This book provides a selection of modern and sophisticated methodologies for the analysis of large and complex univariate and multivariate categorical data. It gives an overview of a substantive and broad collection of topics in the analysis of categorical data, including association, marginal and graphical models, time series and fixed effects models, as well as modern methods of estimation such as regularization, Bayesian estimation and bias reduction methods, along with new simple measures for model interpretability. Methodological innovations and developments are illustrated and explained through real-world applications, together with useful R packages, allowing readers to replicate most of the analyses using the provided code. The applications span a variety of disciplines, including education, psychology, health, economics, and social sciences.
Author(s): Maria Kateri, Irini Moustaki
Series: Statistics for Social and Behavioral Sciences
Publisher: Springer
Year: 2023
Language: English
Pages: 322
City: Cham
Preface
Contents
Contributors
1 Log-Linear and Log-Multiplicative Association Models for Categorical Data
1.1 Introduction
1.2 Preliminaries
1.2.1 Hierarchical Log-linear models
1.3 Association Models for Two-Way Tables
1.3.1 Estimation and Goodness-of-Fit of AMs
1.3.2 Example: Who Takes Which MOOCs
1.4 Graphical Models
1.4.1 Graphs for Log-linear Models
1.4.2 Graphs of the RC(M) Association Model
1.5 High Dimensional Tables
1.5.1 Graphs for High Dimensional Association Models
1.5.2 Algebraic Details and Properties
1.5.3 Pseudo-likelihood Estimation
1.5.4 Connection to IRT Models
1.6 Sampling Properties
1.7 Evaluation and Testing
1.7.1 Composite Likelihood Ratio Test for Overall Fit
1.7.2 Composite Likelihood Model Selection Criteria
1.8 Example
1.8.1 Measures of Item Fit for the DASS Data
1.9 Conclusion/Discussion
Appendix: DASS Data
References
2 Graphical Models for Categorical Data
2.1 Introduction
2.2 Independence and Conditional Independence
2.3 Relevant Graph Theory
2.4 Markov Properties
2.5 Graphical Log-linear Models
2.5.1 Notation
2.5.2 Log-linear Models
2.5.3 Hierarchical Log-linear Models
2.5.4 Graphical Log-linear Models
2.5.5 Fitting Log-linear Models
2.5.6 Example: Infant Survival Data
2.6 Directed Graphical Models
2.6.1 Directed Acyclic Graphs
2.6.2 Directed Markov Properties
2.6.3 Models
2.6.4 Example: Infant Survival Data (Continued)
2.7 Graphical Chain Models
2.7.1 Chain Graphs
2.7.2 Chain Graph Markov Properties
2.7.3 Models
2.7.4 Example: Infant Survival Data (Continued)
2.8 Further Reading
References
3 Marginal Models: An Overview
3.1 Introduction
3.2 Motivation
3.2.1 Repeated Measurements and Panel Studies
3.2.2 Missing Data and Data Fusion
3.2.3 Graphical Modelling
3.3 Parameterizations of Discrete Probability Distributions
3.3.1 Parameters and Parameterizations
3.3.2 Variation Independence
3.4 Marginal Log-linear Parameterizations
3.4.1 Definition
3.4.2 Basic Properties
3.4.3 Smoothness of Marginal Log-linear Parameters
3.4.4 Collapsibility
3.5 Marginal Log-linear Models
3.6 Alternative Parameterizations of Marginal Log-linear Models
3.7 Marginal Log-linear Parameterization of Conditional Independence Models
3.8 Estimation and Testing
3.8.1 Matrix Formulation of Marginal Models
3.8.2 Characterization of ML Estimators
3.8.3 Likelihood Ratio Tests and Asymptotic Distribution of ML Estimators
3.8.4 Algorithms for Finding ML Estimators
3.8.4.1 Lagrangian Methods
3.8.4.2 Fisher Scoring
3.8.4.3 Software
3.8.5 The GEE Method
3.8.5.1 Remarks on the GEE Method
3.9 Areas of Application
3.9.1 Directed Graphical Models
3.9.2 Path Models
3.9.3 Latent Variable Models
3.9.4 Further Applications and Extensions
References
4 Bayesian Inference for Multivariate Categorical Data
4.1 Introduction
4.1.1 Contingency Tables
4.1.2 Log-Linear Models
4.1.3 Hierarchical, Graphical, and Decomposable Log-Linear Models
4.2 Bayesian Inference for Contingency Tables
4.2.1 Distributions Based on the Normal Distribution
4.2.2 Distributions Based on the Dirichlet Distribution
4.2.2.1 Conditional Dirichlet Distribution
4.2.3 Hyper-Dirichlet Distribution
4.2.4 Relationship Between Conditional Dirichlet and Hyper-Dirichlet Distributions
4.3 Posterior Inference
4.3.1 Example 1
4.3.2 Example 2
4.3.3 Convergence of Gibbs Sampler
4.4 Model Determination and Model Averaging
4.4.1 Bayesian Inference Under Model Uncertainty
4.4.2 Computation Under Model Uncertainty
4.4.3 Laplace's Method
4.4.4 Evaluation of Laplace's Method for the Conditional Dirichlet
4.4.5 Bridge Sampling
4.4.6 Numerical Examples
4.4.7 Example: Risk Factors for Coronary Heart Disease
4.5 Further Examples
4.5.1 Example 1: Lymphoma and Chemotherapy
4.5.2 Example 2: Toxaemia in Pregnancy
4.6 Summary
References
5 Simple Ways to Interpret Effects in Modeling Binary Data
5.1 Introduction
5.2 Alternative Models for Binary Data
5.2.1 Identity and Log Link Models for Binary Data
5.2.2 Example: Models for Italian Survey Data
5.3 Alternative Effect Measures for Explanatory Variables
5.3.1 Probability Effect Measures
5.3.2 A Probability Summary for Ordered Comparison of Groups
5.3.3 Example: Measures for Italian Survey Data
5.4 Generalized Additive Model for Binary Data
5.4.1 Example: GAM for Horseshoe Crab Study
5.5 Discussion and Future Research
Appendix
References
6 Mean and Median Bias Reduction: A Concise Review and Application to Adjacent-Categories Logit Models
6.1 Overview
6.2 Boundary Estimates in Categorical Response Models
6.3 Mean Bias Reduction
6.4 Median Bias Reduction
6.5 Inference with Mean and Median Bias Reduction
6.6 Bias Reduction and Parameter Transformation
6.6.1 Maximum Likelihood Estimation and General Parameter Transformations
6.6.2 Mean Bias Reduction and Linear Parameter Transformations
6.6.3 Median Bias Reduction and Component-Wise Parameter Transformations
6.6.4 Mean Bias Reduction and General Parameter Transformations
6.7 Adjacent-Categories Logit Models
6.7.1 Proportional and Non-proportional Odds Models
6.7.2 Equivalence with Baseline-Category Logit Models
6.7.3 Maximum Likelihood Estimation
6.7.4 Exponential Families
6.7.5 Infinite Maximum Likelihood Estimates
6.8 Mean and Median Bias Reduction for ACL Models
6.9 Mean Bias Reduction of Ordinal Superiority Summaries
6.10 Supplementary Material
References
7 Regularization and Predictor Selection for Ordinal and Categorical Data
7.1 Introduction
7.2 Regularization for Ordinal Covariates
7.2.1 Quadratic Smoothing Penalties for Ordinal Predictors
7.2.1.1 Basic Ideas
7.2.1.2 Further Statistical Inference in Generalized Additive Models
7.2.2 Smoothing and Selection
7.2.2.1 Group Lasso and Similar Approaches
7.2.2.2 Forward/Backward Selection in Generalized Additive Models
7.2.3 Level Fusion
7.2.3.1 Fused Lasso and Similar Approaches
7.2.3.2 Stepwise Selection After Recoding
7.3 Numerical Experiments: L1-Regularization vs. Forward Selection
7.3.1 Smoothing and Selection of Covariates
7.3.1.1 Simulation Setup
7.3.1.2 Results and Discussion
7.3.2 Level Fusion and Selection
7.3.2.1 Simulation Setup
7.3.2.2 Results
7.4 Case Study: The ICF
7.5 Nominal Predictors and Categorical Response
7.5.1 Fusion Penalties for Nominal Predictors
7.5.1.1 All-Pairs Penalties
7.5.1.2 The SCOPE and Range Penalty and Tree-Structured Approaches
7.5.2 Regularization for Multi-categorical Response Models
7.5.2.1 Ordinal Responses
7.5.2.2 Nominal Response Models
7.6 Concluding Remarks
References
8 An Overview of ARMA-Like Models for Count and Binary Data
8.1 Introduction
8.2 General Overview
8.3 Some Relevant Models
8.3.1 GARMA
8.3.2 M-GARMA
8.3.3 GLARMA
8.3.4 Poisson Autoregression
8.3.5 BARMA
8.3.6 Discussion
8.4 Weak Stationarity
8.4.1 GARMA
8.4.2 M-GARMA
8.4.3 GLARMA
8.5 Strong Stationarity
8.5.1 Strict Stationarity and Ergodicity for the GARMA Model
8.5.1.1 Perturbed Model
8.5.1.2 Unperturbed Model
8.5.2 Strict Stationarity and Ergodicity for Log-Linear Poisson Autoregression
8.6 Inference
8.7 Applications
8.7.1 Number of Deaths from COVID-19
8.7.2 Returns Sign for J&J Stock
8.8 Concluding Remarks
Appendix
Technical Details
Markov Chain Specification
Perturbation Approach
Feller Conditions
Coupling Construction
Assumptions and Results of the Alternative Markov Chain Approach Without Irreducibility
Main Proofs
Proof of Theorem 8.2
Proof of Theorem 8.4
Computational Aspects
References
9 Advances in Maximum Likelihood Estimation of Fixed-Effects Binary Panel Data Models
9.1 Introduction
9.2 Preliminaries
9.3 Binary Choice Panel Data Models
9.4 Fixed-Effects Approach and Incidental Parameter Problem
9.5 Target-Corrected Estimators
9.5.1 Bias Correction of the ML Estimator
9.5.2 Bias Correction of the Score and Likelihood Functions
9.6 Conditional Inference
9.7 Simulation Study
9.7.1 Simulation Design
9.7.2 Simulation Results
9.8 Empirical Application
9.9 Software
9.10 Conclusions
Appendix
References
