

95extramarital sex, and marital status (divorced, still married) is given in Table 6.3 (p. 217, Agresti). Thedata come from a British survey of a sample of men and women who had petitioned for divorce, and asimilar number of married people.The causal diagram indicates a conditional independence relation: M and G are conditionallyindependent given E and P. Thus, if we broke the arrows connecting E to M and P to M, there would beno path between G and M. A logit model with M as response, then, might have E and P as explanatoryvariables, but not G. This model and the remaining in Table 6.4 (p. 218, Agresti) are fitted below using S.First, I enter the data (see note at end of this section)table.6.3

962 E + P + E:P 1032 1331.266 +E:P 1 12.914043 E + P + G + E:P 1031 1326.718 +G 1 4.54768Adding E:P to a model with E and P reduces the deviance by about 13 points (=18.2 – 5.2). Furtheradding G reduces deviance by 4.5 points ( = 5.2 – 0.7). Thus, we see how Table 6.4 is obtained.I have since found this data set available in the R package vcd. Thus, if you have loaded the vcd library,just issue the command: data(PreSex), to have the PreSex data array available. The functionas.data.frame() can be used to transform it to a data frame.D. Logistic Regression DiagnosticsThis section gives more details on diagnostics for logistic regression. After illustrating each set ofprocedures, I use the two data sets in Subsections 6.2.2 and 6.2.3 in Agresti to demonstrate their use inS.1. Pearson, Deviance and Standardized ResidualsWe already illustrated Pearson residuals in Chapter 3, Section D.1 and Chapter 4, Section F. The sum ofthe squared Pearson residuals is equal to the Pearson chi-squared statistic. For a logistic regressionmodel, with responses as counts out of totals, n i, i=1,…,N, the fitted response value at the ithcombination of the covariates is nπ î i. So, the Pearson and deviance residuals use deviations of theobserved responses from these fitted values. Pearson residuals are divided by an estimate of thestandard deviation of an observed response, and the standardized version is further divided by thesquare root of the 1 − the ith estimated leverage value (ith diagonal of estimated “hat” matrix). Thisstandardized residual is approximately distributed standard normal when the model holds. Thus,absolute values of greater than about three provide evidence of lack of fit.Deviance residuals were illustrated for S in Chapter 4, Section F. These are the signed squareroots of the components of the LR statistic. Standardized deviance residuals are approximatelydistributed standard normal.These residuals can be plotted against fitted linear predictors to detect lack of fit, but as Agrestisays, they have limited use. When n i= 1, individual residuals can be either uninteresting (Pearson) oruninformative (deviance).The heart disease data in Agresti (Table 6.5, p. 221) classifies blood pressure (BP) for a sample of maleresidents aged 40-59, into one of 8 categories. Then, a binary indicator response is whether each mandeveloped coronary heart disease (CHD) during a six-year follow-up period. An independence model isfit initially. This model has BP independent of CHD.BP

97pear.std.indep

98dev.indep

99table.6.7

1003. Summarizing Predictive Power: R and R-squared MeasuresThe function lrm from Harrell’s Design library (available for both R and S-PLUS) computes measures ofpredictive ability of a logistic regression model, including R and R-squared-like measures. Thesemeasures can also be validated using resampling (function validate.lrm). This function calculatesbias-corrected estimates of predictive ability. Bias is incurred when the data used to fit a model is alsoused to assess its predictive ability on a potential new data set. Model selection, assessment, and tuningof a model fit are usually not taken into account when one computes a measure of predictive power.These activities can bias the computed measure upward. Thus, bias-correction is in order.To illustrate the variety of measures computed by these functions, I use lrm to fit the linear logit model tothe blood pressure data. lrm requires binary responses instead of count response, so I form table.6.5out of objects defined previously. Note that I set the arguments x and y to TRUE so that we can usevalidate.lrm later on. First, when loading the libraries, set first=T so that certain functions fromDesign don’t get confused with built-in ones from S-PLUS (done automatically in R).library(Hmisc, first=T) # first=T not needed in Rlibrary(Design, first=T) # R: loads Hmisc upon loading Design#BP

101R1− exp( −( L −LM)/ n)=1−exp( −L/ n)2 0N0where L 0is the deviance (-2 times log likelihood) under the null model, and L Mis the deviance under theunconstrained model, and n is the number of responses. All three of these will differ when using grouped2versus ungrouped data, and the R-squared values will differ. Consider the value of RNfor the glm fit,resLL

102I only show the two indexes of predictive accuracy, Dxy (Somer’s D) and R2, but several others areoutput as well. The index.corrected column is the row index corrected for bias due to over-optimism(i.e., index.orig – optimism). The optimism column is computed as the difference between theaverage of the indexes calculated from the training samples and the average calculated from the testsamples. Here, there is very little over-optimism, so the corrected estimates are very close to the originalindexes.validate.lrm(res.lrm, method = "boot", B = 100)# excerptindex.orig training test optimism index.corrected nDxy 0.272819936 0.270357318 0.27281993603 -0.0024626183 0.27528255435 100R2 0.045469052 0.047315513 0.04546905186 0.0018464612 0.04362259074 100..........snipHarrell has written much about the topic of assessing predictive ability from linear, logistic, and survivalmodels. The above information came from a preprint (Harrell, 1998) of his now-published Springer book.In addition, on the website for one of my courses at UHCL (http://math.cl.uh.edu/~thompsonla/5537), Ihave html documents and S-PLUS scripts illustrating many of the lrm model assessments on GLIMs,with annotation.After using Design and Hmisc, we should detach them.detach("Design") # R: detach("package:Hmisc")detach("Hmisc") # R: detach("package:Design")E. Inference about Conditional Associations in 2 x 2 x K TablesIn this section, I show how to test for conditional independence between response Y and predictor X in 2x 2 x K tables using S, where K is the number of levels of a stratification variable. Agresti illustrates a testusing an appropriate logit model and the Cochran-Mantel-Haeszel Test. Table 6.9 (p. 230, Agresti) givesresults of a clinical trial with eight centers. The study compared two cream preparations, an active drugand a control, on their success in treating an infection. This data set is available on Agresti’s text website(I have changed the name, and changed the extension to “ssc”). I read it in using scan in order to readthe numeric levels (1 and 2) for the response and treatment as characters. Then, I make it a data frameand change the levels to something more meaningful.table.6.9

103}if (l == 2 && dim(x) != c(2, 2))stop("Not a 2 x 2 - table.")if (!is.null(stratum) && dim(x)[-stratum] != c(2, 2))stop("Need strata of 2 x 2 - tables.")lor

104Thus, we reject the null hypothesis of conditional independence between Treatment and Response givenCenter. (The common odds ratio result is discussed later). We can also use the logit model test. This isa test that the coefficient corresponding to Treatment in a logit model is zero, given the model includesCenter (see equation (6.4) in Agresti). If we use glm, we need to make a few modifications to the dataframe. Below, I add the total to each Center/Treatment combination.n

105Exact conditional test of independence in 2 x 2 x k tablesdata: table.6.9.arrayS = 55, p-value = 0.01336alternative hypothesis: true common odds ratio is not equal to 195 percent confidence interval:1.137370 4.079523sample estimates:common odds ratio2.130304If we assume that the log odds ratios have different directions across strata, we can test conditionalindependence using a logit model fit. The test is a likelihood-ratio test of model (6.5) in Agresti (only aCenter effect) compared with a saturated model (Center, Treatment and Center:Treatment). With thistest, we also reject the null hypothesis.res2

106Treatment 1 6.66882 7 9.74632 0.009811426Alternatively, the Woolf test is available in the R package vcd, under woolf.test. The null hypothesis isno three-way (XYZ) interaction, or homogeneous odds ratios across Centers. The function can also beused within S-PLUS with no modification.library(vcd)woolf.test(table.6.9.array)Woolf-test on Homogeneity of Odds Ratios (no 3-Way assoc.)data: table.6.9.arrayX-squared = 5.818, df = 7, p-value = 0.5612G. Using Models to Improve Inferential PowerAs Agresti says, ordinal test statistics in categorical data analysis usually refer to narrower, more relevantalternatives than do nominal test statistics. They also usually have more power when an approximatelinear trend actually exists between the response and a predictor variable. A linear trend model (e.g.,linear logit model) has fewer parameters to estimate than does a nominal model with a separateparameter for each level of the ordinal variable. The LR or Pearson goodness-of-fit statistic used to testthe linear trend parameter of a linear logit model has only 1 degree of freedom associated with it,whereas the test of the corresponding nominal model against an independence model has degrees offreedom equal to one less than the number of levels of the variable. Thus, the observed value of the teststatistic in the former case does not have to be as large (compared to the chi-squared distribution) as thatin the second case in order to reject the independence model over the linear logit model .Agresti uses an example on the treatment of leprosy by sulfones and streptomycin drugs to illustrate thedifference in power to detect a suspected association (Table 6.11, p. 239). The degree of infiltrationmeasures the amount of skin damage (High, Low). The response, the amount of clinical change incondition after 48 weeks, is ordinal with five categories (Worse, …, Marked Improvement). The responsescores are {-1, 0, 1, 2, 3}. Agresti compares the mean change for the two infiltration levels, and notesthat this analysis is identical to a trend test treating degree of infiltration as the response and clinicalchange as the predictor. Thus, a linear logit model has the logit of the probability of high filtration linearlyrelated to the change in clinical condition.In S, first we set up the data, then fit a model with a separate parameter for each change category.table.6.11

107NULL 4 7.277707change 4 7.277707 0 0.000000 0.1219205With p = 0.12, this test does not reject an independence model. Now, we fit a linear logit model usingscores c(-1, 0, 1, 2, 3), and test it against an independence model.resLL.leprosy

108bpower(n=200, p1=.6, p2=.7, alpha=0.05)Power0.3158429where we assume the 0.10 difference is divided as p1 = 0.6 and p2 = 0.7, and n is the sum of n1 and n2(with n1 = n2). bpower only computes power for a two-sided alternative.With n1 = 20 and n2 = 25, we get slightly lower powerbpower(n1=20,n2=25,p1=.6,p2=.7)Power0.1084922binomial.sample.size also computes sample size needed to achieve a given power. The functionbsamsize in package Hmisc does as well. Again, the latter is closer to Agresti’s formula on p. 242, whichhe says is an underestimate of the sample size.To compute sample size required to achieve 90% power with p1 = 0.6 and p2 = 0.7, and an 0.05-leveltest, we need close to 500 subjects for each group.binomial.sample.size(p=.6, p2=.7, alpha=.05,alternative="two.sided", power=.9)p1 p2 delta alpha power n1 n2 prop.n21 0.6 0.7 0.1 0.05 0.9 497 497 1bsamsize(power=.9, p1=.6, p2=.7, alpha=0.05)n1 n2476.0072 476.0072Sample size calculations for logistic regression can easily be programmed into a simple S function,following the formulas on p. 242 in Agresti.Power for a chi-squared test can be computed using the following function in S-PLUS:chipower.f

109As a first example, I use Table 6.13 in Agresti. This table and its “true” joint probabilities from thephysician are input below. Then I fit two models, one with the standard and new therapies independentlyinfluencing the response, and one model with standard therapy alone influencing the response. We willcompute the power for detecting an independent effect of the new therapy on response, over and abovethat from the standard therapy.table.6.13

110As described in Section 6.6 in Agresti, the idea behind probit models is that there is a latent tolerancevalue underlying each binary response. The tolerance is subject-specific. When a linear combination ofthe predictor variables is high enough to exceed an individual’s tolerance, the binary response becomes 1and remains there. Otherwise, it remains at zero. The tolerance is a continuous random variable. Whenit has a normal distribution, we get a probit model. Probit models are used in toxicological experimentswhere the predictor is dosage. So, when the dosage exceeds a threshold (tolerance), the response isdeath.In GLIM terminology, the probit link function is the inverse CDF of the standard normal distribution(equation 6.11 in Agresti). Agresti gives some differences between probit link and logit link for binomialregression. Fitting the probit model is fitting a GLIM, so no new estimation procedures are introduced.Estimates from Newton-Raphson and Fisher scoring will have slightly different estimated standard errorsbecause observed and expected information differ in the probit model. The probit is not the canonical linkfor a binomial model.Agresti fits a probit model to the beetle mortality data in Table 6.14.table.6.14

111log.dose n y fitted.probit fitted.cloglog1 1.691 59 6 3.4 5.72 1.724 60 13 10.7 11.33 1.755 62 18 23.4 20.94 1.784 56 28 33.8 30.35 1.811 63 52 49.6 47.76 1.837 59 53 53.4 54.27 1.861 62 61 59.7 61.18 1.884 60 60 59.2 59.9plot(table.6.14$log.dose,table.6.14$y/table.6.14$n, pch=16, xlab="Log dosage",ylab="Proportion Killed", bty="L", axes=F)axis(1, at=seq(1.7, 1.9, .05))axis(2, at=seq(0,1,.2))lines(table.6.14$log.dose, fitted(res.probit), lty=2)lines(table.6.14$log.dose, fitted(res.cloglog), lty=1)key(x=1.8, y=.4, text=list(c("Complementary log-log", "Probit")),lines=list(type=c("l", "l"), lty=c(1,2)), border=F,text.width.multiplier=.8)# R: legend(x=1.8,y=.4,legend=c("Complementary log-log", "Probit"),lty=c(1,2),cex=.85, text.width=1)1.00.8Proportion Killed0.60.4Complementary log-logProbit0.21.70 1.75 1.80 1.85Log dosageJ. Conditional Logistic Regression and Exact Distributions1. Small-sample Conditonal Inference for 2x2 Contingency tablesSmall-sample conditional tests for 2 x 2 tables use Fisher’s Exact test. See the discussion in Chapter 3.2. Small-sample Conditonal Inference for Linear Logit ModelsSmall-sample conditional inference for the linear logit model is an exact conditional trend test, and theconditional distribution of the cell counts under the null hypothesis of β = 0 is multiple hypergeometric.Agresti applies an exact test to Table 5.3 on maternal alcohol consumption and infant malformation. This

112data set has a lot of cases, but the table is sparse in that there are few successes or cases. Thus,ordinary maxmimum likelihood inference (with its asymptotic basis) breaks down.To do conditional logistic regression in S, one can use the coxph function (or the clogit function in R,which is just a wrapper for coxph). However, with more than about 500 cases (in my experience), thefunction hangs “forever”. Thus, we turn to approximations to conditional logistic regression for largesamples. There are S functions available for doing approximate conditional logistic regression on ascalar parameter of interest. One is cond.glm (normal approximation) from Alessandra Brazzale (seehttp://www.ladseb.pd.cnr.it/~brazzale/lib.html#ins). Another is cox.test (for a normal approximation)and cpvalue.saddle and cl.saddle (for a saddle-point approximation) from Chris Lloyd (Lloyd, 1999).These latter functions are apparently available on the Wiley website (http://www.wiley.com), but I havenot been able to locate them yet. I contacted Professor Lloyd, via the Wiley website, for copies.First, I show how to use coxph and clogit for an exact conditional test. However, the problem is muchtoo large to use this method. So, I do not give results. The response value must be binary (“cases”indicate successes). Here I create a data frame with case as the response, and alcohol as a numericexplanatory variable.# recall from chapter 5malformed

113conditional estimate of the coefficient and its standard error. A summary function gives hypothesis testsabout this coefficient.# R: library(cond)(fit

114Modifications within R would be similar.To use cl.saddle on table.5.3, which is quite large, we type the following. I choose to compute B=20points of the conditional log-likelihood, within the range –1 to 1 of the parameter value. The termargument specifies the parameter of interest.attach(table.5.3)(approx.points

115mantelhaen.test(xtabs(count~month+race, data=table.6.15), exact=T, alternative=”less”)Exact conditional test of independence in 2 x 2 x k tablesdata: xtabs(count ~ race + promote + month, data = table.6.15)S = 0, p-value = 0.02566alternative hypothesis: true common odds ratio is less than 195 percent confidence interval:0.0000000 0.7795136sample estimates:common odds ratio0The result using S-PLUS and the correct=F argument ismantelhaen.test(design.table(table.6.15), correct=F)Mantel-Haenszel chi-square test without continuity correctiondata: design.table(table.6.15)Mantel-Haenszel chi-square = 4.5906, df = 1, p-value = 0.0321A 95% two-sided confidence interval on the odds ratio is printed automatically.# R onlymantelhaen.test(xtabs(count~race+promote+month, data=table.6.15), exact=T, alternative= "two.sided")Exact conditional test of independence in 2 x 2 x k tablesdata: xtabs(count ~ race + promote + month, data = table.6.15)S = 0, p-value = 0.05625alternative hypothesis: true common odds ratio is not equal to 195 percent confidence interval:0.000000 1.009031sample estimates:common odds ratio0A final example is given where the table displays “separation”. That is, the cases all correspond to onelevel of the risk factor. In these situations, as Agresti says, maximum likelihood estimation gives infiniteestimates. Thus, exact inference is needed. Since the number of observations is quite large, we can’tuse coxph. Even if we take advantage of the fact that we really have a 2x2 table in Cephalexin andDiarrhea (as mentioned by Agresti, bottom of p. 256), we still have over 1,000 observations. However,with the 2x2 table, we can use Fisher’s Exact Test. Note that S-PLUS won’t handle this data set becauseit is too large (> 200 counts total). However, R handles it.table.6.16

116odds ratioInfNote that we get the confidence interval as well.K. Bias-reduced Logistic RegressionIn cases of quasi or complete separation, a different kind of estimation can be done instead of ordinarymaximization. Bias-reduced logistic regression (Firth, 1993, as cited in R help file for package brlr)maximizes a penalized likelihood with penalty function, Jeffreys invariant prior. This leads to less biasedestimates that are always finite. According to Firth, the bias reduction (toward zero) can be quitenoticeable for problems that display separation. Here, we fit the main-effects only logistic regressionmodel from subsection J to the diarrhea data, using glm (ML) and brlr (penalized ML). Note thedifference in the coefficient estimate for Cephalexin, as well as its smaller standard error from thepenalized estimation. The large magnitude of the Ceph estimate and its corresponding large standarderror from the glm fit indicate an infinite maximizer.library(brlr)fit

117Chapter 7 – Logit Models for Multinomial ResponsesA. Summary of Chapter 7, AgrestiIn Chapter 7, Agresti discusses logit models for multinomial responses, both nominal responsesand ordered responses. Multinomial logit models for nomial responses use a separate binary logit modelfor each pair of response categories (some of which are redundant), and are fit using maximum likelihoodestimation subject to the individual response probabilities simultaneously satisfying their separate logitmodels.There are several types of models for ordinal response categories. Section 7.2 discussescumulative logit models, which use the logits of the cumulative probabilities PY ( ≤ j| x ) =π ( ) ... ( )1x + + π jx , j = 1,…,J. So, the cumulative logit is the log of the odds of responding in category upto j. A cumulative logit model models all J – 1 cumulative logits simulataneously. A popular cumulativelogit model is a proportional odds model. This model assumes the J – 1 logits have different intercepts,increasing in j, but otherwise the coefficients are identical. Thus, the J – 1 response curves have thesame shape, but are shifted horizontally from one another. The term proportional odds applies becausethe log of the odds ratio of cumulative probabilities (comparing the conditions in one covariate vector toanother) is proportional to the distance between the covariate vector. The same proportionality constantapplies to each logit. Proportional odds models are fit using maximum likelihood estimation.A latent variable interpretation of the proportional odds model assumes an underlying(unobserved) continuous response variable that has a logistic distribution. When this response fallsbetween the (j – 1)th and jth cutpoints on the response scale, the observed discrete response value is j.Effects of explanatory variables are invariant to the choice of cutpoints for the response so that thecoefficients are identical across logits.Section 7.3 discusses cumulative link models for ordinal response categories. Here, we use alink function that relates the cumulative probabilities to a linear predictor involving the covariates. Theuse of the logit link is the cumulative logit model. The probit link gives the cumulative probit model.These models assume that the ordinal response categories have an underlying continuous distributionthat is related to the particular link function. They also assume that the distributions of each covariatesetting are stochastically ordered, so that the cumulative probabilities given one covariate setting arealways at least as great or at least as less as the cumulative probabilities of another covariate setting. Ifthis is not so, then one covariate setting may differ in dispersion of responses than another covariatesetting.Other models for ordinal responses are discussed in Section 7.4. These include adjacentcategorieslogit models, continuation-ratio logit models, and mean response models. The adjacentcategorieslogits are the J – 1 logits formed by the log odds of the probability of response falling incategory j given that the response falls within either category j or j + 1, j = 1,…,J – 1. Adjacent-categorieslogit models differ from cumulative logit models in that effects of explanatory variables refer to individualresponse categories and not cumulative groups. They also do not assume an underlying latentcontinuous response.Continuation-ratio logits, as given in Agresti, are the logits of the conditional probabilities that theresponse falls in the jth category given that it falls at least in the jth category for j = 1,…,J – 1. The fulllikelihood is the product of multinomial mass functions, which can each be represented as a product ofbinomial mass functions, leading to an easy way to estimate these models.Finally, mean response models for ordered responses are linear regression models for ordinalresponses, represented by numerical scores. The linear probability model is a special case. They canbe fit using maximum likelihood estimation, where the sampling model is product multinomial.To test conditional independence in I x J x K tables, where one factor Z is conditioned, one canuse multinomial models such as cumulative logit models or baseline-category logit models depending onwhether the response Y is ordinal or nominal, respectively. The test of homogenous XY associationacross levels of Z is then a test of the inclusion of a term or terms involving X.Generalized Cochran-Mantel Haenszel (CMH) tests can be used to test conditionalindependence for I x J x K tables with possibly ordered categories. With both X and Y ordinal, the

118generalization is Mantel’s M 2 statistic. Some versions of the generalized CMH statistic correspond toscore tests for various multinomial logit models.Discrete-choice multinomial logit models use a different set of values for a categoricalexplanatory variable for different response choices. Conditional on the choice set (e.g., choosing whichtype of transportation one prefers), the probability of selecting option j is a logistic function of theexplanatory variables. The model can also incorporate explanatory variables that are characteristic of theperson doing the choosing. Fitting these models using maximum likelihood can be done using the coxphfunction in R or S-PLUS, as the likelihood has the same form as the Cox partial likelihood. This can beseen from equation (7.22) in Agresti.B. Nominal Responses: Baseline-Category Logit ModelsWhen responses are nominal, for a given covariate pattern, x, the vector of counts at the Jresponse categories is assumed multinomial with probability vector { π1( x),..., π J( x )} . The J – 1 uniquelogits pair each of J – 1 response categories with a baseline category (say, the Jth category). As shownon p. 268 of Agresti, from these J – 1 logits that compare each category with a baseline, we can estimatelogits comparing any two categories.Agresti uses the Alligator Food Choice data set to fit a multinomial logit model. The measuredresponse is the category of primary food choice of 219 alligators caught in four Florida lakes (fish,invertebrate, reptile, bird, other). The explanatory variables are all categorical: L = lake of capture, G =gender, S = size ( ≤ 2.3 m, > 2.3 m). The data can be entered into S using the following commands. Thelevels argument codes the first level indicated as the baseline category.food.labs

119deviance(fit4)-deviance(fitS)deviance(fit5)-deviance(fitS)deviance(fit0)-deviance(fitS)Collapsing over gender gives# options(contrasts=c("contr.treatment","contr.poly"))fitS

120hancock 2.3 9.1 0.4 1.1 2.3 3.1oklawaha 2.3 12.8 7.0 5.5 0.8 1.9trafford 2.3 8.6 5.6 5.9 3.1 5.8george 2.3 14.5 3.1 0.5 1.8 2.2Each of the 4 binary logit models has a set of estimated effects or coefficients. These can be extractedusing the summary function, as here. Based on the way the factor levels were given in the definition oftable.7.1, the category fish is the baseline category, and each logit model compares the odds ofselecting another food to the odds of selecting fish.library(MASS) # needed for vcov functionsummary(fit3, cor = F)Coefficients:(Intercept) size lakehancock lakeoklawaha laketraffordinvert -1.549021 1.4581457 -1.6581178 0.937237973 1.122002rep -3.314512 -0.3512702 1.2428408 2.458913302 2.935262bird -2.093358 -0.6306329 0.6954256 -0.652622721 1.088098other -1.904343 0.3315514 0.8263115 0.005792737 1.516461Std. Errors:(Intercept) size lakehancock lakeoklawaha laketraffordinvert 0.4249185 0.3959418 0.6128465 0.4719035 0.4905122rep 1.0530583 0.5800207 1.1854035 1.1181005 1.1163849bird 0.6622971 0.6424863 0.7813123 1.2020025 0.8417085other 0.5258313 0.4482504 0.5575446 0.7765655 0.6214372To estimate response probabilities using values on the explanatory variables that (together) did notappear as a data record, you can use predict, with newdata, a data frame containing the labeledexplanatory values. In S-PLUS (but not R), when we have categorical variables entered as factors in theoriginal fit, we have to specify the prediction values as being levels of those factors. This can be done byusing factor along with the original levels. So, to estimate the probability that a large alligator in LakeHancock has invertebrates as the primary food choice, we use in S-PLUS# S-PLUS: options(contrasts=c("contr.treatment", "contr.poly"))predict(fit3, type="probs", newdata=data.frame(size=factor(">2.3", levels=c(">2.3","2.3", lake="hancock"))fish invert rep bird other0.57018414 0.02307664 0.07182898 0.14089666 0.19401358Obviously, the specification in R is how we would want to do it!To get the estimated response probabilities for all combinations of levels of the predictors, use a call toexpand.grid as the newdata. For example,predictions

121cbind(expand.grid(size = size.labs, lake = lake.labs), predictions)size lake fish invert rep bird other1 2.3 hancock 0.5701841 0.02307664 0.07182898 0.140896663 0.194013583 2.3 oklawaha 0.4584248 0.24864188 0.19484366 0.029424140 0.068665475 2.3 trafford 0.2957470 0.19296047 0.20240167 0.108228505 0.200662307 2.3 george 0.6574619 0.13968168 0.02389991 0.081046954 0.09790956The same multinomial model can be fit using lcr in R package ordinal from P. Lindsey. However, thatfunction is much more complicated to use than multinom because it fits more flexible models. We willsee it in the next sections for ordinal responses.Another function that is as easy to use as multinom is vglm from R package VGAM (and soon to bepackage for S-PLUS 6.X, although it exists for the Unix versions). The commands for fitting the maineffects model above islibrary(vgam)fit.vglm

122Now, we fit the main effects modellibrary(MASS)fit.polr

123measurements and time-varying covariates. To use the function, we have to set up the response andcovariate objects, then put them within a repeated object. These additional functions come from thermutil library, which is loaded with ordinal.library(ordinal)First, we must have a numeric response, as well as numeric covariates. So, we use the mentalC variableof table.7.5. It also must begin at 0, not 1. (Factor covariates must be transformed to dummy coding oranother numeric coding scheme. One can do this using the function wr, in the rmutil library.)# set up response vectory

124Call:nordr(w, dist = "prop", mu = ~ses + life, pmu = c(-0.5, 1, -0.3),pintercept = c(1, 2))proportional odds model-Log likelihood 49.54895AIC 54.54895Iterations 17Location coefficientsLocation function:~ses + lifeestimate s.e.(Intercept) -0.2817 0.6423ses[.i] 1.1111 0.6109life[.i] -0.3189 0.1210Intercept contrastsestimate s.e.b[2] 1.495 0.3898b[3] 2.491 0.5012Correlation matrix1 2 3 4 51 1.0000 -0.4449 -0.5924 -0.2557 -0.22292 -0.4449 1.0000 -0.1859 0.1885 0.23633 -0.5924 -0.1859 1.0000 -0.2312 -0.30034 -0.2557 0.1885 -0.2312 1.0000 0.71685 -0.2229 0.2363 -0.3003 0.7168 1.0000detach("table.7.5")To obtain a score test, we extract the likelihood attribute from the fitted object fit.lcr, and compare it tothe likelihood from fitting a model with separate sets of coefficients per response. This latter model canbe conveniently fitted using lcr as well.fit

125P(Y > 2)0.0 0.2 0.4 0.6 0.8 1.0SES=0SES=10 1 2 3 4 5 6 7 8 9Life Events IndexWe see that the probability of moderate or impaired mental health, P(Y > 2), is predicted to be greaterwith a higher life events index, and furthermore that a low SES has a higher predicted probability thanhigher SES.Fitting interaction models is straightforward. Different link functions other than the logit link can be usedin the function lcr and the function vglm, which is from the R package VGAM (coming soon to S-PLUS 6.x,but available for S-PLUS under L/Unix). This is illustrated in the next section. The function vglm can alsofit partial proportional odds models mentioned on page 282 of Agresti. This is done by setting the zeroargument of the family function, cumulative. See the help file for details.D. Ordinal Responses: Cumulative Link ModelsTo illustrate the use of different link functions within an ordinal categorical model, Agresti uses a life tableby race and sex (Table 7.7). The percentages within the four populations indicate that the underlyingcontinuous cdf of life length increases slowly at low to moderate ages, then increases sharply at olderages. This pattern suggests a complementary log-log link function.Several different link functions can be used within a proportional odds model. The function lcr in the Rpackage ordinal fits a cumulative complementary log-log link model (as well as many other linkfunctions).First, we set up the data. I created a numeric response, LifeC, as well as a categorical one. lcrrequires a numerical response.table.7.7

126library(ordinal)y

127over 65 73.0 55.4 84.9 73.7The function vglm will support dispersion effects, but it requires some work. In theory, it is possible towrite your own family function to do this, by modifying the cumulative family.E. Ordinal Responses: Adjacent-Categories Logit ModelsAdjacent-categories logit models fit J – 1 logits involving the log odds of the probability that a responsefalls in category j given that it falls in either category j or j + 1, j = 1,…,J. These logits are modeled aslinear functions of explanatory variables asπj( x)log = αj+ β ′ x, j = 1,..., J −1(7.1)π +( x)j 1Library functions exist for directly fitting these models in R. However, as Agresti shows on p. 287, theycan also be fit using the equivalent baseline-category logit model. Thus, they can be fit in S-PLUS aswell.Agresti uses the Job Satisfaction Example to illustrate adjacent-categories logit models. The responsevariable is job satisfaction in categories (Very Dissatisfied, Little Satisfied, Moderately Satisfied, VerySatisfied). The explanatory variables are gender (1 = females) and income (< $5,000, $5,000 – $15,000,$15,000 – $25,000, > $25,000). Scores of 1 to 4 are used for income. The data set is available onAgresti’s web site. Here we read it into R/S-PLUS and create an ordered factor out of the response.table.7.8

128fit.vglm

129y

130The "Model L.R." given above is supposed to be the model likelihood ratio chisquare according to Harrell(1998). However, if you examine what glm gives, you can see that model L.R. is actually equal to –2LogLH(model with intercept + x). The d.f. above gives the number of d.f. used with the addition of x inthe model (i.e., 1). What Agresti gives (p. 291) is the model residual deviance. That is, he gives –2LogLH(model with intercept only) – (–2LogLH(model with intercept + x). His d.f. correspond to theresulting d.f. when going from an intercept model (df = 4) to a model with x (df = 3). These are the df andLR statistic given directly by glm when modeling a linear logit, as shown later.To fit the j = 2 model:fit=malformed * x 0.010986 0.0013020 8.44 0Thus, when y>=malformed (j = 2), the linear logit is –5.70 + .017x. When y=all (j = 1), the linear logit is –3.247 + .0064x. The less desirable outcome is more likely as the concentration increases.Notice that the values for model L.R. in the individual model sum to the model L.R. for the interactionmodel above. However, the d.f. do not add.1. Odds RatiosTo get selected odds ratios for the j = 2 model, first issue the datadist command and reissue the lrmcall, as follows:

131x.u

132fit$null.deviance[1] 259.1073fit$null.deviance-fit$deviance[1] 253.3298# this is what lrm gave us as model L.R.summary(fit)Null Deviance: 259.1073 on 4 degrees of freedomResidual Deviance: 5.777478 on 3 degrees of freedomThe difference of the above sets of values gives Null Deviance-Residual Deviance = 253.3298 and df=1.These are the Model L.R. and Model df reported by lrm.For the j = 2 model, take the second and third columns of table.7.9a, and use the second column asthe success:n2

133(Intercept1) -2.404463 0.1078119Concen 0.007184 0.0003578Correlation matrix1 2 31 1.0000 0.6162 -0.79632 0.6162 1.0000 -0.77383 -0.7963 -0.7738 1.0000Note the different estimated concentraton effect.4. Using vglm from library VGAMUsing vglm for continuation-ratio models is similar to using it for adjacent-categories logit models. We justchange the family function to sratio (or cratio). First, I illustrate sratio, as this matches Agresti’sdefinition of continuation-ratio logit. Both sratio and cratio have an argument, reverse, which fits thecorresponding logits in reverse order (see the help files).For the data, we use the contingency table format (table.7.9a), and cbind the response columns.x=2])Dispersion Parameter for sratio family: 1Residual Deviance: 11.83839 on 6 degrees of freedomLog-likelihood: -730.3872 on 6 degrees of freedomNumber of Iterations: 4Note that the Residual Deviance reported is the sum of the individual Residual Deviances (the likelihoodratiofit statistics) reported by glm.Now, the family function cratio fits the logits, logit(P(y > j | y ≥ j), which is not the same as logit(P(y = j | y≥ j). In fact it is the logit of the complement, under the conditioning. Here is that fit.x

134Number of linear predictors: 2Names of linear predictors: logit(P[Y>1|Y>=1]), logit(P[Y>2|Y>=2])Dispersion Parameter for cratio family: 1Residual Deviance: 11.83839 on 6 degrees of freedomLog-likelihood: -730.3872 on 6 degrees of freedomNumber of Iterations: 4It is not surprising to see that the estimates are all negated, as we are fitting the logits of thecomplementary probabilities.As a side note, as mentioned in Harrell (1998) as well as Agresti (Chapter 7 notes), the continuation-ratiomodel is a discrete version of the Cox proportional hazards model. Thus, one could probably fit thesemodels using either coxph or cph, which is in the Design library. It is left to the reader to make theconnection.G. Ordinal Responses: Mean Response ModelsA mean response model for ordinal responses is a linear regression model with ordinal responsesrepresented by numerical scores. As in ordinary linear regression with continuous response, theconditional mean is assumed linearly related to the explanatory variables. The response distribution isassumed product multinomial (i.e., independent multinomial at each fixed set of covariates).This type of model can be fit using something like the lpmreg function of Chapter 4.Also, using the Poisson/multinomial connection, we might try glm with a poisson(identity) family. Or,we might try aov, as we are fitting a linear regression model. The functions glm and aov give similarcoefficient estimates. However, they are not exactly the same as those assuming the productmultinomialmodel. Also, the standard errors are wrong.fit.aov

135Multinomial-Poisson Homogeneous ModelA better alternative for fitting mean response models using ML estimation is to use J. Lang’s mph.fitfunction for R, which fits Multinomial-Poisson Homogenous models. For the mean response model, weassume product-multinomial sampling, given the totals from the eight populations.To use the function mph.fit, we need Table 7.8 in the follow form, as a 8 x 4 matrix of counts.table.7.8a

136fit

1372 0 0 3 5 13 0 0 0 7 34 0 0 1 9 6Next, we need the sample size for each population (income x gender combination), and the sampleproportions for each population.n

138F ( π ) = α + 0⋅ β + 1 ⋅ β =ν ( π ,..., π )T1 g x1(1,1) 4(1,1)TF ( π ) = α + β + 4 ⋅ β =ν ( π ,..., π )8 g x1(2,4) 4(2,4)representing the 2 x 4 combinations of Gender and Income. The 4 x 1 vector ν contains the scores forthe 4 job satisfaction categories.Now, note that the matrixQ ⎡ ⎤ on p. 602 contains the 1 x 32 row vectors∂ Fk( π)= ⎣ ∂πji| ⎦∂ Fk( π)∂π= (01,...,0 ( k −1)4, ν1,..., ν4,0,...,0 32),ji |and the design matrix is..1 gender income1 1 11 1 21 1 31 1 41 0 11 0 21 0 31 0 4All of the above is input into S via the following commands. For the F( ) functions, we use the sampleproportions instead of the probabilities, π .js

139sqrt(diag(Covb))[1] 0.2311374 0.1369473 0.0681758I compute a Wald statistic for entire model using the formula on p. 603 in Agresti.as.numeric(t(Fp-X%*%b)%*%InvVF%*%(Fp-X%*%b))5.242789H. Generalized Cochran-Mantel Haenszel Statistic for Ordinal CategoriesThe mantelhaen.test function in the R package ctest, which comes with R, can handle I x J x K tables,but does not take special advantage of ordinal categories (The same built-in function in S-PLUS can onlyhandle 2 x 2 x 2 tables, but the statistic itself is easily calculable using formula that is alreadyprogrammed into the R function. A calculation is available in the S-PLUS scripts for this manual). To testfor conditional independence of job satisfaction and income given gender, treating job satisfaction andincome with scores {1, 3, 4, 5} and {3, 10, 20, 35}, respectively, we need to use equation (7.21) in Agresti.This statistic has an approximate chi-squared distribution with 1 df.With nominal categories for job satisfaction and income, mantelhaen.test calculates (7.20) in Agrestiusing input from xtabs.mantelhaen.test(xtabs(freq~jobsatf+income+gender, data=table.7.8))Cochran-Mantel-Haenszel testdata: xtabs(freq ~ jobsatf + income + gender, data = table.7.8)Cochran-Mantel-Haenszel M^2 = 12.5314, df = 9, p-value = 0.185But, this gives a different answer than that given by SAS in Table 7.12 of Agresti. I don’t know why, as itis clear from the R code that it is computing (7.20).The test of nonzero correlation uses the ordinal scores for both row and column variables and computes(7.21). This computation is easy in either R or S-PLUS.First, we add the scores to the data frametable.7.8$jobsatS

140(sum(rowsums*u^2) - (sum(rowsums*u)^2)/n)*(sum(colsums*v^2) -(sum(colsums*v)^2)/n)/(n-1)}, u=c(1,3,4,5), v=c(3,10,20,35))The statistic is(sum(Tk-ETk)^2)/sum(varTk)[1] 6.156301with p-value1-pchisq((sum(Tk-ETk)^2)/sum(varTk),df=1)[1] 0.01309447The row mean scores differ association treats rows as nominal and columns as ordinal. The test statisticcan be computed in R or S-PLUS using the formula in the notes for Chapter 7 (p. 302, Agresti).However, the matrices in this formula do not have the appropriate dimensions for multiplication. So, onemay have to check the original paper. In any case, the mantelhaen.test function can be modified easilyto incorporate the B matrix given on p. 302, to get the new statistic.

141Chapter 8 –Loglinear Models for Contingency TablesA. Summary of Chapter 8, AgrestiLoglinear models are used for modeling cell counts in a contingency table. As Agresti notes, thesemodels are usually used when we have a multivariate response, not a univariate response, as the modeltreats all classification factors as responses. With a univariate response, methods such as logit modelsor multinomial logit models are better alternatives, and there is a correspondence between logit modelswith categorical explanatory variables and loglinear models (Section 8.5, Agresti). Loglinear modelsmodel the expected cell frequencies as log linear combinations of effects (model parameters) due to eachclassification factor by itself and possibly due to interactions among classification factors. Certaincontrasts involving parameters (such as differences between parameters for a given factor) areinterpreted as log odds of making one response on that variable, relative to another response. Contrastsinvolving interaction parameters can have interpretations in terms of log odds ratios. Interactionparameters by themselves are most useful in an association interpretation. That is, a three-factorinteraction parameter is zero if there is no three-factor association. For identifiability of all parameters,arbitrary constraints are made. This makes the individual parameter estimates not unique acrossconstraints, but estimated contrasts that encode log odds ratios are the same across constraints.A generalized linear model interpretation of loglinear models treats the N = IJ cell counts of an I x Jtable as independent observations from a Poisson random component with corresponding means equalto the expected cell counts. The same model can have a multinomial interpretation with two categoricalresponses and N total observations. These statements also apply to three-way tables.In a three-way table with response variables X, Y, and Z, several types of potential independence canbe present. Mutual independence of all three variables results in all interaction parameters in the loglinearmodel being zero. In a multinomial interpretation, this means all joint cell probabilities equal the productsof the corresponding marginal probabilities. Joint independence of one variable, Y, and the combinedclassifications of the other two (X and Z) results in a loglinear model with only one possible nonzerointeraction parameter, that between X and Z. Finally, variables X and Y are conditionally independent ofvariable Z if independence holds within each partial table conditional on a given value of Z. This wouldresult in a loglinear model with two possible nonzero interaction parameters: that describing associationbetween X and Z and that describing association between Y and Z. Relationships among the types ofindependence appear in Table 8.1. in Agresti.A loglinear model for no three-factor interaction in a three-way table is called a homogenousassociation model. This means that the conditional odds ratios between any two variables are identical ateach category of the third variable. Its parameters have interpretations in terms of conditional oddsratios. Of theIP2× JP2possible odds ratios, there are (I – 1)(J – 1) nonredundant odds ratios describingthe association between variables X and Y, at each of K levels of a third conditioning variable.Conditional independence of X and Y (i.e., no three-factor interaction) means that all of the odds ratiosare equal to 1.0. The logs of these odd ratios are functions of the parameters of the homogeneousassociation model, as shown in equation (8.14) in Agresti, and the functions do not depend on the level ofthe conditioning variable.Higher dimensional tables have straightforward extensions from the three-way table.Chi-squared goodness-of-fit tests can be used to compare nested models. This usually means usingthe likelihood ratio chi-squared statistic, which has an asymptotic chi-squared distribution when theexpected frequencies are large (with fixed number of cells). The degrees of freedom equal the differencein dimension between the null and alternative hypothesis.Loglinear models can be fit using one of two methods for doing maximum likelihood estimation,Newton-Raphson or Iterative proportional fitting (IPF). Newton-Raphson is an iterative procedure thatsolves a weighted least squares equation at each iteration. At each iteration of the IPF algorithm, thefitted values satisfy the model and eventually converge to match the sufficient statistics (leading toMLEs). However, IPF does not automatically produce the estimated covariance matrix of the parametersas a byproduct. ML parameter estimates have asymptotic normal distributions and asymptotic standarderrors estimated by the inverse of the Information matrix of the log likelihood.

142A generalized loglinear model generalizes mean modeling for loglinear models such as PoissonGLIMs. Specifically, the link function relating the response mean to the linear predictor is not strictlylogarithmic and not necessarily invertible, and takes the formC(log Aµ ) = Xβfor matrices C and A, which are not necessarily invertible. With A and C identity matrices, we getordinary loglinear models. For example, the logit model for a three-way table that postulatesindependence of the response and the two explanatory variables, each with two levels (i.e., the logitmodel only has an intercept) has A equal to an 8 x 8 identity matrix so that = ( µ , µ ,..., µ ) and⎡1 0 −1 0 0 0 0 0 ⎤⎢0 1 0 −1 0 0 0 0⎥C = ⎢⎥⎢ 0 0 0 0 1 0 − 1 0 ⎥⎢⎥⎣0 0 0 0 0 1 0 −1⎦µ111 112 222With X a vector of four ones, and β = α , we get four row logits (two per each level of the secondexplanatory variable) that are all equal.B. Loglinear Models for Three-way TablesWith a three-way table, there are three response variables for a loglinear model. We can test the fitof a saturated model (three-way interaction) against a homogeneous association model (all pairwiseassociations) or that of homogeneous association versus conditional independence of two variablesgiven a third. Agresti uses a data set on alcohol, cigarette, and marijuana use to fit these various models.We can set up the data astable.8.3

143data.frame(table.8.3[,-4], ACM=c(aperm(fitted(fitACM))),AC.AM.CM=c(aperm(fitted(fitAC.AM.CM))), AM.CM=c(aperm(fitted(fitAM.CM))),AC.M=c(aperm(fitted(fitAC.M))), A.C.M=c(aperm(fitted(fitA.C.M))))marijuana cigarette alcohol ACM AC.AM.CM AM.CM AC.M A.C.M1 Yes Yes Yes 911 910.383057 909.2395630 611.17749 539.982542 No Yes Yes 538 538.616089 438.8404236 837.82251 740.226073 Yes No Yes 44 44.616840 45.7604179 210.89632 282.091254 No No Yes 456 455.385590 555.1595459 289.10370 386.700075 Yes Yes No 3 3.616919 4.7604165 19.40246 90.597396 No Yes No 43 42.383881 142.1595764 26.59754 124.193927 Yes No No 2 1.383159 0.2395833 118.52373 47.328818 No No No 279 279.614380 179.8404236 162.47627 64.87991Because we set param = T, we can get estimates of the model parameters (the lambdas), usingfit$param. These can be used to get estimates of conditonal odds ratios, as per equation (8.14) inAgresti (see Thompson (1999, p. 27)). However, it may be simpler to just compute the odds ratios fromthe array of fitted values. To do this, we use the apply function on the array. For example, for thehomogeneous association model (AC, AM, CM), which has fitted counts close to the observed counts, wehave the following conditional odds ratios:fit.array

144[1,] 1 3[2,] 2 4, , 2[,1] [,2][1,] 1 3[2,] 2 4sum.array(junk)[,1] [,2][1,] 2 4[2,] 6 8Now, we compute the marginal odds ratios. The current dimensions of fit.array are A,C,M.odds.ratio(sum.array(fit.array)) # AC (sum over M, so 3 needs to be listed first)[1] 17.70244odds.ratio(sum.array(fit.array, perm=c(1,2,3))) # CM (sum over A, so 1 is first)[1] 25.13620odds.ratio(sum.array(fit.array, perm=c(2,1,3))) # AM (sum over C, so 2 is first)[1] 61.87182As loglm is a front-end to loglin, there is no need to show the results from loglin. loglin requiresinput of a table in the form of an array and also has an argument margin, where the model is specifiedusing a list of numeric vectors. As an example, to fit the homogeneous association model, we dologlin(fitted(fitACM), margin=list(c(1,2), c(2,3), c(1,3)), param=T,fit=T)$lrt:[1] 0.3738868$pearson:[1] 0.4011037$df:[1] 1$margin:$margin[[1]]:[1] "alcohol" "cigarette"$margin[[2]]:[1] "cigarette" "marijuana"$margin[[3]]:[1] "alcohol" "marijuana"$fit:, , YesYes NoYes 910.382996 44.617058No 3.616851 1.383149, , NoYes NoYes 538.61639 455.3836No 42.38408 279.6159

145The margin argument gives the associations allowed for the model. Here, we choose all pairwiseassociations. lrt and pearson give the associated goodness-of-fit statistics, and fit gives the fittedcounts.The same model can be fit using glm with a poisson family. Here, the fit algorithm is Newton-Raphson.options(contrasts=c("contr.treatment","contr.poly")) # dummy coding for factors(fit.glm X^2)Likelihood Ratio 0.3742223 1 0.5407117Pearson 0.4011002 1 0.5265216Comparison of nested models can be done using the anova method. For example,anova(fitAC.M, fitAC.AM.CM, fitAM.CM, fitA.C.M)LR tests for hierarchical log-linear modelsModel 1:count ~ alcohol + cigarette + marijuanaModel 2:count ~ alcohol + cigarette + marijuana + alcohol:cigaretteModel 3:count ~ alcohol + cigarette + marijuana + alcohol:marijuana + cigarette:marijuanaModel 4:count ~ alcohol + cigarette + marijuana + alcohol:cigarette + alcohol:marijuana +cigarette:marijuanaDeviance df Delta(Dev) Delta(df) P(> Delta(Dev)

146Model 1 1286.0200195 4Model 2 843.8267822 3 442.1932373 1 0.00000Model 3 187.7544556 2 656.0723267 1 0.00000Model 4 0.3742223 1 187.3802338 1 0.00000Saturated 0.0000000 0 0.3742223 1 0.54071gives the likelihood ratio tests comparing hierarchical loglinear models given in the list of arguments.Each item in the Delta(Dev) column compares Deviances between the current row and the previousrow. So, the test of conditional independence between A and C, which compares models AM.CM andAM.CM.AC is 187.75 – 0.37 = 187.38. According to the output, the only model that fits well among thefour is the homogeneous association model (Model 4). It’s deviance is close enough to the deviance forthe saturated model (zero) to give a nonsignificant p-value.The parameter estimates returned by loglin and loglm (fit$param) come from a parameterization suchthat the constant component describes the overall mean, each single factor sums to zero, each twofactor parameter sums to zero both by rows and columns, etc. Thus, the parameterization is not suchthat the parameters in the last row and the last column are zero, and will not match those from SAS on p.325 in Agresti. To obtain parameter estimates and standard errors from loglm matching those from SAS,we can use plug-in calculations, as mentioned by Agresti. That is, take the fitted expected counts fromloglm output and plug them into the formula (8.25) on p. 339 in Agresti. (This is illustrated below)However, we can get the parameter estimates and standard errors directly from the glm fit. In order thatour estimates match the SAS results given by Agresti on p. 325, we must first issue the commandoptions(contrasts= c("contr.treatment", "contr.poly")), and then ensure that our factor levelsmatch those of SAS. An inspection of the model.matrix from fit.glm shows the dummy coding usedby contr.treatmentmodel.matrix(fit.glm)(Intercept) marijuana cigarette alcohol marijuana:cigarette marijuana:alcohol cigarette:alcohol1 1 0 0 0 0 0 02 1 1 0 0 0 0 03 1 0 1 0 0 0 04 1 1 1 0 1 0 05 1 0 0 1 0 0 06 1 1 0 1 0 1 07 1 0 1 1 0 0 18 1 1 1 1 1 1 1By the manner in which table.8.3 was set up, we see that the coding is 0 = Yes and 1 = No. We want 1= Yes and 0 = No. Fortunately, we don’t have to redefine the factors in table.8.3. We just need to usethe contrasts argument of glm, as follows.fit.glm2

147marijuana:alcohol 2.986014 0.46454749 6.427791cigarette:alcohol 2.054534 0.17406289 11.803401(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: 2851.461 on 7 degrees of freedomResidual Deviance: 0.3739859 on 1 degrees of freedomNumber of Fisher Scoring Iterations: 3For loglm, we compute equation (8.25) using the model.matrix function to get X. Note that we still needto modify the 0/1 coding using contrasts argument to match the coding in Agresti.options(contrasts=c("contr.treatment","contr.poly"))contrasts# ensure we have treatmentX

148fitG.I.L.S

149locationbelt Urban RuralNo 10358.93 6150.193Yes 10959.23 6697.643, , gender = Female, injury = Yeslocationbelt Urban RuralNo 1009.7857 964.3376Yes 713.3783 797.4979, , gender = Male, injury = Yeslocationbelt Urban RuralNo 834.0684 1056.8071Yes 389.7681 508.3566We can examine the three-way interaction by seeing how the conditional odds ratios of any two variablesof the trio (gender, location, belt use) change at the levels of the third variable. But, they are the sameregardless of injury status. (Output is given from R)fit.array

150Yes 0.5799411 0.5799412apply(fit.array,c(1,3),odds.ratio)# IL (same for each combination of GS)Female MaleNo 2.134127 2.134127Yes 2.134127 2.134127apply(fit.array,c(2,3),odds.ratio)# IS (same for each combination of GL)Female MaleUrban 0.4417123 0.4417123Rural 0.4417122 0.4417123Computing the dissimilarity matrix to check goodness-of-fit of this model to the data is simple in S.Fitted.values

151these remaining coefficients is a consequence of having only two levels per variable and also of usingtreatment contrasts. To see that this is so, read subsection 8.5.2 in Agresti, and recall that we set the lastcoefficient in each row and column of a pairwise interaction to zero for constraints.The fact that there is a correspondence between estimates from the two models, which assume twodifferent sampling distributions for responses, follows from the correspondence of the likelihoods withrespect to the parameters in question.F. Contingency Table StandardizationTable standardization is useful for comparing tables with different marginal totals or matching sampletable data to standardized marginal distributions. Next, we show how to “rake” Table 8.15 by forcing allrow and column totals to equal 100. This is done using the tip on p. 346 of Agresti. We make the log ofthe observed count an offset in a glm and create a pseudo response that satisfies the model and themarginal totals.First I input the data, coding factors with levels in the order most natural.table.8.15

152Chapter 9 –Building and Extending Loglinear ModelsA. Summary of Chapter 9, AgrestiThis chapter extends loglinear models to ordinal variables and correlated variables, and also dealswith special issues like collapsibility and empty cells.With higher order tables, the analysis is easier if we can collapse over some dimensions (variables).Collapsibility describes the conditions under which a variable can be collapse over (i.e., ignored). Forthree-way tables with variables X, Y, and Z, we can collapse over Z while keeping the XY association thesame for the marginal table as well as the conditional table (on levels of Z) if either Z and X areconditionally independent given Y or Z and Y are conditionally independent given X. For four-way tableswith variables W, X, Y, and Z, if we collapse over Y, and association terms involving XY and WY are zero,then the partial WX odds ratios, conditioning on Y, are the same as the marginal WX odds ratios aftercollapsing. The general principle describing these examples is on p. 360 of Agresti.Loglinear models treat all variables as responses. However, sometimes certain marginal totals arefixed by sampling design. If the corresponding loglinear terms are not included in the model, then thefitted marginal totals will not necessarily match the observed totals for these combinations. This isbecause the likelihood will not match the observed total to the fitted total in the likelihood equation(equation 8.22 in Agresti). In sum, the highest interaction term containing all explanatory variables shouldbe included in a loglinear model.Nested loglinear models can be compared using the likelihood ratio statistic in equation (9.3) inAgresti, or using the Pearson statistic in equation (9.4). These have asymptotic chi-squared distributions,and are used to test the significance of an additional term or variable in the model. To check the fit of agiven model, one can examine cell residuals.Just as with logit models, if there are ordinal responses, then greater power results from fittingspecific ordinal trends than either ignoring ordinality or fitting general association terms. The lattersituation may not even be possible without fitting a saturated model. The linear-by-linear association (L xL) model assigns ordered row scores to rows and ordered column scores to columns, and includes, inaddition to main effect terms for the rows and columns, an interaction term in the row and column scores.The interaction term has a single unknown parameter which represents positive or negative linearassociation, depending on the sign of the parameter. Odds ratios are a function of the distance betweencorresponding rows and columns and the magnitude of the interaction parameter.L x L models are best fit when the observed counts show a linear trend in the rows at each columnand a linear trend in the columns for each row. The model is fit using ML estimation, and the correlationbetween rows and columns is the same in both the observed and fitted counts.Association models generalize loglinear models for ordinal responses to multi-way tables and tomixed ordinal and nominal responses. The row effects model and column effects model have either rowsor columns as nominal responses, respectively, and the other as ordinal. The nominal variable hasparameter scores instead of fixed scores. Ordinal responses within multi-way tables are straightforwardgeneralizations of their two-way table analogs.Other types of models that emphasize the estimation of association are Multiplicative Row andColumn effects models (RC Models), correlation models, and correspondence analysis. The RC modelmodifies the L x L model by replacing the scores with parameters. Correlation models are similar to RCmodels. In these models, each cell probability is augmented with an association term, dependent on rowand column scores. Correlation models can have either fixed or parameter scores. Correspondenceanalysis is mostly a graphical technique to represent associations in contingency tables.The Poisson loglinear model can also be used to model the rate of occurrence of an event over anexposure time or space, dependent on covariates. The exposure is treated as an offset. Also, due to theconnection between ML estimation using a Poisson likelihood for numbers of events and a negativeexponential likelihood for survival time, one can model survival times using a Poisson loglinear model.Sparse tables have many empty cells where there are no observations. A sampling zero occurs in acell when the data do not contain observations corresponding to that cell. A structural zero occurs whenit is impossible for the data to have an observation for the cell. For unsaturated models, MLEs existwhen all cell counts are positive, but do not exist when some counts in the sufficient marginal tables are

153zero. They also may not exist when all sufficient counts are positive, but at least one cell is zero.However, even if a point estimate is infinite, one endpoint of a likelihood-ratio confidence interval isusually finite. Adding a constant like 0.5 to each cell to prevent sampling zeroes for fitting an unsaturatedmodel is not recommended, especially for large numbers of cells, because it influences estimates andtest statistics conservatively.As an alternative to large-sample goodness-of-fit tests (which may not apply when a table is sparse),one can use exact tests or Monte Carlo approximations to exact tests.B. Model Selection and ComparisonAgresti uses the Student Survey example to illustrate model selection and comparison. We will use abackwards stepwise procedure to help in comparing models by comparing likelihood ratio chi-squarestatistics.We start by inputing the datatable.9.1

154Step: AIC= 45.78count ~ cigarette + alcohol + marijuana + sex + race + cigarette:alcohol +cigarette:marijuana + cigarette:sex + alcohol:marijuana +alcohol:sex + alcohol:race + marijuana:sex + marijuana:race + sex:raceDf Deviance AIC- cigarette:sex 1 16.73504 44.73504 NA 15.78347 45.78347- marijuana:race 1 18.95695 46.95695- alcohol:sex 1 19.18046 47.18046- alcohol:race 1 20.33869 48.33869- marijuana:sex 1 25.56924 53.56924- alcohol:marijuana 1 107.79480 135.79480- cigarette:alcohol 1 201.21688 229.21688- cigarette:marijuana 1 513.50057 541.50057Step: AIC= 44.74count ~ cigarette + alcohol + marijuana + sex + race + cigarette:alcohol +cigarette:marijuana + alcohol:marijuana + alcohol:sex +alcohol:race + marijuana:sex + marijuana:race + sex:raceDf Deviance AIC NA 16.73504 44.73504- marijuana:race 1 19.90859 45.90859- alcohol:race 1 21.29033 47.29033- alcohol:sex 1 22.02148 48.02148- marijuana:sex 1 25.81125 51.81125- alcohol:marijuana 1 108.77579 134.77579- cigarette:alcohol 1 204.11536 230.11536- cigarette:marijuana 1 513.73033 539.73033Call: glm(formula = count ~ cigarette + alcohol + marijuana + sex + race +cigarette:alcohol + cigarette:marijuana + alcohol:marijuana +alcohol:sex + alcohol:race + marijuana:sex + marijuana:race + sex:race, family =poisson, data = table.9.1)Coefficients:Value Std. Error t value(Intercept) 2.2651798 0.12467553 18.168600cigarette -0.2822714 0.05491143 -5.140485alcohol -1.4163287 0.12121797 -11.684149marijuana 1.2671910 0.12443051 10.183925sex 0.1090531 0.04581267 2.380413race -1.1940934 0.05467500 -21.839842cigarette:alcohol 0.5136335 0.04351564 11.803423cigarette:marijuana 0.7119723 0.04095963 17.382295alcohol:marijuana 0.7486558 0.11619789 6.442938Value Std. Error t valuealcohol:sex 0.07321130 0.03190131 2.2949310alcohol:race 0.11284083 0.05166169 2.1842266marijuana:sex -0.06805431 0.02261757 -3.0089132marijuana:race 0.07958278 0.04495553 1.7702554sex:race 0.03582621 0.03976789 0.9008829(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: 4818.051 on 31 degrees of freedomResidual Deviance: 16.73504 on 18 degrees of freedomThe output is lengthy, but it indicates the eliminated coefficient (which I’ve bolded) at each step. Notethat we don’t quite get to Agresti’s Model 6, but stop at Model 5. Model 6 would correspond to theremoval:Df DevianceAIC

155- marijuana:race 1 19.90859 45.90859We now rename the result of the stepwise search:fit.AC.AM.CM.AG.AR.GM.GR

15624 No No No female other 12 14.31255445 -0.8848276025 Yes Yes Yes male other 30 35.77995571 -1.7412161926 No Yes Yes male other 1 1.75353441 -0.5885741827 Yes No Yes male other 1 0.27593320 1.4173616628 No No Yes male other 0 0.10552401 -0.3285553129 Yes Yes No male other 19 18.51435109 0.1486450130 No Yes No male other 18 15.65328710 0.7438181231 Yes No No male other 8 2.88487421 3.2662966732 No No No male other 17 19.03254028 -0.74988607D. Modeling Ordinal AssociationsWhen both responses in a loglinear model are ordinal, a linear-by-linear association model can be usedto fit a linear trend in ordered row and column scores. These scores may be assumed to represent anunderlying continuous distribution. For an I x J table, the loglinear model adds an interaction term in thescores, as followsX Ylog µ = λ + λ + λ + βuv , i= 1,..., I; j = 1,..., Jij i j i jThe parameter, β , describes the linear association. It is positive or negative with the sign of β . Oddsratios depend on β , as well as the score distance between the corresponding rows and columns. When{ u = i}and { v = j}, the local odds ratios for adjacent rows and columns are uniform, and equalijexp( β ) . This is called the uniform association model.Agresti fits a uniform association (UA) model to data from the 1991 General Social Survey on opinionsabout premarital sex and birth control for teenagers. The four categories for opinions about premaritalsex range from “Always wrong” to “Not wrong at all”. The four categories for opinions about teenage birthcontrol sex range from “Strongly disagree” to “Strongly agree”. The scores used are {1, 2, 3, 4} for bothrows and columns, initially.The data are available from Agresti’s web site. I have copied the data into a text file called sex.txt. Wefirst read in the data, and give the columns names.table.9.3

157options(contrasts=c("contr.treatment","contr.poly"))(fit.ua

158Degrees of Freedom: 16 Total; 9 ResidualResidual Deviance: 127.6529A LR test of the null hypothesis of independence (i.e., H : 00β = ) has df =1:anova(fit.ind, fit.ua, test = "Chi") # output from S-PLUSAnalysis of Deviance TableResponse: countTerms Resid. Df Resid. Dev Test Df Deviance Pr(Chi)1 premar + birth 9 127.65292 premar + birth + u1:v1 8 11.5337 +u1:v1 1 116.1192 0E. Association ModelsAssociation models generalize loglinear models to ordinal responses, and include the linear-by-linearassociation models. The focus of these models is on estimating association. Row and column effectsmodels have rows or columns, respectively, as nominal responses, and model an ordinal effect at each ofthe rows or columns. Multi-way tables with some ordinal responses generalize the two-way tables,adding three or higher way interactions.1. Row and Column Effects ModelsThe row effect model treats rows (X) as nominal and columns (Y) as ordinal. Each row has a parameterassociated with it, called the row effect. The model for the log expected count in cell ij isX Ylog µ = λ+ λ + λ + µvij i j i jwhere the µiare the unknown row effects (i = 1,…, I), and the vjare known column scores. Thus, themodel has I – 1 more parameters than an independence model. An independence model here wouldimply that all row effects are equal. That is, the linear effect of the column variable on the log expectedcount is the same within each row. The column effects model is defined similarly.For the row effects model, the distance between adjacent-categories logits is the same across rows.However, the level of the logit will differ between rows i and k by µi− µk. Thus, this model is called theparallel odds model.The Political Ideology Example (Table 9.5 in Agresti) uses a table that cross-classifies Party Affiliation(Democrat, independent, Republican) and Political Ideology (Liberal, Moderate, Conservative) for asample of voters in a Wisconsin primary.A row effects model is fit to Table 9.5, with rows as Party Affiliation. When we construct the table, thevariable c.Ideo are the scores (1, 2, 3) for Political Ideology.table.9.5

159I coded the levels of the factor variables so that the Republican and Conservative levels had zero-valuedcoefficients, for identifiability. This is done to match the SAS results in Table 9.6 in Agresti.To fit a row effects model:options(contrasts = c("contr.treatment", "contr.poly"))(fit.RE

160An extension of association models to multi-way tables permits higher-order interaction models that aremore parsimonious than their corresponding nominal models. In particular, the three-way interactionmodel for a three-way table is not saturated. For a three-way table, with X and Y ordinal, with scores{ u i} and { v j} respectively, we model heterogeneous linear-by-linear XY association aslog µ = λ+ λ + λ + λ + β uv + λ + λX Y Z XZ YZijk i j k k i j ik jkIn this model, the log XY odds ratios are uniform within each level of Z, but differ across levels of Z. Ifβk= β for all k, then the log odds ratios the same across k as well. The model is then called thehomogeneous linear-by-linear association model.The data in Table 9.7 in Agresti are used to fit these models. The table cross-classifies smoking status(S), breathing test results (B) and age (A) for workers in industrial plants.The data are read in, and the levels are set so that the last category is zero. I also define scores for allthree variables. These are denoted with a “c” prefix.table.9.7

161A homogeneous L x L model is fit usingfit.homo

162Age55-59Wheeze -1.4479318 0.1233793 -11.735617Age60-64Wheeze -1.6556074 0.1337443 -12.378905c.Breath:c.Wheeze 3.676197 0.1998956 18.39058(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: 25889.47 on 35 degrees of freedomResidual Deviance: 6.801743 on 7 degrees of freedomThus, the estimated BW local odds ratio at level k of Age is 3.676 – 0.131k.3. Conditional Tests for Ordinal Models in Multi-way TablesFor small samples, the asymptotic chi-squared approximations to goodness-of-fit tests, such as thelikelihood-ratio test, do not hold. Instead, theoretically, the p-value for a test of the goodness of fit of amodel can be computed from a conditional distribution of the test statistic, by conditioning on estimates ofcertain nuisance parameters. Conditioning on these sufficient statistics leaves the conditional distributionfree of these unknown parameters. So, the p-value is the probability of the test statistic exceeding itsobserved value, conditional on the sufficient statistic taking on its observed value.The functions in exactLoglinTest give Monte Carlo estimates of conditional p-values for tests ofPoisson log-linear models. The function mcexact conditions on all sufficient statistics for parameters in agiven model, and simulates multi-way tables from the conditional distribution satisfying the observedvalues of the sufficient statistics. The function approximates the conditional distribution, which is ageneralized hypergeometric by default, by sampling from it via either importance sampling (method =“bab”) or MCMC (method = “cab”). The details of the procedures can found in the citations that arelisted in the documentation.The default test statistics used are the LR test statistic of a model against the saturated model (modeldeviance) and the Pearson chi-squared statistic.For example, to fit model (9.12) in Agresti using a conditional test, we use mcexact with importancesampling.library(exactLoglinTest)set.seed(1)fit.mc

163pvalue 0.45862023 0.45123625mcse 0.01003939 0.00998465“Number of iterations” gives the number of simulations actually obtained. “T degrees of freedom” is atuning parameter, and df is the model df. “Proportion of valid tables” here is just proportion of simulationsdone, out of maxiter.In the second half of the output, we have the estimated p-values. We can compare these to their largesampleversionspchisq(fit.mc$dobs, df=7, lower=F)[1] 0.4498103 0.4491093They are close, which is expected because of the large sample size.However, before we “believe” these estimates, we might examine the trace plots and the autocorrelationfunction of the iterations, as well as the logs of the importance weights (if we use importance sampling).We saved these results by specifying savechain=T. They are saved as the chain attribute of fit.mc.layout(matrix(c(1,1,2,3),2,2, byrow=T))plot(fit.mc$chain[,1], type="l", xlab="iteration", ylab="deviance")acf(fit.mc$chain[,1])library(MASS)truehist(fit.mc$chain[,3], xlab="log importance weight")

164deviance0 200 400 600 800 10000 2000 4000 6000 8000 10000iterationSeries fit.mc$chain[, 1]ACF0.0 0.2 0.4 0.6 0.8 1.00.00 0.05 0.10 0.150 10 20 30 40Lag-109100 -108900 -108700log importance weightThe trace plot shows good mixing (“randomness”), and the acf shows low autocorrelations. Also, many ofthe logs of the importance weights are similar.To continue the sampling, we can use the update method for bab (update.bab). However, based on thediagnostic plots, and the low Monte Carlo standard errors (in summary output) we probably have enoughiterations. Large, sparse tables may benefit most from these methods, and we will examine them later inthis chapter.F. Association Models, Correlation Models, and Correspondence Analysis1. Multiplicative Row and Column Effects ModelThe row effects model and the column effects model (as well as the L x L model) are special cases of theRow-and-column-effects model (Goodman’s RC model), which has parameter scores for both rows andcolumns. The model multiplies the two sets of parameters in the model equation,

165X Ylog µ = λ + λ + λ + βµνij i j i jwhere µiand νjare parameter scores. So, instead of having to fix arbitrary scores, we can estimateparameter scores. Agresti notes that finding MLEs is not always easy because the likelihood is notalways concave. An iterative modeling fitting algorithm is suggested by Goodman, where one alternatesbetween fitting row effects and column effects models, using the estimated scores from each in turn. Inthis way, one can ideally get MLEs of the row and column scores. Actually, if corner-point constraints areused for the parameter scores, then one gets the differences between each score and the last score (ifthe last parameter score is set to zero for identifiability). Also, one set of either the row or columnparameter score estimates is multiplied by the estimate for β . Thus, the method does not appear to beuseful as a way to estimate parameter scores.In the R package VGAM, there is a function grc, which is a front end for the function rrvglm (reducedrankvector generalized linear models), also in the same package. This function will fit an RC model, butnot estimate the scores as per Agresti’s (9.13) representation of the RC model. Instead, it estimates amatrix of “interaction terms”, δij, i = 1,…, I – 1; j = 1,…, J – 1, where δij= aci j= β ( µi−µ I)( νj− νJ), inthe language of model (9.13) in Agresti. Separating δ ijinto these three components appears to beimpossible. However, we can still fit the model, and compare its deviance to that of an L x L model. Wewill see later that correspondence analysis will give us MLEs of the row and column scores.When I prepare the data for grc, I use “reverse” levels for the factors so that the last category of therespective factor is zeroed out instead of the first category.table.9.9

166In the output, the Row and Column coefficients refer to the λ Xiand λ Y j’s. Because we set the lastcategory to zero, Row2 refers to SES E and Row6 refers to SES A. Also, Col2 refers to Moderate, andCol4 refers to Well. The coefficients labeled A through E refer to the first five cjvalues (the last is 0). Acall to summary (i.e., summary(fit.rc)) gives some more information, as well as standard errors.To get the matrix of δ ij’s, we first compute a biplot (without plotting it). The use of biplot reflects thefact that the cjvalues are involved in a latent variable interpretation in the rrvglm context. However, weonly use biplot here to get the interaction term estimates.res

167(fit.LL

168First canonical correlation(s): 0.1615119MH scores:well mild moderate impaired-1.611765 -0.1822571 0.08506316 1.470761SES scores:A B C D E F-1.127146 -1.151843 -0.3717666 0.07545902 1.018916 1.775564Compare these estimates with those on p. 381 in Agresti.We could fit a saturated model (i.e., M = 3) by using nf = 3 or by using functions for correspondenceanalysis, described next.3. Correspondence AnalysisCorrespondence analysis is a graphical method for describing associations among categorical variables.The rows and columns of a contingency table are represented by points on a graph. The greater themagnitude of the projections of the points onto an axis of the graph, the greater the association describedby that axis. Using the same notation as for equation (9.1) above, correspondence analysis uses theadjusted scoresx = λ µ , y = λνik k ik jk k jkThen, the graph of the first two dimensions plots ( xi1, xi2) for each row, and ( yj1, yj 2) for each column.There are many functions for doing correspondence analysis in S-PLUS and R. I will try to illustratedifferent aspects of each.The library multiv in both S-PLUS and R has function ca for doing correspondence analysis and functionplaxes to help with creating a plot like that in Figure 9.4 of Agresti. It needs an array or table, sotable.9.9b.array

169evals are the squared correlations, the λkabove. The second and third are almost zero, showing lesserimportance of the second and third dimensions in explaining variability. rproj and cproj are the row andcolumn scores on each of the three dimensions.To plot the scores, use# plot of first and second factorsplot(fit.ca$rproj[,1], fit.ca$rproj[,2],type="n",ylim=c(-.1,.1),xlim=c(-.3,.3),xlab="",ylab="",axes=F)text(fit.ca$rproj[,1], fit.ca$rproj[,2], labels=dimnames(table.9.9b.array)$SES)text(fit.ca$cproj[,1], fit.ca$cproj[,2], labels=dimnames(table.9.9b.array)$MH)# Place additional axes through x=0 and y=0:my.plaxes(fit.ca$rproj[,1], fit.ca$rproj[,2],size=.15)# R: my.plaxes.f(fit.ca$rproj[,1], fit.ca$rproj[,2],Length=.15)where my.plaxes is a modification of plaxes# S-PLUSmy.plaxes

170fit.corresp$cscore %*% diag(fit.corresp$cor)[,1] [,2] [,3]A -0.18204759 0.01965179 0.027712258B -0.18603645 0.01028853 -0.026939901C -0.06004474 0.02157969 -0.010480869D 0.01218753 -0.04121676 0.009748570E 0.16456713 -0.04381688 -0.008190311F 0.28677477 0.06237172 0.003614761Plotting is done via plot(fit.corresp)The R package CoCoAn will do (constrained) correspondence analysis via the function CAIV. The calltakes an array (table) and here would belibrary(CoCoAn)fit.CAIV

171Coefficients:Value Std. Error t value(Intercept) -6.3120972 0.5064590 -12.463196Valve -0.3298665 0.4381267 -0.752902Age 1.2209479 0.5136586 2.376964(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: 10.84053 on 3 degrees of freedomResidual Deviance: 3.222511 on 1 degrees of freedomNumber of Fisher Scoring Iterations: 4Thus, the estimated rate for the older age group (coded 1, by the ordering of the levels) is exp(1.221) =3.4 times that for the younger group. The LR confidence intervals arelibrary(MASS)exp(confint(fit.rate))2.5 % 97.5 %(Intercept) 0.0005868836 0.004396417Valve 0.2988564722 1.709356104Age 1.3239556759 10.392076295The chi-squared statistic issum(resid(fit.rate, type = "pearson")^2)[1] 3.113503which is not a significantly poor fit at the .05 level.Fitted values are obtain byattach(table.9.11)mhat

172summary(fit.id, cor=F)Coefficients:Value Std. Error t valueI(codes(Valve) * Exposure) -0.001936109 0.001315309 -1.4719796I(rev(codes(Age)) * Exposure) 0.003965548 0.001440062 2.7537340Exposure 0.000479430 0.003249681 0.1475314(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: Inf on 4 degrees of freedomResidual Deviance: 1.093067 on 1 degrees of freedomNumber of Fisher Scoring Iterations: 4H. Modeling Survival TimesAgresti notes the connection between ML estimation using a Poisson likelihood and a survival-timelikelihood using an exponential hazard function. So, we can use methods for fitting loglinear models to fitthese types of survival models.The random variable is the time to some event. For the ith subject, this is t i. If the ith subject does notexperience the event during the study time, then their recorded observation is censored. Let wi= 0 if theith observation is censored, and 1 otherwise. Then, if the exponential hazard is modeled asht ( ; x) = λ exp( β ′ x)for covariates in x, then the survival-time log likelihood as a function of β is identical to the log likelihoodfor independent Poisson variates w iwith expected values µi= thti(i; xi). Thus, we fit the Poissonloglinear model log( µiti)= log λ + β ′ xito responses { wi}.In the Lung Cancer Survival example, Agresti uses a piecewise constant hazard rate instead of aconstant hazard rate. The hazard is constant in two-month intervals of follow-up time. The data containcounts of deaths within each follow-up interval, by factors Histology (I, II, III) and Stage of disease (1, 2,3). When we read in the data set, the variable Deaths is the number of deaths at each cell of the table, orthe sum of the { w } for the ith histology, jth stage, and kth time interval.ijkAs usual, we reverse the levels of factors so that the glm coding sets the last category to zero instead ofthe first.table.9.13

173fit.surv

174I. Empty Cells and SparsenessWhen some cells in a table are empty, ordinary ML estimation may give infinite estimates of modelparameters, or very large estimates with large standard errors. Also, goodness-of-fit statistics may nothave asymptotic chi-squared distributions. Alternative tests include exact small-sample tests and MonteCarlo approximations to exact tests.For example, for the pharmaceutical clinical trials example, with five centers, two centers have nosuccesses for either the treatment or placebo groups. A logit model with factors Treatment and Center,with no intercept gives nonsense estimates for the Center 1 and 3 effects because the Center-Responsemarginal table has zeroes.table.9.16

175However, we can use the survival library to get an exact likelihood ratio test of the treatment effect.Recall that to use clogit (R) or coxph (S-PLUS) for conditional logistic regression, the data must be in0’s and 1’s (i.e., success, failure) instead of counts out of totals. I modify the data first, and rename thetable table.9.16a. I call the success/failure factor, case.temp

176Chapter 10 – Models for Matched PairsA. Summary of Chapter 10, AgrestiMatched pairs data contain observations on two response variables where each observation fromone response variable pairs with one and only one observation from the other response variable. Thus,the two response variables are dependent. This chapter examines analyses of matched-pairs data withcategorical responses.The first section considers binary outcomes. Marginal homogeneity occurs when the marginalprobabilities for one response equal the corresponding marginal probabilities for the other response. Thisimplies that the off-diagonal joint probabilities are also equal (symmetry). Mcnemar’s Test can be used tocompare the corresponding dependent proportions.“Subject-specific” tables have separate tables for each matched pair. They allow probabilities tovary by pair. (Thus, a better term might be pair-specific tables). Population-averaged tables sum thesubject-specific tables and correspond to marginal models. Subject-specific tables are used to fit logitmodels that allow each pair to have their own probability distribution. However, with these models, thereare usually more parameters than pairs, leading to problems with ML estimation. Solutions includeconditional ML estimation and random effects models.Extensions to matched-pairs with multinomial responses and additional explanatory variables usethe random effects approach. This is because the conditional ML approach can only estimate within-pairor within-cluster effects. Between-cluster effects (covariates that are constant within any given cluster)will cancel out of the conditional likelihood. Marginal models for matched-pairs for multinomial responsesare discussed in Section 10.3.Instead of modeling the marginal probabilities, one can model the joint cell probabilities. Cellprobabilities that satisfy symmetry (i.e., π ab= π bafor a≠b) also satisfy marginal homogeneity, but notnecessarily vice versa (unless quasi-symmetry also holds). The equivalent loglinear model has pairwiseassociation terms independent of the order of the subscripts (e.g., λ ab= λ bafor a≠b). Because of itssimplicity, a symmetry model fits well in very few applications. A quasi-symmetry loglinear model permitsthe main-effect terms in the symmetry model to differ, so that λ ab= λ bafor log µ aband log µ ba, but themain-effect terms differ so that log µab≠ log µba, as it is for symmetry (see equation 10.19 in Agresti).Y Xλ − λ are equal to conditional MLEs ofThe differences in the MLEs of the main effects parameters { j j }the { β j } in a baseline-category logit model applied to a population-averaged table. Finally, a squaretable satisfies quasi-independence if the row and column variables are independent given that the rowand column outcomes differ. Ordinal versions of quasi-symmetry and quasi-independence are given inSections 10.4.6 – 10.4.8.Matched-pairs models are used to analyze agreement between two observers. An independencemodel assumes no agreement. Quasi-independence and quasi-symmetry models imply someassociation. A numerical index of agreement is Cohen’s Kappa, ranging between 0 = agreement bychance only and 1 = perfect agreement.Paired preferences are common in analyses using preference data. The data may consist of an Ix I table whose cells contain the number of preferences for the column designation versus the rowdesignation. The Bradley-Terry logit model can be used to model influences on the probability that onemember is preferred to another. The model can be extended to handle comparisons on ordinal scales.Matched-sets are the generalization of matched-pairs to three or more responses. There areanalogs for marginal homogeneity, symmetry, and quasi-symmetry for matched-sets. Marginalhomogeneity is the equality of marginal probabilities in the multi-way table. Symmetry is the equality ofsymmetric joint probabilities, where symmetric means permuted responses. The log-linear quasisymmetrymodel is the symmetry model, plus factors representing main effect terms for the marginals.

177B. Comparing Dependent ProportionsIn a 2 x 2 table, if π1 +denotes the marginal “success” probability for response 1, and π + 1denotesthe marginal “success” probability for response 2, then to test for marginal homogeneity ( H0 : π1 += π+1),one can use the difference between the corresponding sample proportions. The square of the scorestatistic for this test (equation 10.4 in Agresti) is the test statistic of Mcnemar’s Test, which has chisquareddistribution with 1 df. Mcnemar’s test does not use the counts on the main diagonal becausethey are irrelevant to decide whether the marginal probabilities π1 +and π + 1differ. Confidence intervalson the difference ( π1 − π1), based on large-sample theory, do depend on the main diagonal counts.+ +The Prime Minister Approval Rating example compares the proportion of survey respondentswho approved of the UK Prime Minister on two different occasions (the responses).table.10.1

178C. Conditional Logistic Regression for Binary Matched PairsA subject-specific table (or, more accurately, a pair-specific table) can be used to fit a logit model withpair-specific probabilities. Specifically, for binary response Yitfrom the tth observation from the ith pair[ PY ( )]logit = 1 = α + β xit i twhere x1= 0 and x2= 1. With this model, for each pair, the odds of success for observation 2 areexp( β ) times the odds for observation 1, and this odds ratio is common across pairs. Conditional on theαiand β , pairs are independent of each other, and within a pair, observations are independent of eachother. This means that the association within a pair is described completely by αi, and pairs are onlyassociated via the common effect, β .Conditional logistic regression can be used to find the MLE for β , where one conditions onsufficient statistics for the αi, which are the pairwise success totals { Si = yi1 + yi2}(see section 10.2.3 inAgresti). This distribution only depends on β when at least one of the yitis 1. Thus, inference on βonly depends on outcomes in different categories at the two observations.An alternative way to find the MLE for β is to treat the α ias random effects, with identicalnormal distributions. Integrating the likelihood with respect to the distributions of theα iyields a marginallikelihood for β , which is maximized to get the MLE. The model is called a generalized linear mixedmodel. The MLE is the same for either the conditional ML estimation or ML estimation via thegeneralized linear mixed model.Agresti illustrates a case-control study where 144 cases have myocardial infarction (MI) and 144controls, matched to cases by age and gender, do not. Thus, there are 144 pairs. Each member of apair was also asked whether they had diabetes (1 = yes, 0 = no). The response in this model is MI, whichis fixed. The predictor is diabetes, which is random. The study is thus retrospective, but we can still getan estimate of the XY odds ratio, exp( β ) in a logistic regression model, by using the fixed responses(see chapter 2). The model for subject t in pair i is[ PY ( )]logit = 1 = α + β xit i itWe can get the MLE of β using conditional logistic regression or by integrating out random effects. Thepopulation-averaged table is in Table 10.3 in Agresti. The pairs for the subject-specific table are in Table10.4. We input the data into S using the subject-specific form. The following data frame has thevariables: pair (either subject 1 or subject 2 of the matched pair), MI (myocardial infarction; yes=1, no=0),diabetes (yes=1, no=0).table.10.3

179summary(fit.CLR)n= 288coef exp(coef) se(coef) z pdiabetes 0.838 2.31 0.299 2.8 0.0051exp(coef) exp(-coef) lower .95 upper .95diabetes 2.31 0.432 1.29 4.16Rsquare= 0.029 (max possible= 0.5 )Likelihood ratio test= 8.55 on 1 df, p=0.00345Wald test = 7.85 on 1 df, p=0.00508Score (logrank) test = 8.32 on 1 df, p=0.00392In Section 10.2.6, Agresti shows how to fit this model using only software for logistic regression.We can also fit a generalized linear mixed model (GLMM), where the αiare random effects with normaldistribution, mean 0 and unknown standard deviation. The MASS library fits GLMMs with its functionglmmPQL via penalized quasi-likelihood estimation. It uses the function lme, so the specification of itsarguments is very similar to lme, and its output is identical. Two additional R packages have functions forfitting GLMMs. glmmML in package glmmML fits GLMMs (binomial and Poisson families only) with random(normal) intercepts via maximum likelihood estimation. The repeated package also contains a glmmfunction, which fits via ML estimation. Gauss-Hermite quadrature is used to integrate with respect to therandom effects.To use glmmPQL, we need to specify a fixed-effect formula and a random-effect formula. The fixed-effectformula will be like an ordinary S formula: response ~ predictor. Here, it is MI ~ diabetes becausediabetes is the predictor. To specify a random intercept within pair, we use ~ 1 | pair for the randomargument. The 1 indicates a random intercept, and “| pair” means “within pair”. Thus, the call for bothS-PLUS and R islibrary(MASS)fit.glmmPQL

180In the output, under “Random effects”, we get the standard deviation of the normal distribution of therandom effects, which is quite small at 0.001827267. This may imply a very low association betweenmembers of the same pair, as it means that the αiare probably very similar (Recall that the associationis described completely by the αi, by definition of the model). “Residual”, I believe, is an estimate of thescale, which for the binomial is fixed at 1.0. We also get estimates of the fixed effects, notably theestimate of the effect of diabetes (0.8038742). The PQL estimate is slightly different from the conditionalML estimate from coxph.The function glmmML only allows a random intercept, and as such is less general than glmmPQL, althoughit is fine for this particular problem. The arguments for glmmML differ in that cluster defines the randomfactor. Note that the cluster argument will be evaluated in the calling environment, not within the datalist. So, you cannot just set cluster=pair for analyzing table.10.3 above (you must include the dataframe as well). Using glmmML, we get the same numerical estimate of β as from glmmPQL. The estimateof the standard deviation of the random effects (7.12e-06) is much smaller, but the resulting conclusionabout low association is the same.library(glmmML)glmmML(MI ~ diabetes, cluster=table.10.3$pair, family = binomial, data = table.10.3)coef se(coef) z Pr(>|z|)(Intercept) -0.1942 0.1364 -1.423 0.15500diabetes 0.8039 0.2835 2.836 0.00457Standard deviation in mixing distribution: 7.12e-06Std. Error: 0.05206Residual deviance: 390.9 on 285 degrees of freedom AIC: 396.9The definitions of AIC are the same in both functions (i.e., -2*loglik + 2*npar). Thus, the difference inthe AIC values across functions is due to the representation of the log likelihood, as glmmML gives -195.4561 for the log likelihood.The function glmm in the repeated library also fits random effects model, and allows several choices oferror distributions besides binomial. The argument nest is for specifying the pair variable.fit.glmm|z|)(Intercept) -1.942e-01 1.365e-01 -1.423 0.1548diabetes 8.039e-01 2.837e-01 2.834 0.0046 **sd 6.829e-07 1.196e-01 5.71e-06 1.0000---Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1(Dispersion parameter for binomial family taken to be 1)Null deviance: 399.25 on 287 degrees of freedomResidual deviance: 390.80 on 285 degrees of freedomAIC: 396.8Number of Fisher Scoring iterations: 4Normal mixing variance: 4.662868e-13Thus, glmm gives the same estimates as glmmML.

181D. Marginal Models for Square Contingency TablesThis section discusses marginal models for matched-pairs with ordinal and nominal multinomialresponses. For an I x I table, marginal homogeneity is PY (1= a) = PY (2= a)for responses Y t, t = 1, 2and categories a= 1,..., I .1. Ordinal ClassificationsThe cumulative logit model for matched-pairs is[ ( )]logit PY≤ j = α + βx, t= 1, 2; j= 1,..., I−1t j twhere x1= 0 and x 2= 1. In this model, the odds of outcome Y 2≤ j is exp( β ) times the odds ofoutcome Y 1≤ j . Marginal homogeneity corresponds to β = 0 .In the Premarital and Extramarital Sex example, subjects responded with opinions aboutpremarital sex (1 = always wrong, 2 = almost always wrong, 3 = sometimes wrong, 4 = never wrong) andabout extra-marital sex. Agresti fits a cumulative marginal logit model (equation 10.14) to these data,using maximum likelihood estimation. One way to do this is to use J. Lang’s mph.fit function for R,which uses a method of constrained maximum likelihood estimation. The model specification follows thatin Section 11.2.5 in Agresti. Briefly, if π denotes the complete set of multinomial joint probabilities, thena marginal logit model has the generalized loglinear form( π ) β (10.1)Clog A = XThe components of (10.1) are used in mph.fit.In using the function, I follow the notation of Lang’s documentation. First, I will fit a marginalhomogeneity model, which constrains the corresponding row and column marginal probabilities to beequal. This constraint is specified in the function h.fct, which is an optional argument to mph.fit. Asyou might perceive, the h function is the gradient of the Lagrangian.The observed counts are in y. The matrix Z describes the strata in the data. Here, we do not havestratification, so Z is a vector of ones. ZF = Z because the sample size is assumed fixed (see thedocumentation). M1, M2, and C.matrix are used in the constraint function. M1 and M2 are used to get rowand column totals of the expected counts. C.matrix is a constraint matrix that specifies marginalhomogeneity.y

1823,8,byrow=T)h.fct

183CMarg

184from the Northeast, Midwest, South, and West regions of the U.S. in 1980 to any of those same regionsin 1985. The large majority remained in the same region for both years.To compute a LR test using glm, we need the cell counts and a set of dummy variables thatcorrespond to the constraints of marginal homogeneity. Table A.17 in Agresti shows the set of dummyvariables. In that table, corresponding marginal expected frequencies are constrained to be equal andare thus represented by only three variables: m1, m2, and m3 (m4 = n – m1 – m2 – m3), instead of sixvariables, differentiating row from column totals. Then, the observed counts are modeled as functions ofthe expected marginal frequencies as well as expected cell counts. For example, the observed (1, 4) cellis modeled as m1 – (m11 + m12 + m13), where mij gives the expected count in the ijth cell.So, we create a data frame with this matrix of dummy variables and the observed counts.dummies

185NE 11607.00000 98.08285 265.6867 93.96975MW 88.73298 13677.00000 379.0733 232.34523S 276.47479 350.80060 17819.0000 287.25936W 92.53157 251.26811 269.7747 10192.00000Of course, with such a large value of the statistic, we reject marginal homogeneity.The R function mph.fit can also be used to get the LR test. Example 3 of Lang’s documentation for thefunction gives the necessary steps, which can be applied almost verbatim to the Migration data. It is leftto the reader to try mph.fit on this data set, recalling that h.fct is the constraint function.A Wald test of marginal homogeneity can be computed using Bhapkar’s statistic (equation 10.16 inAgresti). This statistic is actually very easy to calculate if we follow Wickens (1989, pp. 313-314) andnote that what we are testing is a set of simultaneous linear combinations of the cell probabilities. Theselinear combinations areπ + π + π = π + π + ππ π π π π ππ π π π π π12 13 14 21 31 4112+32+42=21+23+2413+23+43=31+32+34which imply marginal homogeneity if they hold simultaneously. If we represent the cell probabilities in a16 x 1 vector, π , then the following matrix, A, gives the three linear combinations above if post-multipliedby πA

186[1] 0E. Symmetry, Quasi-symmetry, and Quasi-independenceAn I x I table satisfying symmetry has πab = πba, a≠b. This implies the loglinear modellog µ ab= λ + λ a+ λ b+ λ abwhere all λab= λba. Note that the main-effect terms are the same for the two expected frequencies µaband µba. Quasi-symmetry allows the main-effect terms to differ so thatlog µ = λ + λ X + λ Y + λ ≠ log µ = λ + λ X + λ Y + λab a b ab ba b a abFor this model, the odds ratios on one side of the main diagonal are identical to the “mirror-image” oddsratios on the other side. The loglinear model for a table displaying quasi-independence adds a parameterfor each cell on the main diagonal:log µ = λ+ λ X+ λ Y+ δ Ia ab a b a( = b )When δa> 0 , µaais greater than under independence. Thus, this model would be used for a table thatshowed independence on the off-diagonal cells, but had larger counts on the main diagonal.Agresti uses the Migration example to illustrate fitting these three models via maximum likelihoodestimation. We can use the definitions above to create a data frameresidence80

18710 MW S 225 2,311 S S 17819 3,312 W S 270 3,413 NE W 63 1,414 MW W 176 2,415 S W 286 3,416 W W 10192 4,4options(contrasts = c("contr.treatment", "contr.poly"))(fit.symm

188matrix(fitted(fit.qsymm), ncol = 4, byrow = T, dimnames =list(rev(levels(residence80)), rev(levels(residence85))))NE MW S WNE 11607.00000 95.78862 370.4375 123.7739MW 91.21138 13677.00000 501.6825 311.1061S 167.56253 238.31746 17819.0000 261.1200W 63.22609 166.89392 294.8800 10192.0000From equation (10.25) in Agresti, a test of the null hypothesis of marginal homogeneity is given by1 - pchisq(fit.symm$deviance - fit.qsymm$deviance, df = fit.symm$df - fit.qsymm$df)[1] 0showing strong evidence of marginal heterogeneity.A fit of the quasi-symmetry model for the Migration data is given in an example in the documentation forthe R package exactLoglinTest. In fact, the data are included in the library as residence.dat.library(exactLoglinTest)data(residence.dat)residence.daty res.1985 res.1980 sym.pair1 11607 NE NE 12 100 MW NE 23 366 S NE 34 124 W NE 45 87 NE MW 26 13677 MW MW 57 515 S MW 68 302 W MW 79 172 NE S 310 225 MW S 611 17819 S S 812 270 W S 913 63 NE W 414 176 MW W 715 286 S W 916 10192 W W 10The variable sym.pair serves the same purpose as symm above. We can use mcexact to check theasymptotic p-value.(fit.qsymm.exact

189To fit a quasi-independence model to the Migration data, we can add four dummy variables to the dataframe that represent the main diagonal cells.table.10.6$D1

190table.10.5$symm

191Value Std. Error t value(Intercept) 1.6094379 0.4472136 3.5988126symm3,4 1.6327093 0.4883751 3.3431460symm2,4 1.7419086 0.4842847 3.5968686symm1,4 3.2108947 0.4560560 7.0405708symm3,3 0.1823216 0.6055301 0.3010941symm2,3 1.1472015 0.5123936 2.2389066symm1,3 2.8054296 0.4603858 6.0936490symm2,2 -0.2231436 0.6708204 -0.3326428symm1,2 1.9299608 0.4781430 4.0363673symm1,1 3.3603754 0.4549115 7.3868777tau -4.1303475 0.4499372 -9.1798320(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: 888.3778 on 15 degrees of freedomResidual Deviance: 15.51736 on 5 degrees of freedomNumber of Fisher Scoring Iterations: 5The estimated value of τ implies that the estimated probability that premarital sex is considered morewrong (i.e., closer to 1 than 4) than extramarital sex is exp(-4.13) = 0.016 times the estimated probabilitythat extramarital sex is considered more wrong.3. Quasi-Uniform AssociationEven after conditioning on the event that the two responses differ, sometimes independence still does nothold. Sometimes there is a monotone pattern to the probabilities. The loglinear modelX Ylog µ = λ + λ + λ + βuu + δ I( a=b)ab a b a b apermits linear-by-linear association off the main diagonal. For equal-interval scores, it implies uniformlocal association, given that responses differ.Fitting this model to the above example requires first defining scores for the row variable, then creating aDelta vector respresenting δa.table.10.5$scores.a

192Delta 0.43837201 0.2737843 1.601158scores:scores.a 0.61632154 0.1142078 5.396493(Dispersion Parameter for Poisson family taken to be 1 )Null Deviance: 888.3778 on 15 degrees of freedomResidual Deviance: 5.797136 on 7 degrees of freedomNumber of Fisher Scoring Iterations: 4Thus, off the main diagonal, the estimated local odds ratio is exp(0.616) = 1.85, which is slightly lowerthan Agresti’s result of 1.88.G. Measuring Agreement Between ObserversMatched-pairs models are used to measure agreement between ratings from two observers. Anindependence model would imply no agreement, whereas a quasi-independence or quasi-symmetrymodel implies some agreement in ratings.Section 10.5 in Agresti uses the Pathologists’ Tumor Ratings data to illustrate fitting differentmodels to assess degree of agreement between the two pathologists. The baseline model is theindependence model. The quasi-independence and quasi-symmetry models allow estimation of the oddsof agreement.The data set is available in the R package exactLoglinTest. We dump the object and source it into an S-PLUS session. So, for example# Rlibrary(exactLoglinTest)data(pathologist.dat)dump("pathologist.dat", file="c:/program files/insightful/splus61/users/cda/pathologist.R")# S-PLUSsource("pathologist.R")However, there are three differences between pathologist.dat and Table 10.8 in Agresti. First, Agrestionly uses the first 4 levels of each variable. Second, the A and B labels are reversed. Third, two of thecounts differ across the two data sets. We will fix the last two now, then subset the data in the glm call.We can switch the labels by changing the namesnames(pathologist.dat)

193options(contrasts = c("contr.treatment", "contr.poly"))(fit.ind

194[3,] 0.7646757 1.2353243 36.000000 4.366551e-005[4,] 1.8673623 3.0167010 13.115937 1.000000e+001A quasi-symmetry model would account for any association on the off-diagonal. To fit that model, weneed to create a symm factor. To ensure that we use the correct levels, we create a new data framewithout the ratings of 5. Then, we drop the unused level 5 in the factors.pathologist.dat2

195H. Kappa Measure of AgreementA numerical index that summarizes agreement is Cohen’s Kappa. Kappa compares the probability ofagreement, ∑ π , to that expected if the two sets of ratings were independent, a aa∑ π π a a + + a, as aproportion of the difference between perfect agreement (1.0) and chance agreement.κ =∑∑∑πa− π π1−π πaa a a+ + aaa+ + aKappa equals 0 when agreement equals that expected under independence, and equals 1.0 whenperfect agreement occurs. With ordered categories, one can weight disagreements in ratings differentlydepending on the difference between the ratings. Weighted Kappa isκ =w∑ ∑∑ ∑∑∑ ww π −1−a b ab ab a b ab a+ + babwπ πab a+ + bπ πFleiss and Cohen give weightsw a b I2 2ab= 1 −( − ) ( − 1) , which are larger for ratings closer together.We can compute Kappa and its ASE in S-PLUS using the functionKappa

196path.tab

197options(contrasts=c("contr.treatment", "contr.poly"))fit.BT

198µabX Y X Ylog = ( λ + λa + λb + λab ) − ( λ + λb + λa + λab)µbaX Y X Y= ( λ −λ ) −( λ − λ ) = β −βa a b b a bWe fit this quasi-symmetry model using the method from Section E of this chapter. First, I set up the dataset. The symm factor represents the interaction term.losing.team

199matrix(round(fitted(fit.BTQS),1),nr=7,nc=7,byrow=T,dimnames=list(win.team,losing.team))J. Bradley-Terry Model with Order EffectWe would like to see if there is a significant home team advantage. That is, is a team more likely to win ifthey play in their home city? We can incorporate a home team advantage for the team called a, via alogit model for the probability that team a beats team b when a is the home team:Πlog α ( βaβb)1− Π = + −*ab*abWhen α > 0 , there is a home team advantage. If the teams were equally matched ( βa= βb), the homeexp( α) 1+ exp( α)of winning.team has probability [ ]As mentioned in Agresti, the 42 pair sets (where order matters) can be viewed as 42 independentbinomial samples, where response is the number of successes (wins) for the home team out of the totalplayed. So, for example,response

200Thus, the estimate of the home team advantage parameter is positive, not surprisingly, implying a hometeam advantage, and its ASE is not too large.The fitted probabilities are obtained usingfitted.probs

201the family has a low income, (2) when the woman is not married and doesn’t want to marry the man, and(3) when the woman wants it for any reason. Gender was also recorded. So, there are four binaryresponses.Agresti first fits a complete symmetry model, with the same complete symmetry pattern within eachgender. So, there is a different intercept for each gender. The data are read below. Notice the order ofthe levels of the question variables.table.10.13

202Residual Deviance: 10.16 AIC: 112.8To see what the model predicts, we can look at the fitted probabilities.fitted.qs

203Chapter 11 – Analyzing Repeated Categorical ResponseDataA. Summary of Chapter 11, AgrestiRepeated categorical responses may come from repeated measurements over time on eachindividual or from a set of measurements that are related because they belong to the same group orcluster (e.g., measurements made on siblings from the same family, measurements made on a set ofteeth from the same mouth). Observations within a cluster are not usually independent of each other, asthe response from one child of a family, say, may influence the response from another child, because thetwo grew up together. Matched-pairs are the special case of each cluster having two members.Using repeated measures within a cluster can be an efficient way to estimate the mean responseat each measurement time without estimating between-cluster variability. Many times, one is interestedin the marginal distribution of the response at each measurement time, and not substantially interested inthe correlation between responses across times. Estimation methods for marginal modeling includemaximum likelihood estimation and generalized estimating equations. Maximum likelihood estimation isdifficult because the likelihood is written in terms of the I T multinomial joint probabilities for T responseswith I categories each, but the model applies to the marginal probabilities. Lang and Agresti give amethod for maximum likelihood fitting of marginal models in Section 11.2.5. Modeling a repeatedmultinomial response or repeated ordinal response is handled in the same way.Generalized estimating equations (GEE) are a multivariate generalization of quasi-likelihoodestimation. If there are T measurements per subject, then the quasi-likelihood method specifies a modelfor the mean and variance (as a function of the mean) for each measurement, without requiringknowledge of the distribution of the measurements (as ML estimation does). A “working” covariancestructure for the responses must be specified. The estimates of parameters in the mean response aresolutions of quasi-likelihood equations called generalized estimating equations. The estimates areconsistent even if the covariance matrix is incorrectly specified. However, this does not imply that onecan use an identity matrix for the correlation structure. The standard errors would be incorrect (probablyunderestimated). They are adjusted using the empirical dependence.However, as GEE does not have a likelihood function, then inference procedures based on alikelihood cannot be done. Instead, inference only uses Wald statistics and their asymptotic normality.Transitional models are special cases of logit or loglinear models for repeated categoricalobservations. When the repeated observations can be assumed to follow a Markov chain, an appropriateMarkov chain model can be fit. The first-order Markov property is satisfied when the prediction of a futureresponse only depends on the response immediately prior to it, and no other responses occurring beforethat previous response. Estimation of Markov chain models is via IRLS. Estimation can also be doneusing IPF if the model is a loglinear model. When there are explanatory variables, a regressive logisticmodel can be used (for a binary response) where the regressors are explanatory variables and alsoprevious responses.B. Comparing Marginal Distributions: Multiple ResponsesWith T repeated binary measurements, one may be interested in comparing the probability ofsuccess at each of the T times. The marginal logit model for T responses islogit[ PY ( = 1)] = α + β , t= 1,..., T(11.1)tBecause there are T probabilities with T + 1 parameters, one of the parameters must be constrained tobe zero. In that case, the model is saturated, and MLEs of the marginal probabilities are the sampleproportions with yt= 1. The multinomial likelihood is written in terms of the joint probabilities of response.Each sample cell proportion is the MLE of the joint probability corresponding to that cell. Marginalt

204homogeneity corresponds to PY (1= 1) = ... = PY ( T= 1) . That is, there are T identical responsedistributions across the T time periods. A likelihood ratio test compares the likelihood maximized undermarginal homogeneity to the ML fit under model (11.1). Maximum likelihood estimation under marginalhomogeneity is discussed in Section 11.2.5 in Agresti and in Section D in Chapter 10 of this manual, forillustrating Lang’s mph.fit function. The generalized loglinear form of the marginal logit model is given in(11.8) of Agresti, and repeated in (10.1) of this manual. For the case of a 2 T table, there are T marginalprobabilities. The model of marginal homogeneity with no covariates has the form given at the bottom ofp. 464 in Agresti when T = 2. The Lagrangian includes constraints that set the marginal probabilitiesequal to one another, as well as identifiability constraints. The fitting method of Lang gives consistentestimates of the regression coefficients, β , if the marginal model holds, regardless of the form of the jointdistribution. This is because the marginal parameters β and the joint parameters π are orthogonal.Thus, for consistency of the marginal parameter estimates, the method is robust to the specification of thejoint distribution.Agresti tests marginal homogeneity for the Crossover Drug Comparison Example (Table 11.1) whereeach subject used each of three drugs at three different times for the treatment of a chronic condition.The binary response at each time was favorable/unfavorable response. We first use the functionmph.fit for R (available from J. Lang. See Agresti p. 649) to test marginal homogeneity.First, I source in the appropriate files.source("C:/Program Files/R/rw1080/CDA/mph.r")source("C:/Program Files/R/rw1080/CDA/mph.supp.fcts.r")I happen to set up the data as a data frame, but we only use the count vector within mph.fit for the test.table.11.1

205h.fct

206I have copied the appropriate data lines and column headers from Agresti’s CDA data set web site, andpasted them into a file, which I’ve renamed “depress.txt”. The time variable is transformed using logbase 2.table.11.2

207To begin, we define the indices g to index the four groups (1=Severe/New; 2=Severe/Standard,3=Mild/New, 4=Mild/Standard). We also define the indices t 1to t 3to index the outcome values at thethree measurement times. Then, we set the vector π equal to the 32 cell probabilities { πgt123t t}. Forexample, the probability of a normal outcome at all three times from an individual on the new drug, withsevere depression is π1111. The marginal probability of normal response for this individual at time 0 is theanti-logit of α + β1+ β2.We define the A and C matrices to get the expression on the left of (11.2). For example, to write themarginal probability of Normal response at time 0, for group 1, we compute the sum of cell probabilities,∑ ∑ . Thus, the row of A that corresponds to this sum has a 1 in each of the spots that1 1πt3= 0 t 1,1,2=0 t2,t3corresponds to each of the terms of the sum. Each row of the matrix C has exactly one element with a –1and exactly one element with a +1, the remaining being 0. This matrix forms the logits (a ratio of logs ofsums of cell probabilities). Finally, the matrix X has one column of all ones (for the intercept), twocolumns for the group designation, and a final column for the value of time (0, 1, 2). Using thesedefinitions, we can run mph.fit for R.First, we must “unstack” the data frame table.11.2 so that time is three variables (log week 0, 1, and 2)instead of one.table.11.2a

2081 time treat diagnose1 1 0 1 12 1 1 1 13 1 2 1 14 1 0 0 15 1 1 0 16 1 2 0 17 1 0 1 08 1 1 1 09 1 2 1 010 1 0 0 011 1 1 0 012 1 2 0 0The sampling matrix has a column for each group and a row for each of the 32 cells, where the rows arein the same order as the Y vector. In each column, there is a 1 for each cell that corresponds to a cell forthe group for that column.Z

2090,0,0,0, 1,0,0,0, 0,0,0,0, 1,0,0,0, 0,0,0,0, 1,0,0,0, 0,0,0,0, 1,0,0,0, # g1 t0=00,0,0,0, 0,0,0,0, 1,0,0,0, 1,0,0,0, 0,0,0,0, 0,0,0,0, 1,0,0,0, 1,0,0,0, # g1 t1=00,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, 1,0,0,0, 1,0,0,0, 1,0,0,0, 1,0,0,0, # g1 t2=00,1,0,0, 0,0,0,0, 0,1,0,0, 0,0,0,0, 0,1,0,0, 0,0,0,0, 0,1,0,0, 0,0,0,0, # g2 t0=10,1,0,0, 0,1,0,0, 0,0,0,0, 0,0,0,0, 0,1,0,0, 0,1,0,0, 0,0,0,0, 0,0,0,0, # g2 t1=10,1,0,0, 0,1,0,0, 0,1,0,0, 0,1,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, # g2 t2=1# complement0,0,0,0, 0,1,0,0, 0,0,0,0, 0,1,0,0, 0,0,0,0, 0,1,0,0, 0,0,0,0, 0,1,0,0, # g2 t0=00,0,0,0, 0,0,0,0, 0,1,0,0, 0,1,0,0, 0,0,0,0, 0,0,0,0, 0,1,0,0, 0,1,0,0, # g2 t1=00,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, 0,1,0,0, 0,1,0,0, 0,1,0,0, 0,1,0,0, # g2 t2=00,0,1,0, 0,0,0,0, 0,0,1,0, 0,0,0,0, 0,0,1,0, 0,0,0,0, 0,0,1,0, 0,0,0,0, # g3 t0=10,0,1,0, 0,0,1,0, 0,0,0,0, 0,0,0,0, 0,0,1,0, 0,0,1,0, 0,0,0,0, 0,0,0,0, # g3 t1=10,0,1,0, 0,0,1,0, 0,0,1,0, 0,0,1,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, # g3 t2=1# complement0,0,0,0, 0,0,1,0, 0,0,0,0, 0,0,1,0, 0,0,0,0, 0,0,1,0, 0,0,0,0, 0,0,1,0, # g3 t0=00,0,0,0, 0,0,0,0, 0,0,1,0, 0,0,1,0, 0,0,0,0, 0,0,0,0, 0,0,1,0, 0,0,1,0, # g3 t1=00,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,1,0, 0,0,1,0, 0,0,1,0, 0,0,1,0, # g3 t2=00,0,0,1, 0,0,0,0, 0,0,0,1, 0,0,0,0, 0,0,0,1, 0,0,0,0, 0,0,0,1, 0,0,0,0, # g4 t0=10,0,0,1, 0,0,0,1, 0,0,0,0, 0,0,0,0, 0,0,0,1, 0,0,0,1, 0,0,0,0, 0,0,0,0, # g4 t1=10,0,0,1, 0,0,0,1, 0,0,0,1, 0,0,0,1, 0,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, # g4 t2=1# complement0,0,0,0, 0,0,0,1, 0,0,0,0, 0,0,0,1, 0,0,0,0, 0,0,0,1, 0,0,0,0, 0,0,0,1, # g4 t0=00,0,0,0, 0,0,0,0, 0,0,0,1, 0,0,0,1, 0,0,0,0, 0,0,0,0, 0,0,0,1, 0,0,0,1, # g4 t1=00,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,0, 0,0,0,1, 0,0,0,1, 0,0,0,1, 0,0,0,1), # g4 t2=0nr=24,nc=32, byrow=TRUE)When multiplied by the log of the entries in Aπ , C.matrix gives the logits of the 12 marginalprobabilities.C.matrix

210link4 -1.32492541 -1.77220372 0.1580641 1.8976785link5 -0.94446161 -0.86527205 0.1129916 -0.4220890link6 -0.16034265 0.04165961 0.1320449 -1.3442489link7 0.11441035 0.36019844 0.1542620 -1.3096890link8 1.29928298 1.26713007 0.1434996 0.1283795link9 3.52636052 2.17406086 0.1851307 3.8787830link10 0.05001042 -0.52647349 0.1481831 3.2436321link11 0.35364004 0.38045817 0.1208956 -0.1390165link12 0.73088751 1.28738984 0.1554322 -2.4992189CONVERGENCE STATISTICS...iterations = 100norm.diff = 0.371203norm.score = 0.000702893Original counts used.FITTING PROGRAM USED: mph.fit, version 1.0, 6/5/02Warning message:the condition has length > 1 and only the first element will be used in: if (a$L !="NA")The warning message given is apparently a bug in the mph.summary function, but would not affect results.It can be eliminated by changing the lineif (a$L != "NA") {toif (!all(is.na(a$L))) {in the mph.summary function.Here we see that this model gives a LR statistics of around 34.6. A much smaller LR statistic comes fromadding an interaction term in treatment and time, implying that the effect of time varies over time.X$trtxtime

211link8 1.29928298 1.38743932 0.1510355 -0.3417647link9 3.52636052 2.87709738 0.2337484 1.3582751link10 0.05001042 -0.04385680 0.1698035 0.6448222link11 0.35364004 0.43879603 0.1201678 -0.4368103link12 0.73088751 0.92144887 0.1647290 -1.0296818CONVERGENCE STATISTICS...iterations = 100norm.diff = 1.25822norm.score = 9.27206e-10Original counts used.FITTING PROGRAM USED: mph.fit, version 1.0, 6/5/02The beta coefficients correspond to the columns of the X matrix. Thus, at time t, the estimated odds ofNormal response for the new drug are exp(–0.06 + 1.01t) times the odds for the standard drug, for eachdiagnosis level.D. Marginal Modeling: Maximum Likelihood Approach. Modeling a RepeatedMultinomial ResponseWith a repeated multinomial response with I categories, there are I – 1 logits for each observation ormeasurement. For observation t, the logit takes the formTlogit ( t) = α + β x , j = 1,..., I −1(11.3)j j j tFor example, for an ordinal response, a proportional odds model uses logit ( t) logit [ P( Y j)]cumulative link, and βj= β .j= ≤ , aThe data in Table 11.4 in Agresti give the results of a randomized clinical trial comparing an activehypnotic drug with a placebo in patients who have insomnia. The response is the reported time to fallasleep (with categories < 20 minutes, 20 – 30 minutes, 30 – 60 minutes, or > 60 minutes). For this dataset, I copied the data from Agresti’s website, pasted it into a text file called “insomnia.txt”, and added afirst line of variable headings. We will not be using the final variable called “count” in this data set.We read in the data set using read.table.table.11.4

212[,1] [,2] [,3] [,4][1,] 120 120 120 120[2,] 120 120 120 120, , 2[,1] [,2] [,3] [,4][1,] 119 119 119 119[2,] 119 119 119 119Now, we get the marginal counts using xtabs, and divide these counts by n, which is an array of thesame size (thus the division can be done).xtabs(~time+outcome+treat, data=table.11.4)/n# note that we divide by n, , treat = 0outcometime 1 2 3 40 0.11666667 0.16666667 0.29166667 0.425000001 0.25833333 0.24166667 0.29166667 0.20833333, , treat = 1outcometime 1 2 3 40 0.10084034 0.16806723 0.33613445 0.394957981 0.33613445 0.41176471 0.15966387 0.09243697We next fit the proportional odds model in equation (11.6) in Agresti, using mph.fit. This model permitsinteraction between occasion and treatment.The data set has a four-category (I = 4) ordinal response at T = 2 occasions. We follow the same stepsas for section C. First, we “unstack” the data frame table.11.4 so that time is two variables (initial,follow-up) instead of one.table.11.4a

213count is for treat=1: active treatment, for initial and follow-up time < 20 minutes (coded 1). The secondcount is for treat=0, with the same initial and follow-up time.Y

214[28,] 0 1[29,] 1 0[30,] 0 1[31,] 1 0[32,] 0 1Each row of the A matrix, when multiplied by the π vector, results in one of the cumulative marginalprobabilities or its complement. For example, the first row of A.matrix below, when multiplied by πgives the probability that a randomly chosen observation from group 1 (active) takes < 20 minutes to fallasleep at initial time. The third row gives its complement.A.matrix

215nr=12,nc=24,byrow=T)The function L.fct computes the right hand side of equation 11.8 in Agresti.L.fct

216GEE methodology is a generalization of quasi-likelihood for multivariate responses. As with quasilikelihood,one specifies the mean for the set of responses and a link function relating the linear predictorto the mean, as well as the variance as a function of the mean and a guess at the correlation structureamong the responses. Estimation is accomplished via a set of estimating equations, yielding consistentestimates of the coefficients in the linear predictor regardless of the correctness of the correlationstructure. However, the standard errors of the estimates will be affected by the specification of thecorrelation structure. Thus, it is advantageous to try to specify an approximately accurate correlationstructure.Common correlation structures are independence (no correlation), exchangeable (all pairs ofresponses have the same correlation), and unstructured. The unstructured correlation matrix requiresthe most new parameters to be estimated, and will not be efficient if there are many responses.GEE differs from ML estimation in that no multivariate distribution has to be specified for themultivariate response. Thus, there is no likelihood with GEE. However, this also means there is nolikelihood-based inference. Instead, Wald statistics and their asymptotic distributions are used.Agresti uses GEE to fit a model to the Longitudinal Mental Depression Example from Table 11.2 (SectionC, this chapter). First, we read in the data, and change the binary variables to factors.# S-PLUS or Rtable.11.2

217The version of yags for S-PLUS 6.1 does not have a data argument. Thus, it is necessary to attach thedata set.library(yags)attach(table.11.2b) # S-PLUS onlyoptions(contrasts=c("contr.treatment", "contr.poly"))fit.yags

218gee S-function, version 5.2 modified 99/06/03 (1998)Model:Link:LogitVariance to Mean Relation: BinomialCorrelation Structure: ExchangeableCoefficients:Estimate Naive S.E. Naive z Robust S.E. Robust z(Intercept) -0.02809866 0.1625499 -0.1728617 0.1741791 -0.1613205diagnose -1.31391033 0.1448627 -9.0700418 0.1459630 -9.0016667treat -0.05926689 0.2205340 -0.2687427 0.2285569 -0.2593091time 0.48246420 0.1141154 4.2278625 0.1199383 4.0226037treat:time 1.01719312 0.1877051 5.4191018 0.1877014 5.4192084Estimated Scale Parameter: 0.985392Number of Iterations: 2Working Correlation[,1] [,2] [,3][1,] 1.000000000 -0.003432732 -0.003432732[2,] -0.003432732 1.000000000 -0.003432732[3,] -0.003432732 -0.003432732 1.000000000R also has a package called geepack, with largely identical arguments. However, it allows different linkfunctions for the mean, scale, and correlation. That is, the scale and correlation can depend oncovariates, being related to them via link functions.library(geepack)fit.geese

219The output only gives the robust s.e.s (called san.se, here).F. Marginal Modeling: GEE Approach. Modeling a Repeated Multinomial OrdinalResponseFor repeated ordinal responses, GEE can also be used to estimate parameters from the logit model. Forexample, for the the proportional odds cumulative logit model, we get parameter estimates for thecutpoints, as well as the coefficients for the covariates. We can also get parameter estimates for theassociation parameters in the covariance matrix, relating the responses to one another.The estimating equations are similar to those for univariate responses. The general form is givenin section 11.4.4 of Agresti for multinomial responses. For ordinal responses in particular, the responsevariable yichanges. Suppose we have an ordinal response with I = 4 categories and two responses perindividual. Then, each yiis length 2 x (4 – 1) = 6, with elements ( yi1(1), yi 1(2),..., yi2(3)). The elementyit(1) is equal to 1 if the tth response for individual i is greater than 1. The element yit(2) equals 1 if thethe tth response for individual i is greater than 2, etc. Thus, an individual with responses 3 and 4 at times1 and 2 has yi= (1,1,0,1,1,1) .For the proportional odds model, the vector µigives the probabilities associated with yi. Thus,for the example, µi= ( Py (i1 > 1),..., Py (i2> 3)) , where Py ( ) ( 1 exp( T )) 1it> j= + αj +xiβ −. Also, thecovariance structure of the elements of y iis put into Vi. Ordinary GEE (GEE1) uses the estimatingequations in section 11.4.4 of Agresti to estimate the regression coefficients and cut-points, usingsomething like Fisher scoring (with intial estimates for the covariance parameters). Then to estimate thecovariance parameters, a method of moments update is done, using the Fisher scoring estimates for theregression coefficients. These two steps are repeated until “convergence”. The “sandwich” estimator ofthe covariance matrix for the regression coefficients is given in section 11.4.4 of Agresti.A different estimation procedure, called GEE2, builds a set of estimating equations for thecovariance parameters too, updating both sets of parameters simultaneously. Differences between thetwo estimation methods are found in the Heagerty and Zeger (1996) reference in Agresti. Certain R andS-PLUS functions (namely, ordgee in package geepack, and the function ord.EE for S-PLUS Unix) claimto use “the” method of Heagerty and Zeger, which is apparently GEE1, although Heagerty and Zegercompare both GEE1 and GEE2. According to computations from these functions and from SAS’sGENMOD, they give different estimates than SAS does. I illustrate this for the insomnia data from table11.4 in Agresti, that was analyzed in Section D of this chapter.The function ordgee requires the data to be sorted by individual and response within individual. So, afterreading in the data, we modify table.11.4 as follows.table.11.4ord

220fit.ordgee

221Scale Parameter:1.013369Number of iterations : 6Number of observations : 478Number of clusters : 239These estimates match those from SAS. Also, notice that our scale parameter estimate is close to one,which is the theoretical value for a multinomial model.G. Markov Chains: Transitional ModelingDiscrete-time Markov chains are discrete-time stochastic processes with a discrete state space. Thatmeans that the random variable (potentially) changes states at discrete time points (say, every hour), andthe states come from a set of discrete (many times, also finite) possible states. So, if the state of therandom variable at time t is represented by y t, then the stochastic process is represented by{ y0, y1, y2,...}.This section deals with first-order Markov chains. Thus, for all t, the conditional distribution ofyt + 1given { y0, y1, y2,..., y t} is identical to the distribution of yt + 1given only yt. In words, this means thatin order to predict the state at the next time point, we only need to consider the current state. Any statesprior to the current state add nothing to predictability of the next state. This is sometimes called theMarkov property.The conditional probability, PY (t= j| Yt−1= i) = πji |( t), is called a one-step transition probability,because it is the probability of transition from state i at time t – 1 to state j at time t one time-step away.The sum of these probabilities over j is equal to 1 because at time t, the process must take on one of thepossible states, even if it remains at its current state. Because of the Markov property, the jointdistribution for a first-order Markov chain depends only on the one-step transition probabilities and on themarginal distribution for the initial state of the process.Because association is only present between pairs of adjacent, consecutive states, a first-orderMarkov chain can be fit to a sample of realizations from the chain by fitting the loglinear model( YY0 1, YY1 2,..., YT− 1YT)for T realizations. This model says that, for example, at any combination of states atthe time points 2,…, T, the odds ratios describing the association between Y0and Y1are the same.To illustrate how these models are fit, we fit several loglinear models to the Respiratory Illness data(Table 11.7 in Agresti). This was a longitudinal study of the effects of air pollution on respiratory illness inchildren, with yearly assessments done on children from age 9 to age 12. At each measurement, a childwas classified according to the presence or absence of wheeze. Thus, the data represent a four-waycontingency table, with two categories per dimension (presence, absence).First, we read in the data.table.11.7

222library(MASS)(fit.loglm1 X^2)Likelihood Ratio 122.9025 8 0Pearson 129.5971 8 0The LR statistic is very high at 122.9, on 8 degrees of freedom.A second-order MC model assumes that the prediction at 12 years is independent of the response at 9years, given both the responses at 10 years and at 11 years. This model is fit by(fit.loglm2 X^2)Likelihood Ratio 23.86324 4 8.507772e-05Pearson26.67938 4 2.307740e-05Again, the LR statistic is high at 23.9, on 4 degrees of freedom.The loglinear model with all pairwise associations gives the following.(fit.loglm3 X^2)Likelihood Ratio 1.458592 5 0.9177982Pearson 1.446029 5 0.9192127To get the parameter estimates, we call fit.loglm3$param. To get the conditional log odds ratios ofTable 11.8 in Agresti, we do the same manipulating that we did in Chapter 8. For example, at eachcombination of 9 and 10-year responses, the odds of wheezing at 12 years old areexp(fit.loglm3$param$Y11.Y12[1,1] + fit.loglm3$param$Y11.Y12[2,2] -(fit.loglm3$param$Y11.Y12[1,2] + fit.loglm3$param$Y11.Y12[2,1]))[1] 6.358584times higher than the odds of wheezing at 11 years old. (The anti-log of this odds ratio gives the log oddsratio in Table 11.8).The loglinear model with all pairwise associations, but constrained to have the one-year pair associationsidentical, and the greater-than-one-year associations identical fits well too. To fit this model, we createtwo new variables that represent, respectively, the one-year association and the greater-than-one-yearassociation. These variables just count the number of such associations.oneyear

223We will treat the two new variables as continuous instead of categorical. So, we can’t use loglm to dothe fitting (which is IPF). We will use glm instead. With glm, the factors in the model are not consideredcategorical.options(contrasts=c("contr.treatment", "contr.poly")) # S-PLUS only(fit.glm4

224table.11.9

Recall that a LRT on any single coefficient is obtained by fitting another model without that coefficient,and subtracting residual deviances.225

226Chapter 12 – Random Effects: Generalized Linear MixedModels for Categorical ResponsesA. Summary of Chapter 12, AgrestiWhen observations occur in clusters, we can model the dependence within a cluster using(latent) random effects. With this model, there are as many random effects as observations. They takethe same value for each observation in a cluster, but take different values across clusters. A cluster canbe an observation itself, in which case the random effects will differ across observations. In a randomeffects model, the random effects apply to a sample of clusters instead of to a fixed set of clusters. Thatis, the cluster effects are considered randomly sampled from a distribution of such effects, and the actualrealization of clusters will tend to differ across samples.Random effects models are similar to the marginal models of Chapter 11 in that both modelsincorporate dependence among observations. However, with the marginal models the dependence isaveraged over, and the marginal probability distribution is of more interest. With random effects models,we model the probability distribution within each cluster. Also, the (positive) correlation betweenobservations within a cluster is assumed to have arose because the random effects vary across clusters.Generalized linear mixed models (GLIMMs) extend generalized linear models (GLIMs) toincorporate random effects. Conditional on the random effect, a GLIMM resembles a GLIM. Forobservation t in cluster i, the linear predictor isT Tg ( µ ) = x β + z u(12.1)it it it ifor link function g and conditional mean µ it. The random effect vector, u i , is assumed to have amultivariate normal distribution N(, 0 Σ), where Σ describes the covariances among the elements of u i .Conditional on u i , the observations {y it} are independent over i and t. With the random effect appearingon the scale of the explanatory variables, subject-specific random effects can represent heterogeneitycaused by omitting certain explanatory variables. Random effects can also represent randommeasurement error in the explanatory variables.Because the u i are random effects, their inclusion only increases the number of parameters inthe model by the number of unique elements in their covariance matrix. On the other hand, if they werefixed effects, then the number of parameters would increase with the number of clusters. In that case,one can condition on the sufficient statistics for u i , making the resulting conditional likelihood independentof them. Maximizing this likelihood for β gives consistent estimates. Advantages of using a conditionallikelihood include not having to specify a distribution for the random effects. Also, if the clusters are notrandomly sampled (such as for a retrospective study), then conditional likelihood estimation isappropriate. However, conditioning removes the source of variability needed to estimate between-clustereffects (such as explanatory variable effects that are not repeated within clusters). Recall that conditionalMLE only allows estimation of within-cluster effects (such a treatment effect within pairs in a matchedpairsdesign). Also, a conditional model will not allow prediction of the individual random effects. Finally,sufficient statistics for the u i have to exist to use the approach.The fixed effects parameters β , both within-cluster and between-cluster, in a GLIMM haveconditional interpretations, given the random effect. For example, in a cross-over drug trial, where eachsubject takes both the control drug and experimental drug, the drug effect (a within-subject effect),conditional on the subject’s random effect is the coefficient that multplies the drug dummy variable. Also,in a multi-center drug trial, with a random center effect, where each center has patients who take thecontrol drug as well as patients who take the experimental drug, the drug effect (a between-center effect),again conditional on a given center’s random effect is the coefficient that multiples the drug dummyvariable. This is only true when the control and experimental patient come from the same center. Thus,comparisons are done at a fixed value of the random effect. In contrast, effects in marginal models are

227averaged over all clusters. When the link function is nonlinear, these effects are often smaller inmagnitude than their analogous cluster-specific effects. The approximate relationship between the twoparameters in a logistic-normal model has been shown to be that the marginal model coefficient is about2 1/2[1 + 0.6 σ ]− of the conditional coefficient, making it smaller in magnitude. Agresti’s Figure 12.1illustrates this attenuation.With a multinomial outcome (ordinal or nominal), a random effects model is represented via theobvious extension. That is, one adds a random effect to the linear predictor in the same way as for abinomial outcome.B. Logit GLIMM for Binary Matched PairsWhen g in (12.1) is the logit link, and µitis the conditional expectation of a Bernoulli random variablegiven u i, then we have the logit GLIMM. Clustering occurs if the data are paired, so that each (matched)pair is a cluster. Suppose cluster i of n contains the matched pair ( yi1, yi2), where yit= 1 or 0, and givenu i,[ ]logit PY ( = 1| u)= α + β x+ u(12.2)it i t i2where x1= 0 and x 2= 1, and ui~ N(0, σ ) . Then, we have a logit GLIMM for binary matched pairs.The two elements of the pair are independent given u i. So, their only dependence occurs through thisrandom effect. If we compute the log odds ratio of responding 1 in each of the two pairs, we get β . Thatis,[ ] [ ]logit PY ( = 1| u) − logit PY ( = 1| u)= βi2 i i1iwithin each cluster. Thus, β is called the cluster-specific log odds ratio. The model in (12.2) is called arandom intercept model because the intercept of the linear predictor is a random effect instead of a fixedconstant.The marginal distribution ofYtnyi=1 it{ exp( α βt ) [ 1 exp( α βt )]}= ∑ is binomial with index n and probability parameterE + x + U + + x + U , with expectation with respect to U a2N(0, σ ) randomvariable. The model implies a non-negative correlation between Y 1and Y2, with greater association2resulting from greater heterogeneity (larger σ ). Greater heterogeneity means that all the u iare spreadapart, where there are both large positive and large negative values. Large positive values of u imake itmore likely that ( yi1, yi2)are both 1. Large negative values of u imake it more likely that ( yi1, yi2)areboth 0. This correspondence between the two elements of the match pair traverses to a correspondencebetween the two sums Y1and Y2, leading to higher positive association.The 2 x 2 population-averaged table represents the frequency distribution of ( yi1, yi2). When the samplelog odds ratio is non-negative, then the MLE of β can be shown to be equal to log( n21 / n12), which is theconditional MLE, using conditional maximum likelihood estimation as discussed in Agresti’s chapter 10.2A sample log odds ratio that is negative means that the estimate of the variance, σ , is zero, implyingindependence. Then, the MLE of β is identical to that from the marginal model for matched pairs.

228Agresti uses the Prime Minister Ratings data to illustrate a random intercept model. The matched pairsare the clusters. The data are available in table form from Agresti’s website. I copied the data andplaced it into a text file called primeminister.txt. Then, I read it into R/S-PLUS.(table.12.1

229(Intr)occasion -0.376Standardized Within-Group Residuals:Min Q1 Med Q3 Max-2.1613715 -0.4746219 0.3097340 0.4117406 2.0973320Number of Observations: 3200Number of Groups: 1600The MLE for the (cluster-specific) log odds ratio of approval (beta) is –0.569, with SE = 0.073, so that theestimated odds of approval at the second survey are exp(–0.569) = 0.57 times that at the first survey.The estimate of the standard deviation of the random effects is 3.26.Other R functions that fit GLIMMs use other types of approximations (namely, maximization of thenumerically integrated likelihood -- integrated over the normal distrubtion of the random effects). Therepeated library has a function called glmm.library(repeated)fit.glmm|z|)(Intercept) 1.3190 0.1111 11.87 < 2e-16 ***occasion -0.5540 0.1332 -4.16 3.18e-05 ***sd 4.6830 0.1759 26.62 < 2e-16 ***---Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1(Dispersion parameter for binomial family taken to be 1)Null deviance: 4373.2 on 3199 degrees of freedomResidual deviance: 3501.1 on 3197 degrees of freedomAIC: 3507.1Number of Fisher Scoring iterations: 20In the call, the argument nest specifies the cluster variable.In the output, the MLE for the (cluster-specific) log odds ratio of approval (beta) is –0.554, with SE =0.133, so that the estimated odds of approval at the second survey are exp(–0.554) = 0.57 times that atthe first survey. The estimate of the standard deviation of the random effects is 4.68. These numbersare closer to what SAS NLMIXED gets for Agresti. (In fact, it is the same fitting algorithm as NLMIXED).The R package glmmML also fits GLIMMs by maximizing the numerically integrated likelihoodlibrary(glmmML)(fit.glmmML|z|)(Intercept) 1.3404 0.2495 5.372 7.78e-08occasion -0.5563 0.1353 -4.113 3.91e-05Standard deviation in mixing distribution: 5.138Std. Error: 0.4339Residual deviance: 3502 on 3197 degrees of freedom AIC: 3508

230In the call, the argument cluster specifies the cluster variable. Apparently, the variable is not taken fromthe data frame, and must be referenced (e.g., by using the $).In the output, the MLE for the (cluster-specific) log odds ratio of approval (beta) is –0.556, with SE =0.135, so that the estimated odds of approval at the second survey are exp(–0.556) = 0.57 times that atthe first survey. The estimate of the standard deviation of the random effects is 5.14. These numbersare closest to what SAS NLMIXED gives. Using the approximate relationship given in Section A, thisimplies a marginal effect of about –0.135.Another R function called glmm in package GLMMGibbs fits GLIMMs via MCMC sampling.The extension of (12.2) to more than 2 observations in each cluster is sometimes called theRasch Model. In this case the model would look like[ ]logit PY ( = 1| u)= β + u(12.3)it i t ifor t = 1,…,T. Note that the log odds ratio of responding 1 at t = 1 versus t = 2 , for a given subject i is thedifference, β1 − β2. This is not the same as the population-averaged log odds ratio, which would belogit PY ( = 1) − logit PY ( = 1) , and is the subject of marginal models.[ ] [ ]1 2C. Examples of Random Effects Models for Binary Data1. Small Area Estimation of Binomial ProportionsSmall area estimation models are random effects models. These models treat each area as a clusterwith its own random effect coming from a common distribution of the random effects. The modelestimates each area’s mean by borrowing information from other areas. The borrowing takes placethrough the parameters of the common distribution. The resulting estimates of the area means areshrunk toward the population mean. The amount of shrinkage is controlled by the size of the estimatedvariance of the random effects distribution, and the size of the area. A lower variance and lower sizeresult in less shrinkage.Let Yitbe tth binary response for the ith area, t = 1,…,T i. Conditional on the random effect, u i~2N(0, σ ) ,logit PY ( = 1| u)= α + u(12.4)[ ]it i iso that the estimate of πi= PY (it= 1| ui)is a function of the predicted random effect, u î, which isestimated by its marginal posterior mean.Table 12.2 in Agresti gives data on simulated voting proportions in the 1996 presidential election for eachof the 50 states. The states are the areas.After reading in the data (which are available on Agresti’s web site for the text), we fit a random interceptmodel using glmmPQL from the MASS library (available in both S-PLUS and R). The response is theobserved proportion, with “case” weights as the number of observations per state.table.12.2

231fit.glmmPQL

232We can add covariates to model (12.3), and then consider the multiple questions or items for anindividual as part of a single cluster. Agresti illustrates fitting this model using the abortion data in Table10.13. In that data set, each subject (cluster) answered three yes/no questions. Subjects were classifiedby sex. A random effects logistic model for the response assumes a positive correlation betweenresponses from the three questions by a single individual.We start with the previous data frame called table.10.13 (see chapter 10, above).table.10.13gender question1 question2 question3 count1 M Y Y Y 3422 F Y Y Y 4403 M N Y Y 64 F N Y Y 145 M Y N Y 116 F Y N Y 147 M N N Y 19etcWe reshape the data so that it is in “long” format instead of “wide” format. In R, one can use the functionreshape or stack for this; in S-PLUS, one can use the function menuStackColumns. reshape is definitelyeasier to use than menuStackColumns.menuStackColumns(target=table.10.13a,source=table.10.13,group.col.name="question",target.col.spec="response", source.col.spec=c("question1","question2","question3"),rep.source.col.spec=c("gender","count"),rep.target.col.spec=list(gender="gender",count="count"),show.p=F)#R: table.10.13a

233Linear mixed-effects model fit by maximum likelihoodData: table.10.13bAIC BIC logLik32091.19 32130.92 -16039.59Random effects:Formula: ~ 1 | id(Intercept) ResidualStdDev: 4.308437 0.4507951Variance function:Structure: fixed weightsFormula: ~ invwtFixed effects: response ~ gender + questionValue Std.Error DF t-value p-value(Intercept) -0.5108888 0.1710876 3698 -2.98612 0.0028gender 0.0064619 0.2215622 1848 0.02917 0.9767questionquestion2 0.3067315 0.0722624 3698 4.24469

234∫∫∏PY ( = y , Y = y , Y = y ) = PY ( = y , Y = y , Y = y | u) f( u)dui1 i1 i2 i2 i3 i3 i1 i1 i2 i2 i3 i3i i i3= PY ( = y | u) f( u)dut=1it it i i iThe probabilities PY (it= yit | ui) are defined in equation (12.10) in Agresti (p. 504), and the density of u isnormal. So, we numerically integrate. The function in R/S-PLUS to integrate is called integrate, and isvery easy to use, theoretically. Another function that works correctly for this problem is in the R packagermutil, and is called int. For either, we first define the function to be numerically integrated. Here, Icompute the probability of a positive response for each of the three questions, for a female respondent.Then, to get the expected frequency for that cell of the contingency table, we multiply the estimatedprobability by 1037, the number of female observations.First, we get the estimates of the parameters from the fit. Then, we set a variable called female to 1, toindicate a female.beta.coef

235The data are already in the object table.11.2 (see above). Here, we fit a random intercept model usingthe R function glmmML in the glmmML package, the R function glmm in the repeated package, and thefunction glmmPQL in the MASS library.First, the glmmML R function, because it gives the “correct” MLEs.library(glmmML)(fit.glmmML|z|)(Intercept) -0.02795 0.1641 -0.1703 8.65e-01diagnose -1.31521 0.1546 -8.5046 0.00e+00treat -0.05970 0.2225 -0.2684 7.88e-01time 0.48284 0.1160 4.1638 3.13e-05treat:time 1.01842 0.1924 5.2930 1.20e-07Standard deviation in mixing distribution: 0.06581Std. Error: 1.242Residual deviance: 1162 on 1014 degrees of freedom AIC: 1174As Agresti notes, the subject-specific parameter estimates are very close to the GEE and ML marginalestimates. The reason he gives is that the estimate of the standard deviation of the random effectsdistribution is close to zero (0.07), implying little heterogeneity among subjects. Thus, the derivedpairwise correlation among observations on the same subject is not high (see equation (12.6) in Agresti).Now, we fit the same model using glmm in the repeated R package.library(repeated)fit.glmm|z|)(Intercept) -0.02741 0.16514 -0.166 0.868150diagnose -1.33422 0.14783 -9.025 < 2e-16 ***treat -0.06115 0.22378 -0.273 0.784652time 0.48917 0.11560 4.231 2.32e-05 ***sd 0.26182 0.07263 3.605 0.000313 ***treat:time 1.03267 0.19017 5.430 5.62e-08 ***---Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1(Dispersion parameter for binomial family taken to be 1)Null deviance: 1411.9 on 1019 degrees of freedomResidual deviance: 1161.7 on 1014 degrees of freedomAIC: 1173.7

236The function glmmPQL in the MASS library does the same.library(MASS)fit.glmmPQL

237Agresti uses the clinical trial data from Chapter 6 (Table 6.9, repeated in Table 12.5), comparing a drug toa control, to illustrate fitting a logistic-normal model to multicenter clinical trial data. For a subject incenter i using treatment t (1 = drug, 2 = control), let y it= 1 denote success. A general model includes atreatment-by-center interaction:logit PY ( = 1| u, b) = α + ( β + b) / 2 + u[ ]i1i i i i[ ]logit PY ( = 1| u, b) = α − ( β + b) / 2 + ui2i i i i22where ui~ N(0, σa)and bi~ N(0, σb), all independent of one another. With this model, β + biis thecenter-specific log odds ratio for the ith center, and β is the expected center-specific log odds ratio.2Thus, we have random center effects and random treatment-by-center effects. When σb= 0, the logodds ratio between treatment and response is constant over centers.Recall that we read these data into S-PLUS/R as a data frame in Section E of Chapter 6. I make a fewchanges to this data frame so that we have the coding used by Agresti.table.6.9$TreatC

238Number of Observations: 273Number of Groups: 8The estimate of 0.721 (SE = 0.286) is a little off from Agresti’s MLE of 0.739 (SE = 0.300). Note Agresti’sdiscussion of penalized quasi-likelihood in Section 12.6.4.Now, I fit the treatment-by-center interaction as well.fit.glmmPQL.ia

239(Intercept) TreatCa -0.8998236 0.7692297b 0.9721856 0.4584772c 0.1236859 0.6716927d -2.1502939 1.0664178e -1.4833405 0.9778767f -2.4923268 1.1419009g -1.5186057 0.9455523h 0.9443708 0.4365184Now, I get the predicted odds ratios.invlogit

240Tit⎡ exp( xitβ + ui) ⎤ ⎡ 1 ⎤p( yi | ui)= ∏ ⎢ T Tt 1+ exp(it+ ui )⎥ ⎢1+ exp(it+ ui)⎥⎣ x β ⎦ ⎣ x β ⎦2then treat the y i ‘s as multinomial draws, with probabilities given by p( yi) = ∫ p( yi| ui) p( ui| σ ) du ,uand index N. Maximizing this likelihood gives the MLE of N, and the other parameters. This isstraightforward to do in S-PLUS or R, using a function like optim. However, to match the estimateobtained by Agresti, I will also illustrate a conditional approach that Agresti used in Coull and Agresti,1999 (see references in Agresti).First, we should enter the data from Table 12.6. I do this in several steps so that I can the data indifferent forms throughout this section.Here, I enter the 0/1 patterns, separately from the counts.temp

241#S-PLUS: menuStackColumns(source=table.12.6a,target=table.12.6b,source.col.spec=names(table.12.6a)[1:6],group.col.name="Survey",target.col.spec="Capture",rep.source.col.spec="id",show.p=F)#S-PLUS: names(table.12.6b)

242c(result.values[1:6],exp(result.values[7:8]))Survey1 Survey2 Survey3 Survey4 Survey5 Survey6-1.4378836 -0.9094129 -1.7539897 -0.6751367 -1.2273151 -1.0325512 0.9075391 20.9831241The last value (plus 68) is N, and the penultimate value is σ . These are close to those obtained byAgresti, 24+68 and 0.97, respectively.Next, I compute the profile-likelihood confidence interval, using the function plkhci, which is onlyavailable in the R package, Bhat. To obtain the interval in S-PLUS, one could use a brute force methodby inputting different values of the missing cell count, and obtaining the log likelihood.Recall that to use the function plkhci, we create a list with the labels, starting values, and lower andupper bounds. Then, we input this list as an argument, followed by the negative log-likelihood function,and the label of the parameter being profiled. A confidence interval is returned. Here, it is about (74,141), similar to that obtained by Agresti (75, 154).library(Bhat)x

243}Now, we minimize the negative log likelihood and obtain the MLEs of the parameters. The MLE of N isobtained using the expression given above.junk

244table.11.4$treat

245}-sum(log(probs))Now, we use the optim function. Here, I start the optimization at the estimates from polr, but to have itconverge in any reasonable amount of time, you have to start very close to the maximizer, which appearsto be c(-3.4858986, -1.4830128, 0.5606742,1.602,.058,1.081,log(1.9))optim(par=c(fit$zeta,fit$coefficients[c("time","treat","treat:time")],.5), fn=func,control=list(trace=2,REPORT=1), method="BFGS", matrix.i= as.matrix(table.11.4a))$par[1] -3.48824875 -1.48407698 0.56211694 1.60158887 0.05754634 1.08140528[7] 0.64423760$value[1] 592.9728$countsfunction gradient20 4$convergence[1] 0$messageNULLFitting adjacent-categories logit model follows the same idea, by defining the likelihood as a product ofprobabilities within each category.E. Multivariate Random Effects Models for Binary DataOne form of multivariate random effects is the presence of both a random intercept and a random slopefor a regression that “varies” across clusters, like individuals. Another type is nested random effects,where for example, one can have random effects for schools and random effects for students withinschools. Finally, one can have a model with more than one response variable, where each responsevariable has an associated random subject effect, say.1. Bivariate Binary ResponseAgresti gives an example of a bivariate binary response where schoolboys were interviewed twice,several months apart, and asked about their self-perceived membership in the leading crowd (yes or no)and asked about whether they sometimes felt they needed to go against their principles in order to fit in tothat crowd (yes or no). Thus, the two binary responses are called Membership and Attitude. There aresubject random effects for each variable.Following Agresti, for subject i, let yitvbe the response at interview time t on variable v. Then, we usethe logit modellogit [ PY (itv= 1| uiv )]= βtv + uivwhere (u iAtt, u iMem) ~ N(, 0 Σ). Thus, the correlation between Membership and Attitude is considered to bepossibly nonzero.The data in Table 12.8 in Agresti are available on his website. Here I read in the data set after copying itand saving it as a text file called “crowd.txt”.table.12.8

246There is no library function for either R or S-PLUS (yet) that automatically fits multivariate logit modelswith random effects via maximum likelihood or any other method. There are several ways we might dothe fitting. One straightforward method is to maximize the integrated likelihood, where integration is overthe distribution of the bivariate random effects (of course, we use numerical integration). MultivariateGaussian quadrature can be accomplished either via the adapt function in the adapt library or using asimple 2D numerical integration function, called GL.integrate.2D, from Diego Kuonen, and modifiedslightly below.GL.integrate.2D

247att2

248Another option is to take advantage of an existing function for generalized linear mixed models, such asglmmPQL from library MASS, and do some tricks to emulate a multivariate response with multivariaterandom effects. A couple of emails from the R-news newsgroup deal with fitting these models, and onereferences a write-up in the multilevel website (http://multilevel.ioe.ac.uk/softrev/reviewmixed.pdf) whichexplains how to fit a bivariate normal response (not repeated measures) with random effects, using lme. Iapplied this method to glmmPQL (which uses lme) for a bivariate binary response.Here, I set up the original data set so that have one column for “response”, and add two additionalindicator columns that indicate from which variable the response comes. I also add a time indicatorvariable that indicates whether the response is measured at time 1 or time 2.data1

249Structure: General positive-definiteStdDev Corrvar1 2.034878 var1var2 1.578995 0.265Residual 0.771258Variance function:Structure: fixed weightsFormula: ~ invwtFixed effects: response ~ -1 + var1 + var2 + var1:time + var2:timeValue Std.Error DF t-value p-valuevar1 -0.4992500 0.05009042 10191 -9.966977

250malformation given survival. Also, let x dbe the dosage for group d. In the model, we use litter-specificrandom effects: ui ( d )= ( ui ( d )1, ui ( d )2) sampled from a N(, 0 Σ d). The bivariate random effect allowsdiffering amounts of dispersion for the two probabilities. We also fit separate fixed dose effects for thetwo logits, logit( ϖi( d) j)= ui( d)j+ αj+ βjxd, where j is the index for the logit.Here I read in the data from Table 12.9, which I have placed in a text file called toxic.txt. This file hascolumn headings: litter, dose, dead, malformed, and normal. The last 3 columns come directly from thenumbers in Table 12.9 (I will have the data posted on the main website for this manual).table.12.9

251Now, I estimate some starting values, using a continuation-ratio model without random effects. We cando this with vglm from library VGAM in R (discussed in Section 7F of ths manual). Note that I useaggregate to eliminate the litter column.library(VGAM)table.12.9a

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