Building Statistical Models in Python: Develop useful models for regression, classification, time series, and survival analysis

BIRMINGHAM—MUMBAI

Building Statistical Models in Python

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To my parents, Thieu and Tang, for their enormous support and faith in me.

To my wife, Tam, for her endless love, dedication, and courage.

- Huy Hoang Nguyen

To my daughter, Lydie, for demonstrating how work and dedication regenerate inspiration and creativity. To my wife, Helene, for her love and support.

– Paul Adams

To my partner, Kate, who has always supported my endeavors.

– Stuart Miller

Contributors

About the authors

Huy Hoang Nguyen is a mathematician and data scientist with extensive experience in advanced mathematics, strategic leadership, and applied machine learning research. He holds a PhD in Mathematics, as well as two Master’s degrees in Applied Mathematics and Data Science. His previous work focused on Partial Differential Equations, Functional Analysis, and their applications in Fluid Mechanics. After transitioning from academia to the healthcare industry, he has undertaken a variety of data science projects, ranging from traditional machine learning to deep learning.

Paul Adams is a Data Scientist with a background primarily in the healthcare industry. Paul applies statistics and machine learning in multiple areas of industry, focusing on projects in process engineering, process improvement, metrics and business rules development, anomaly detection, forecasting, clustering, and classification. Paul holds an MSc in Data Science from Southern Methodist University.

Stuart Miller is a Machine Learning Engineer with a wide range of experience. Stuart has applied machine learning methods to various projects in industries ranging from insurance to semiconductor manufacturing. Stuart holds degrees in data science, electrical engineering, and physics.

About the reviewers

Krishnan Raghavan is an IT Professional with over 20+ years of experience in software development and delivery excellence across multiple domains and technology ranging from C++ to Java, Python, Data Warehousing, and Big Data tools and technologies.

When not working, Krishnan likes to spend time with his wife and daughter, reading fiction and nonfiction as well as technical books. Krishnan tries to give back to the community by being part of the GDG Pune Volunteer Group, helping the team organize events. Currently, he is unsuccessfully trying to learn how to play the guitar.

You can connect with Krishnan at or via LinkedIn: .

I would like to thank my wife Anita and daughter Ananya for giving me the time and space to review this book.

Karthik Dulam is a Principal Data Scientist at EDB. He is passionate about all things data with a particular focus on data engineering, statistical modeling, and machine learning. He has a diverse background delivering machine learning solutions for the healthcare, IT, automotive, telecom, tax, and advisory industries. He actively engages with students as a guest speaker at esteemed universities delivering insightful talks on machine learning use cases.

I would like to thank my wife, Sruthi Anem, for her unwavering support and patience. I also want to thank my family, friends, and colleagues who have played an instrumental role in shaping the person I am today. Their unwavering support, encouragement, and belief in me have been a constant source of inspiration.

Table of Contents

Preface

Part 1: Introduction to Statistics

1

Sampling and Generalization

Software and environment setup

Population versus sample

Population inference from samples

Randomized experiments

Observational study

Sampling strategies – random, systematic, stratified, and clustering

Probability sampling

Non-probability sampling

Summary

2

Distributions of Data

Technical requirements

Understanding data types

Nominal data

Ordinal data

Interval data

Ratio data

Visualizing data types

Measuring and describing distributions

Measuring central tendency

Measuring variability

Measuring shape

The normal distribution and central limit theorem

The Central Limit Theorem

Bootstrapping

Confidence intervals

Standard error

Correlation coefficients (Pearson’s correlation)

Permutations

Permutations and combinations

Permutation testing

Transformations

Summary

References

3

Hypothesis Testing

The goal of hypothesis testing

Overview of a hypothesis test for the mean

Scope of inference

Hypothesis test steps

Type I and Type II errors

Type I errors

Type II errors

Basics of the z-test – the z-score, z-statistic, critical values, and p-values

The z-score and z-statistic

A z-test for means

z-test for proportions

Power analysis for a two-population pooled z-test

Summary

4

Parametric Tests

Assumptions of parametric tests

Normally distributed population data

Equal population variance

T-test – a parametric hypothesis test

T-test for means

Two-sample t-test – pooled t-test

Two-sample t-test – Welch’s t-test

Paired t-test

Tests with more than two groups and ANOVA

Multiple tests for significance

ANOVA

Pearson’s correlation coefficient

Power analysis examples

Summary

References

5

Non-Parametric Tests

When parametric test assumptions are violated

Permutation tests

The Rank-Sum test

The test statistic procedure

Normal approximation

Rank-Sum example

The Signed-Rank test

The Kruskal-Wallis test

Chi-square distribution

Chi-square goodness-of-fit

Chi-square test of independence

Chi-square goodness-of-fit test power analysis

Spearman’s rank correlation coefficient

Summary

Part 2: Regression Models

6

Simple Linear Regression

Simple linear regression using OLS

Coefficients of correlation and determination

Coefficients of correlation

Coefficients of determination

Required model assumptions

A linear relationship between the variables

Normality of the residuals

Homoscedasticity of the residuals

Sample independence

Testing for significance and validating models

Model validation

Summary

7

Multiple Linear Regression

Multiple linear regression

Adding categorical variables

Evaluating model fit

Interpreting the results

Feature selection

Statistical methods for feature selection

Performance-based methods for feature selection

Recursive feature elimination

Shrinkage methods

Ridge regression

LASSO regression

Elastic Net

Dimension reduction

PCA – a hands-on introduction

PCR – a hands-on salary prediction study

Summary

Part 3: Classification Models

8

Discrete Models

Probit and logit models

Multinomial logit model

Poisson model

The Poisson distribution

Modeling count data

The negative binomial regression model

Negative binomial distribution

Summary

9

Discriminant Analysis

Bayes’ theorem

Probability

Conditional probability

Discussing Bayes’ Theorem

Linear Discriminant Analysis

Supervised dimension reduction

Quadratic Discriminant Analysis

Summary

Part 4: Time Series Models

10

Introduction to Time Series

What is a time series?

Goals of time series analysis

Statistical measurements

Mean

Variance

Autocorrelation

Cross-correlation

The white-noise model

Stationarity

Summary

References

11

ARIMA Models

Technical requirements

Models for stationary time series

Autoregressive (AR) models

Moving average (MA) models

Autoregressive moving average (ARMA) models

Models for non-stationary time series

ARIMA models

Seasonal ARIMA models

More on model evaluation

Summary

References

12

Multivariate Time Series

Multivariate time series

Time-series cross-correlation

ARIMAX

Preprocessing the exogenous variables

Fitting the model

Assessing model performance

VAR modeling

Step 1 – visual inspection

Step 2 – selecting the order of AR(p)

Step 3 – assessing cross-correlation

Step 4 – building the VAR(p,q) model

Step 5 – testing the forecast

Step 6 – building the forecast

Summary

References

Part 5: Survival Analysis

13

Time-to-Event Variables – An Introduction

What is censoring?

Left censoring

Right censoring

Interval censoring

Type I and Type II censoring

Survival data

Survival Function, Hazard and Hazard Ratio

Summary

14

Survival Models

Technical requirements

Kaplan-Meier model

Model definition

Model example

Exponential model

Model example

Cox Proportional Hazards regression model

Step 1

Step 2

Step 3

Step 4

Step 5

Summary

Index

Other Books You May Enjoy

Preface

Statistics is a discipline of study used for applying analytical methods to answer questions and solve problems using data, in both academic and industry settings. Many methods have been around for centuries, while others are much more recent. Statistical analysis and results are fairly straightforward for presenting to both technical and non-technical audiences. Furthermore, producing results with statistical analysis does not necessarily require large amounts of data or compute resources and can be done fairly quickly, especially when using programming languages such as Python, which is moderately easy to work with and implement.

While artificial intelligence (AI) and advanced machine learning (ML) tools have become more prominent and popular over recent years with the increase of accessibility in compute power, performing statistical analysis as a precursor to developing larger-scale projects using AI and ML can enable a practitioner to assess feasibility and practicality before using larger compute resources and project architecture development for those types of projects.

This book provides a wide variety of tools that are commonly used to test hypotheses and provide basic predictive capabilities to analysts and data scientists alike. The reader will walk through the basic concepts and terminology required for understanding the statistical tools in this book prior to exploring the different tests and conditions under which they are applicable. Further, the reader will gain knowledge for assessing the performance of the tests. Throughout, examples will be provided in the Python programming language to get readers started understanding their data using the tools presented, which will be applicable to some of the most common questions faced in the data analytics industry. The topics we will walk through include:

An introduction to statistics

Regression models

Classification models

Time series models

Survival analysis

Understanding the tools provided in these sections will provide the reader with a firm foundation from which further independent growth in the statistics domain can more easily be achieved.

Who this book is for

Professionals in most industries can benefit from the tools in this book. The tools provided are useful primarily at a higher level of inferential analysis, but can be applied to deeper levels depending on the industry in which the practitioner wishes to apply them. The target audiences of this book are:

Industry professionals with limited statistical or programming knowledge who would like to learn to use data for testing hypotheses they have in their business domain

Data analysts and scientists who wish to broaden their statistical knowledge and find a set of tools and their implementations for performing various data-oriented tasks

The ground-up approach of this book seeks to provide entry into the knowledge base for a wide audience and therefore should neither discourage novice-level practitioners nor exclude advanced-level practitioners from the benefits of the materials presented.

What this book covers

Chapter 1, Sampling and Generalization, describes the concepts of sampling and generalization. The discussion of sampling covers several common methods for sampling data from a population and discusses the implications for generalization. This chapter also discusses how to setup the software required for this book.

Chapter 2, Distributions of Data, provides a detailed introduction to types of data, common distributions used to describe data, and statistical measures. This chapter also covers common transformations used to change distributions.

Chapter 3, Hypothesis Testing, introduces the concept of statistical tests as a method for answering questions of interest. This chapter covers the steps to perform a test, the types of errors encountered in testing, and how to select power using the Z-test.

Chapter 4, Parametric Tests, further discusses statistical tests, providing detailed descriptions of common parametric statistical tests, the assumptions of parametric tests, and how to assess the validity of parametric tests. This chapter also introduces the concept of multiple tests and provides details on corrections for multiple tests.

Chapter 5, Non-parametric Tests, discuss how to perform statistical tests when the assumptions of parametric tests are violated with class of tests without assumptions called non-parametric tests.

Chapter 6, Simple Linear Regression, introduces the concept of a statistical model with the simple linear regression model. This chapter begins by discussing the theoretical foundations of simple linear regression and then discusses how to interpret the results of the model and assess the validity of the model.

Chapter 7, Multiple Linear Regression, builds on the previous chapter by extending the simple linear regression model into additional dimensions. This chapter also discusses issues that occur when modeling with multiple explanatory variables, including multicollinearity, feature selection, and dimension reduction.

Chapter 8, Discrete Models, introduces the concept of classification and develops a model for classifying variables into discrete levels of a categorical response variable. This chapter starts by developing the model binary classification and then extends the model to multivariate classification. Finally, the Poisson model and negative binomial models are covered.

Chapter 9, Discriminant Analysis, discusses several additional models for classification, including linear discriminant analysis and quadratic discriminant analysis. This chapter also introduces Bayes’ Theorem.

Chapter 10, Introduction to Time Series, introduces time series data, discussing the time series concept of autocorrelation and the statistical measures for time series. This chapter also introduces the white noise model and stationarity.

Chapter 11, ARIMA Models, discusses models for univariate models. This chapter starts by discussing models for stationary time series and then extends the discussion to non-stationary time series. Finally, this chapter provides a detailed discussion on model evaluation.

Chapter 12, Multivariate Time Series, builds on the previous two chapters by introducing the concept of a multivariate time series and extends ARIMA models to multiple explanatory variables. This chapter also discusses time series cross-correlation.

Chapter 13, Survival Analysis, introduces survival data, also called time-to-event data. This chapter discusses the concept of censoring and the impact of censoring survival data. Finally, the chapter discusses the survival function, hazard, and hazard ratio.

Chapter 14, Survival Models, building on the previous chapter, provides an overview of several models for survival data, including the Kaplan-Meier model, the Exponential model, and the Cox Proportional Hazards model.

To get the most out of this book

You will need access to download and install open-source code packages implemented in the Python programming language and accessible through PyPi.org or the Anaconda Python distribution. While a background in statistics is helpful, but not necessary, this book assumes you have a decent background in basic algebra. Each unit of this book is independent of the other units, but the chapters within each unit build upon each other. Thus, we advise you to begin each unit with that unit’s first chapter to understand the content.

If you are using the digital version of this book, we advise you to type the code yourself or access the code from the book’s GitHub repository (a link is available in the next section). Doing so will help you avoid any potential errors related to the copying and pasting of code.

Download the example code files

You can download the example code files for this book from GitHub at https://github.com/PacktPublishing/Building-Statistical-Models-in-Python. If there’s an update to the code, it will be updated in the GitHub repository.

We also have other code bundles from our rich catalog of books and videos available at https://github.com/PacktPublishing/. Check them out!

Conventions used

There are a number of text conventions used throughout this book.

Code in text: Indicates code words in text, database table names, folder names, filenames, file extensions, pathnames, dummy URLs, user input, and Twitter handles. Here is an example: Mount the downloaded WebStorm-10*.dmg disk image file as another disk in your system.

A block of code is set as follows:

A = [3,5,4]

B = [43,41,56,78,54]

permutation_testing(A,B,n_iter=10000)

Any command-line input or output is written as follows:

pip install SomePackage

Bold: Indicates a new term, an important word, or words that you see onscreen. For instance, words in menus or dialog boxes appear in bold. Here is an example: Select System info from the Administration panel.

Tips or important notes

Appear like this.

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Part 1:Introduction to Statistics

This part will cover the statistical concepts that are foundational to statistical modeling.

It includes the following chapters:

Chapter 1, Sampling and Generalization

Chapter 2, Distributions of Data

Chapter 3, Hypothesis Testing

Chapter 4, Parametric Tests

Chapter 5, Non-Parametric Tests

1

Sampling and Generalization

In this chapter, we will describe the concept of populations and sampling from populations, including some common strategies for sampling. The discussion of sampling will lead to a section that will describe generalization. Generalization will be discussed as it relates to using samples to make conclusions about their respective populations. When modeling for statistical inference, it is necessary to ensure that samples can be generalized to populations. We will provide an in-depth overview of this bridge through the subjects in this chapter.

We will cover the following main topics:

Software and environment setup

Population versus sample

Population inference from samples

Sampling strategies – random, systematic, and stratified

Software and environment setup

Python is one of the most popular programming languages for data science and machine learning thanks to the large open source community that has driven the development of these libraries. Python’s ease of use and flexible nature made it a prime candidate in the data science world, where experimentation and iteration are key features of the development cycle. While there are new languages in development for data science applications, such as Julia, Python currently remains the key language for data science due to its wide breadth of open source projects, supporting applications from statistical modeling to deep learning. We have chosen to use Python in this book due to its positioning as an important language for data science and its demand in the job market.

Python is available for all major operating systems: Microsoft Windows, macOS, and Linux. Additionally, the installer and documentation can be found at the official website: https://www.python.org/.

This book is written for Python version 3.8 (or higher). It is recommended that you use whatever recent version of Python that is available. It is not likely that the code found in this book will be compatible with Python 2.7, and most active libraries have already started dropping support for Python 2.7 since official support ended in 2020.

The libraries used in this book can be installed with the Python package manager, pip, which is part of the standard Python library in contemporary versions of Python. More information about pip can be found here: https://docs.python.org/3/installing/index.html. After pip is installed, packages can be installed using pip on the command line. Here is basic usage at a glance:

Install a new package using the latest version:

pip install SomePackage

Install the package with a specific version, version 2.1 in this example:

pip install SomePackage==2.1

A package that is already installed can be upgraded with the --upgrade flag:

pip install SomePackage –upgrade

In general, it is recommended to use Python virtual environments between projects and to keep project dependencies separate from system directories. Python provides a virtual environment utility, venv, which, like pip, is part of the standard library in contemporary versions of Python. Virtual environments allow you to create individual binaries of Python, where each binary of Python has its own set of installed dependencies. Using virtual environments can prevent package version issues and conflict when working on multiple Python projects. Details on setting up and using virtual environments can be found here: https://docs.python.org/3/library/venv.html.

While we recommend the use of Python and Python’s virtual environments for environment setups, a highly recommended alternative is Anaconda. Anaconda is a free (enterprise-ready) analytics-focused distribution of Python by Anaconda Inc. (previously Continuum Analytics). Anaconda distributions come with many of the core data science packages, common IDEs (such as Jupyter and Visual Studio Code), and a graphical user interface for managing environments. Anaconda can be installed using the installer found at the Anaconda website here: https://www.anaconda.com/products/distribution.

Anaconda comes with its own package manager, conda, which can be used to install new packages similarly to pip.

Install a new package using the latest version:

conda install SomePackage

Upgrade a package that is already installed:

conda upgrade SomePackage

Throughout this book, we will make use of several core libraries in the Python data science ecosystem, such as NumPy for array manipulations, pandas for higher-level data manipulations, and matplotlib for data visualization. The package versions used for this book are contained in the following list. Please ensure that the versions installed in your environment are equal to or greater than the versions listed. This will help ensure that the code examples run correctly:

statsmodels 0.13.2

Matplotlib 3.5.2

NumPy 1.23.0

SciPy 1.8.1

scikit-learn 1.1.1

pandas 1.4.3

The packages used for the code in this book are shown here in Figure 1.1. The __version__ method can be used to print the package version in code.

Figure 1.1 – Package versions used in this book

Having set up the technical environment for the book, let’s get into the statistics. In the next sections, we will discuss the concepts of population and sampling. We will demonstrate sampling strategies with code implementations.

Population versus sample

In general, the goal of statistical modeling is to answer a question about a group by making an inference about that group. The group we are making an inference on could be machines in a production factory, people voting in an election, or plants on different plots of land. The entire group, every individual item or entity, is referred to as the population. In most cases, the population of interest is so large that it is not practical or even possible to collect data on every entity in the population. For instance, using the voting example, it would probably not be possible to poll every person that voted in an election. Even if it was possible to reach all the voters for the election of interest, many voters may not consent to polling, which would prevent collection on the entire population. An additional consideration would be the expense of polling such a large group. These factors make it practically impossible to collect population statistics in our example of vote polling. These types of prohibitive factors exist in many cases where we may want to assess a population-level attribute. Fortunately, we do not need to collect data on the entire population of interest. Inferences about a population can be made using a subset of the population. This subset of the population is called a sample. This is the main idea of statistical modeling. A model will be created using a sample and inferences will be made about the population.

In order to make valid inferences about the population of interest using a sample, the sample must be representative of the population of interest, meaning that the sample should contain the variation found in the population. For example, if we were interested in making an inference about plants in a field, it is unlikely that samples from one corner of the field would be sufficient for inferences about the larger population. There would likely be variations in plant characteristics over the entire field. We could think of various reasons why there might be variation. For this example, we will consider some examples from Figure 1.2.

Figure 1.2 – Field of plants

The figure shows that Sample A is near a forest. This sample area may be affected by the presence of the forest; for example, some of the plants in that sample may receive less sunlight than plants in the other sample. Sample B is shown to be in between the main irrigation lines. It’s conceivable that this sample receives more water on average than the other two samples, which may have an effect on the plants in this sample. The final Sample C is near a road. This sample may see other effects that are not seen in Sample A or B.

If samples were only taken from one of those sections, the inferences from those samples would be biased and would not provide valid references about the population. Thus, samples would need to be taken from across the entire field to create a sample that is more likely to be representative of the population of plants. When taking samples from populations, it is critical to ensure the sampling method is robust to possible issues, such as the influence of irrigation and shade in the previous example. Whenever taking a sample from a population, it’s important to identify and mitigate possible influences of bias because biases in data will affect your model and skew your conclusions.

In the next section, various methods for sampling from a dataset will be discussed. An additional consideration is the sample size. The sample size impacts the type of statistical tools we can use, the distributional assumptions that can be made about the sample, and the confidence of inferences and predictions. The impact of sample size will be explored in depth in Chapter 2, Distributions of Data and Chapter 3, Hypothesis Testing.

Population inference from samples

When using a statistical model to make inferential conclusions about a population from a sample subset of that population, the study design must account for similar degrees of uncertainty in its variables as those in the population. This is the variation mentioned earlier in this chapter. To appropriately draw inferential conclusions about a population, any statistical model must be structured around a chance mechanism. Studies structured around these chance mechanisms are called randomized experiments and provide an understanding of both correlation and causation.

Randomized experiments

There are two primary characteristics of a randomized experiment:

Random sampling, colloquially referred to as random selection

Random assignment of treatments, which is the nature of the study

Random sampling

Random sampling (also called random selection) is designed with the intent of creating a sample representative of the overall population so that statistical models generalize the population well enough to assign cause-and-effect outcomes. In order for random sampling to be successful, the population of interest must be well defined. All samples taken from the population must have a chance of being selected. In considering the example of polling voters, all voters must be willing to be polled. Once all voters are entered into a lottery, random sampling can be used to subset voters for modeling. Sampling from only voters who are willing to be polled introduces sampling bias into statistical modeling, which can lead to skewed results. The sampling method in the scenario where only some voters are willing to participate is called self-selection. Any information obtained and modeled from self-selected samples – or any non-random samples – cannot be used for inference.

Random assignment of treatments

The random assignment of treatments refers to two motivators:

The first motivator is to gain an understanding of specific input variables and their influence on the response – for example, understanding whether assigning treatment A to a specific individual may produce more favorable outcomes than a placebo.

The second motivator is to remove the impact of external variables on the outcomes of a study. These external variables, called confounding variables (or confounders), are important to remove as they often prove difficult to control. They may have unpredictable values or even be unknown to the researcher. The consequence of including confounders is that the outcomes of a study may not be replicable, which can be costly. While confounders can influence outcomes, they can also influence input variables, as well as the relationships between those variables.

Referring back to the example in the earlier section, Population versus sample, consider a farmer who decides to start using pesticides on his crops and wants to test two different brands. The farmer knows there are three distinct areas of the land; plot A, plot B, and plot C. To determine the success of the pesticides and prevent damage to the crops, the farmer randomly chooses 60 plants from each plot (this is called stratified random sampling where random sampling is stratified across each plot) for testing. This