# Logistic Regression Models

Logistic Regression Models, by Joseph Hilbe, arose from Hilbe’s course in logistic regression at statistics.com. The book includes many Stata examples using both official and user-written commands and includes Stata output and graphs.

Hilbe begins with simple contingency tables and covers fitting algorithms, parameter interpretation, and diagnostics. The later chapters include models for overdispersion, complex response variables, longitudinal data, and survey data. The final chapter describes exact logistic regression, available in Stata 10 with the new exlogistic command. Hilbe does not oversimplify controversial issues, like interactions and standardized coefficients.

The prerequisite for most of the book is a working knowledge of multiple regression, but some sections use multivariate calculus and matrix algebra.

Hilbe is coauthor (with James Hardin) of the popular Stata Press book Generalized Linear Models and Extensions. He also wrote the first versions of Stata’s logistic and glm commands.

The fourth printing has been revised: examples in the book now use Stata version 11 code in place of earlier version code, where applicable.

Preface

Chapter 1 Introduction
1.1 The Normal Model
1.2 Foundation of the Binomial Model
1.3 Historical and Software Considerations
1.4 Chapter Profiles

Chapter 2 Concepts Related to the Logistic Model
2.1 2 × 2 Table Logistic Model
2.2 2 × k Table Logistic Model
2.3 Modeling a Quantitative Predictor
2.4 Logistic Modeling Designs

2.4.1 Experimental Studies
2.4.2 Observational Studies

2.4.2.1 Prospective or Cohort Studies
2.4.2.2 Retrospective or Case–Control Studies
2.4.2.3 Comparisons

Exercises
R Code

Chapter 3 Estimation Methods
3.1 Derivation of the IRLS Algorithm
3.2 IRLS Estimation
3.3 Maximum Likelihood Estimation
Exercises
R Code

Chapter 4 Derivation of the Binary Logistic Algorithm
4.1 Terms of the Algorithm
4.2 Logistic GLM and ML Algorithms
4.3 Other Bernoulli Models
Exercises
R Code

Chapter 5 Model Development

5.1 Building a Logistic Model

5.1.1 Interpretations
5.1.2 Full Model
5.1.3 Reduced Model

5.2 Assessing Model Fit: Link Specification

5.2.1 Box–Tidwell Test
5.2.3 Test by Partial Residuals
5.2.4 Linearity of Slopes Test
5.2.6 Fractional Polynomials

5.3 Standardized Coefficients
5.4 Standard Errors

5.4.1 Calculating Standard Errors
5.4.2 The z-Statistic
5.4.3 p-Values
5.4.4 Confidence Intervals
5.4.5 Confidence Intervals of Odds Ratios

5.5 Odds Ratios as Approximations of Risk Ratios

5.5.1 Epidemiological Terms and Studies
5.5.2 Odds Ratios, Risk Ratios, and Risk Models
5.5.3 Calculating Standard Errors and Confidence Intervals
5.5.4 Risk Difference and Attributable Risk
5.5.5 Other Resources on Odds Ratios and Risk Ratios

5.6 Scaling of Standard Errors
5.7 Robust Variance Estimators
5.8 Bootstrapped and Jackknifed Standard Errors
5.9 Stepwise Methods
5.10 Handling Missing Values
5.11 Modeling an Uncertain Response
5.12 Constraining Coefficients
Exercises
R Code

Chapter 6 Interactions
6.1 Introduction
6.2 Binary × Binary Interactions

6.2.1 Interpretation—as Odds Ratio
6.2.2 Standard Errors and Confidence Intervals
6.2.3 Graphical Analysis

6.3 Binary × Categorical Interactions
6.4 Binary × Continuous Interactions

6.4.1 Notes on Centering
6.4.2 Constructing and Interpreting the Interaction
6.4.3 Interpretation
6.4.4 Standard Errors and Confidence Intervals
6.4.5 Significance of Interaction
6.4.6 Graphical Analysis

6.5 Categorical × Continuous Interactions

6.5.1 Interpretation
6.5.2 Standard Errors and Confidence Intervals
6.5.3 Graphical Representation

6.6.1 Binary × Binary
6.6.2 Continuous × Binary
6.6.3 Continuous × Continuous

Exercises
R Code

Chapter 7 Analysis of Model Fit

7.1 Traditional Fit Tests for Logistic Regression

7.1.1 R2 and Pseudo-R2 Statistics
7.1.2 Deviance Statistic
7.1.3 Likelihood Ratio Test

7.2 Hosmer–Lemeshow GOF Test

7.2.1 Hosmer–Lemeshow GOF Test
7.2.2 Classification Matrix
7.2.3 ROC Analysis

7.3 Information Criteria Tests

7.3.1 Akaike Information Criterion—AIC
7.3.2 Finite Sample AIC Statistic
7.3.3 LIMDEP AIC
7.3.4 SWARTZ AIC
7.3.5 Bayesian Information Criterion (BIC)
7.3.6 HQIC Goodness-of-Fit Statistic
7.3.7 A Unified AIC Fit Statistic

7.4 Residual Analysis

7.4.1 GLM-Based Residuals

7.4.1.1 Raw Residual
7.4.1.2 Pearson Residual
7.4.1.3 Deviance Residual
7.4.1.4 Standardized Pearson Residual
7.4.1.5 Standardized Deviance Residual
7.4.1.6 Likelihood Residuals
7.4.1.7 Anscombe Residuals

7.4.2 m-Asymptotic Residuals

7.4.2.1 Hat Matrix Diagonal Revisited
7.4.2.2 Other Influence Residuals

7.4.3 Conditional Effects Plot

7.5 Validation Models
Exercises
R Code

Chapter 8 Binomial Logistic Regression
Exercises
R Code

Chapter 9 Overdispersion
9.1 Introduction
9.2 The Nature and Scope of Overdispersion
9.3 Binomial Overdispersion

9.3.1 Apparent Overdispersion

9.3.1.1 Simulated Model Setup
9.3.1.2 Missing Predictor
9.3.1.3 Needed Interaction
9.3.1.4 Predictor Transformation
9.3.1.6 Existing Outlier(s)

9.3.2 Relationship: Binomial and Poisson

9.4 Binary Overdispersion

9.4.1 The Meaning of Binary Model Overdispersion
9.4.2 Implicit Overdispersion

9.5 Real Overdispersion

9.5.1 Methods of Handling Real Overdispersion
9.5.2 Williams’ Procedure
9.5.3 Generalized Binomial Regression

9.6 Concluding Remarks
Exercises
R Code

Chapter 10 Ordered Logistic Regression
10.1 Introduction
10.2 The Proportional Odds Model
10.3 Generalized Ordinal Logistic Regression
10.4 Partial Proportional Odds
Exercises
R Code

Chapter 11 Multinomial Logistic Regression

11.1 Unordered Logistic Regression

11.1.1 The Multinomial Distribution
11.1.2 Interpretation of the Multinomial Model

11.2 Independence of Irrelevant Alternatives
11.3 Comparison to Multinomial Probit
Exercises
R Code

Chapter 12 Alternative Categorical Response Models
12.1 Introduction
12.2 Continuation Ratio Models
12.3 Stereotype Logistic Model
12.4 Heterogeneous Choice Logistic Model
12.6 Proportional Slopes Models

12.6.1 Proportional Slopes Comparative Algorithms
12.6.2 Modeling Synthetic Data
12.6.3 Tests of Proportionality

Exercises

Chapter 13 Panel Models
13.1 Introduction
13.2 Generalized Estimating Equations

13.2.1 GEE: Overview of GEE Theory
13.2.2 GEE Correlation Structures

13.2.2.1 Independence Correlation Structure Schematic
13.2.2.2 Exchangeable Correlation Structure Schematic
13.2.2.3 Autoregressive Correlation Structure Schematic
13.2.2.4 Unstructured Correlation Structure Schematic
13.2.2.5 Stationary or m-Dependent Correlation Structure Schematic
13.2.2.6 Nonstationary Correlation Structure Schematic

13.2.3 GEE Binomial Logistic Models
13.2.4 GEE Fit Analysis—QIC

13.2.4.1 QIC/QICu Summary–Binary Logistic Regression

13.2.5 Alternating Logistic Regression
13.2.6 Quasi-Least Squares Regression
13.2.7 Feasibility

13.3 Unconditional Fixed Effects Logistic Model
13.4 Conditional Logistic Models

13.4.1 Conditional Fixed Effects Logistic Models
13.4.2 Matched Case–Control Logistic Model
13.4.3 Rank-Ordered Logistic Regression

13.5 Random Effects and Mixed Models Logistic Regression

13.5.1 Random Effects and Mixed Models: Binary Response
13.5.2 Alternative AIC-Type Statistics for Panel Data
13.5.3 Random-Intercept Proportional Odds

Exercises
R Code

Chapter 14 Other Types of Logistic-Based Models

14.1 Survey Logistic Models

14.1.1 Interpretation

14.2 Scobit-Skewed Logistic Regression
14.3 Discriminant Analysis

14.3.1 Dichotomous Discriminant Analysis
14.3.2 Canonical Linear Discriminant Analysis
14.3.3 Linear Logistic Discriminant Analysis

Exercises

Chapter 15 Exact Logistic Regression
15.1 Exact Methods
15.2 Alternative Modeling Methods

15.2.1 Monte Carlo Sampling Methods
15.2.2 Median Unbiased Estimation
15.2.3 Penalized Logistic Regression

Exercises

Conclusion

Appendix A: Brief Guide to Using Stata Commands

Appendix B: Stata and R Logistic Models

Appendix C: Greek Letters and Major Functions

Appendix D: Stata Binary Logistic Command

Appendix E: Derivation of the Beta Binomial

Appendix F: Likelihood Function of the Adaptive Gauss–Hermite Quadrature Method of Estimation

Appendix G: Data Sets

Appendix H: Marginal Effects and Discrete Change

References
Author Index
Subject Index Author: Joseph M. Hilbe
ISBN-13: 978-1-4200-7575-5