STATISTICAL METHODS FOR CATEGORICAL DATA ANALYSIS

Transcription

1 STATISTICAL METHODS FOR CATEGORICAL DATA ANALYSIS Daniel A. Powers Department of Sociology University of Texas at Austin YuXie Department of Sociology University of Michigan ACADEMIC PRESS An Imprint of Elsevier San Diego London Boston New York Sydney Tokyo Toronto

2 Contents PREFACE xiii I Introduction 1.1 Why Categorical Data Analysis? Defining Categorical Variables Dependent and Independent Variables Categorical Dependent Variables Types of Measurement Two Philosophies of Categorical Data The Transformational Approach The Latent Variable Approach An Historical Note Approach of This Book 12 1 A. 1 Organization of the Book 13 Vli

3 VIII CONTENTS 2 Review of Linear Regression Models 2.1 Regression Models Three Conceptualizations of Regression Anatomy of Linear Regression Basics of Statistical Inference Tension between Accuracy and Parsimony Linear Regression Models Revisited Least Squares Estimation Maximum Likelihood Estimation Assumptions for Least Squares Regression Comparisons of Conditional Means Linear Models with Weaker Assumptions Categorical and Continuous Dependent Variables A Working Typology 38 3 Logit and Probit Models for Binary Data 3.1 Introduction to Binary Data The Transformational Approach The Linear Probability Model The Logit Model The Probit Model An Application Using Grouped Data Justification of Logit and Probit Models The Latent Variable Approach Extending the Latent Variable Approach Estimation of Binary Response Models Goodness-of-Fit and Model Selection Hypothesis Testing and Statistical Inference Interpreting Estimates The Odds-Ratio Marginal Effects An Application Using Individual-Level Data Alternative Probability Models The Complementary Log-Log Model Programming Binomial Response Models Summary 85

4 CONTENTS IX 4 Loglinear Models for Contingency Tables 4.1 Contingency Tables Types of Contingency Tables An Example and Notation Independence and the Pearson x 2 Statistic Measures of Association Homogeneous Proportions Relative Risks Odds-Ratios The Invariance Property of Odds-Ratios Estimation and Goodness-of-Fit Simple Models and the Pearson x 2 Statistic Sampling Models and Maximum Likelihood Estimation The Likelihood-Ratio Chi-Squared Statistic Bayesian Information Criterion Models for Two-Way Tables The General Setup Normalization Interpretation of Parameters TopologicalModel Quasi-independence Model Symmetry and Quasi-symmetry Crossings Model Models for Ordinal Variables Linear-by-Linear Association Uniform Association Row-Effect and Column-Effect Models Goodman's RC Model Models for Multiway Tables Three-Way Tables The Saturated Model for Three-Way Tables Collapsibility Classes of Models for Three-Way Tables Analysis of Variation in Association Model Selection Statistical Models for Rates 5.1 Introduction 147

5 CONTENTS 5.2 Log-Rate Models The Role of Exposure Estimating Log-Rate Models Illustration Interpretation Discrete-Time Hazard Models Data Structure Estimation Semipararnetric Rate Models The Piecewise Constant Exponential Model The Cox Model Models for Panel Data Fixed Effects Models for Binary Data Random Effects Models for Binary Data Unobserved Heterogeneity in Event-History Models The Gamma Mixture Model Summary Models for Ordinal Dependent Variables 6.1 Introduction Scoring Methods Integer Scoring Midpoint Scoring Normal Score Transformation Scaling with Additional Information Logit Models for Grouped Data Baseline, Adjacent, and Cumulative Logits Adjacent Category Logit Model Adjacent Category Logit Models and Loglinear Models Ordered Logit and Probit Models Cumulative Logits and Probits The Ordered Logit Model The Ordered Probit Model The Latent Variable Approach Estimation Marginal Effects Summary 222

6 CONTENTS XI 7 Models for Unordered Dependent Variables 7.1 Introduction Multinomial Logit Models Review of the Binary Logit Model General Setup for the Multinomial Logit Model The Standard Multinomial Logit Model Estimation Interpreting Results from Multinomial Logit Models Loglinear Models for Grouped Data Two-Way Tables Three- and Higher-Way Tables The Latent Variable Approach The Conditional Logit Model Interpretation The Mixed Model Specification Issues Independence of Irrelevant Alternatives: The IIA Assumption Sequential Logit Models Summary 252 A The Matrix Approach to Regression A.I Introduction 253 A.2 Matrix Algebra 253 A.2.1 The Matrix Approach to Regression 254 A.2.2 Basic Matrix Operations 255 A.2.3 Numerical Example 259 B Maximum Likelihood Estimation B.I Introduction 261 B.2 Basic Principles 261 B.2.1 Example 1: Binomial Proportion 262 B.2.2 Example 2: Normal Mean and Variance 264 B.2.3 Example 3: Binary Logit Model 266 B.2.4 Example 4: Loglinear Model 272 B.2.5 Iteratively Reweighted Least Squares 275 B.2.6 Generalized Linear Models 277 B.2.7 Minimum x 2 Estimation 281