Tutz G. Regression for Categorical Data

Cambridge University Press, 2012. — x+562 p. — (Cambridge Series in Statistical and Probabilistic Mathematics). — ISBN: 9781107009653.This book introduces basic and advanced concepts of categorical regression with a focus on the structuring constituents of regression, including regularization techniques to structure predictors. In addition to standard methods such as the logit and probit model and extensions to multivariate settings, the author presents more recent developments in flexible and high-dimensional regression, which allow weakening of assumptions on the structuring of the predictor and yield fits that are closer to the data. A generalized linear model is used as a unifying framework whenever possible in particular parametric models that are treated within this framework. Many topics not normally included in books on categorical data analysis are treated here, such as nonparametric regression; selection of predictors by regularized estimation procedures; ternative models like the hurdle model and zero-inflated regression models for count data; and non-standard tree-based ensemble methods. The book is accompanied by an R package that contains data sets and code for all the examples.Categorical Data: Examples and Basic Concepts
Organization of This Book
Basic Components of Structured Regression
Classical Linear Regression
ExercisesBinary Regression: The Logit Model
Distribution Models for Binary Responses and Basic Concepts
Linking Response and Explanatory Variables
The Logit Model
The Origins of the Logistic Function and the Logit Model
ExercisesGeneralized Linear Models
Basic Structure
Generalized Linear Models for Continuous Responses
GLMs for Discrete Responses
Further Concepts
Modeling of Grouped Data
Maximum Likelihood Estimation
Inference
Goodness-of-Fit for Grouped Observations
Computation of Maximum Likelihood Estimates
Hat Matrix for Generalized Linear Models
Quasi-Likelihood Modeling
Further Reading
ExercisesModeling of Binary Data
Maximum Likelihood Estimation
Discrepancy between Data and Fit
Diagnostic Checks
Structuring the Linear Predictor
Comparing Non-Nested Models
Explanatory Value of Covariates
Further Reading
ExercisesAlternative Binary Regression Models
Alternative Links in Binary Regression
The Missing Link
Overdispersion
Conditional Likelihood
Further Reading
ExercisesRegularization and Variable Selection for Parametric Models
Classical Subset Selection
Regularization by Penalization
Boosting Methods
Simultaneous Selection of Link Function and Predictors
Categorical Predictors
Bayesian Approach
Further Reading
ExercisesRegression Analysis of Count Data
The Poisson Distribution
Poisson Regression Model
Inference for the Poisson Regression Model
Poisson Regression with an Offset
Poisson Regression with Overdispersion
Negative Binomial Model and Alternatives
Zero-Inflated Counts
Hurdle Models
Further Reading
ExercisesMultinomial Response Models
The Multinomial Distribution
The Multinomial Logit Model
Multinomial Model as Random Utility Model
Structuring the Predictor
Logit Model as Multivariate Generalized Linear Model
Inference for Multicategorical Response Models
Multinomial Models with Hierarchically Structured Response
Discrete Choice Models
Nested Logit Model
Regularization for the Multinomial Model
Further Reading
ExercisesOrdinal Response Models
Cumulative Models
Sequential Models
Further Properties and Comparison of Models
Alternative Models
Inference for Ordinal Models
Further Reading
ExercisesSemi- and Non-Parametric Generalized Regression
Univariate Generalized Non-Parametric Regression
Non-Parametric Regression with Multiple Covariates
Structured Additive Regression
Functional Data and Signal Regression
Further Reading
ExercisesTree-Based Methods
Regression and Classification Trees
Multivariate Adaptive Regression Splines
Further Reading
ExercisesThe Analysis of Contingency Tables: Log-Linear and Graphical Models
Types of Contingency Tables
Log-Linear Models for Two-Way Tables
Log-Linear Models for Three-Way Tables
Specific Log-Linear Models
Log-Linear and Graphical Models for Higher Dimensions
Collapsibility
Log-Linear Models and the Logit Model
Inference for Log-Linear Models
Model Selection and Regularization
Mosaic Plots
Further Reading
ExercisesMultivariate Response Models
Conditional Modeling
Marginal Parametrization and Generalized Log-Linear Models
General Marginal Models: Association as Nuisance and GEEs
Marginal Homogeneity
Further Reading
ExercisesRandom Effects Models and Finite Mixtures
Linear Random Effects Models for Gaussian Data
Generalized Linear Mixed Models
Estimation Methods for Generalized Mixed Models
Multicategorical Response Models
The Marginalized Random Effects Model
Latent Trait Models and Conditional ML
Semiparametric Mixed Models
Finite Mixture Models
Further Reading
ExercisesPrediction and Classification
Basic Concepts of Prediction
Methods for Optimal Classification
Basics of Estimated Classification Rules
Parametric Classification Methods
Non-Parametric Methods
Neural Networks
Examples
Variable Selection in Classification
Prediction of Ordinal Outcomes
Model-Based Prediction
Further Reading
ExercisesDistributions
Discrete Distributions
Continuous Distributions
Some Basic Tools
Linear Algebra
Taylor Approximation
Conditional Expectation, Distribution
EM Algorithm
Constrained Estimation
Simplification of Penalties
Linear Constraints
Fisher Scoring with Penalty Term
Kullback-Leibler Distance and Information-Based Criteria of Model Fit
Kullback-Leibler Distance
Numerical Integration and Tools for Random Effects Modeling
Laplace Approximation
Gauss-Hermite Integration
Inversion of Pseudo-Fisher MatrixList of Examples

Author Index
Subject Index