Chapter
1.5.2 Bayesian Binomial Inference: Beta Prior Distributions
1.5.3 Example: Opinions about Legalized Abortion, Revisited
1.5.4 Other Prior Distributions
1.6 USING R SOFTWARE FOR STATISTICAL INFERENCE ABOUT PROPORTIONS
1.6.1 Reading Data Files and Installing Packages
1.6.2 Using R for Statistical Inference about Proportions
1.6.3 Summary: Choosing an Inference Method
2 Analyzing Contingency Tables
2.1 PROBABILITY STRUCTURE FOR CONTINGENCY TABLES
2.1.1 Joint, Marginal, and Conditional Probabilities
2.1.2 Example: Sensitivity and Specificity
2.1.3 Statistical Independence of Two Categorical Variables
2.1.4 Binomial and Multinomial Sampling
2.2 COMPARING PROPORTIONS IN 2×2 CONTINGENCY TABLES
2.2.1 Difference of Proportions
2.2.2 Example: Aspirin and Incidence of Heart Attacks
2.2.3 Ratio of Proportions (Relative Risk)
2.2.4 Using R for Comparing Proportions in 2×2 Tables
2.3.1 Properties of the Odds Ratio
2.3.2 Example: Odds Ratio for Aspirin Use and Heart Attacks
2.3.3 Inference for Odds Ratios and Log Odds Ratios
2.3.4 Relationship Between Odds Ratio and Relative Risk
2.3.5 Example: The Odds Ratio Applies in Case-Control Studies
2.3.6 Types of Studies: Observational Versus Experimental
2.4 CHI-SQUARED TESTS OF INDEPENDENCE
2.4.1 Pearson Statistic and the Chi-Squared Distribution
2.4.2 Likelihood-Ratio Statistic
2.4.3 Testing Independence in Two-Way Contingency Tables
2.4.4 Example: Gender Gap in Political Party Affiliation
2.4.5 Residuals for Cells in a Contingency Table
2.4.6 Partitioning Chi-Squared Statistics
2.4.7 Limitations of Chi-Squared Tests
2.5 TESTING INDEPENDENCE FOR ORDINAL VARIABLES
2.5.1 Linear Trend Alternative to Independence
2.5.2 Example: Alcohol Use and Infant Malformation
2.5.3 Ordinal Tests Usually Have Greater Power
2.5.5 Trend Tests for r×2 and 2×c and Nominal–Ordinal Tables
2.6 EXACT FREQUENTIST AND BAYESIAN INFERENCE
2.6.1 Fisher’s Exact Test for 2×2 Tables
2.6.2 Example: Fisher’s Tea Tasting Colleague
2.6.3 Conservatism for Actual (Type I Error); Mid -Values
2.6.4 Small-Sample Confidence Intervals for Odds Ratio
2.6.5 Bayesian Estimation for Association Measures
2.6.6 Example: Bayesian Inference in a Small Clinical Trial
2.7 ASSOCIATION IN THREE-WAY TABLES
2.7.2 Example: Death Penalty Verdicts and Race
2.7.4 Conditional and Marginal Odds Ratios
2.7.5 Homogeneous Association
3 Generalized Linear Models
3.1 COMPONENTS OF A GENERALIZED LINEAR MODEL
3.1.4 Ordinary Linear Model: GLM with Normal Random Component
GENERALIZED LINEAR MODELS FOR BINARY DATA
3.2.1 Linear Probability Model
3.2.2 Logistic Regression Model
3.2.3 Example: Snoring and Heart Disease
3.2.4 Using R to Fit Generalized Linear Models for Binary Data
3.2.5 Data Files: Ungrouped or Grouped Binary Data
3.3 GENERALIZED LINEAR MODELS FOR COUNTS AND RATES
3.3.1 Poisson Distribution for Counts
3.3.2 Poisson Loglinear Model
3.3.3 Example: Female Horseshoe Crabs and their Satellites
3.3.4 Overdispersion: Greater Variability than Expected
3.4 STATISTICAL INFERENCE AND MODEL CHECKING
3.4.1 Wald, Likelihood-Ratio, and Score Inference Use the Likelihood Function
3.4.2 Example: Political Ideology and Belief in Evolution
3.4.3 The Deviance of a GLM
3.4.4 Model Comparison Using the Deviance
3.4.5 Residuals Comparing Observations to the Model Fit
3.5 FITTING GENERALIZED LINEAR MODELS
3.5.1 The Fisher Scoring Algorithm Fits GLMs
3.5.2 Bayesian Methods for Generalized Linear Models
3.5.3 GLMs: A Unified Approach to Statistical Analysis
4.1 THE LOGISTIC REGRESSION MODEL
4.1.1 The Logistic Regression Model
4.1.2 Odds Ratio and Linear Approximation Interpretations
4.1.3 Example: Whether a Female Horseshoe Crab Has Satellites
4.1.4 Logistic Regression with Retrospective Studies
4.1.5 Normally Distributed X Implies Logistic Regression for Y
4.2 STATISTICAL INFERENCE FOR LOGISTIC REGRESSION
4.2.1 Confidence Intervals for Effects
4.2.2 Significance Testing
4.2.3 Fitted Values and Confidence Intervals for Probabilities
4.2.4 Why Use a Model to Estimate Probabilities?
4.3 LOGISTIC REGRESSION WITH CATEGORICAL PREDICTORS
4.3.1 Indicator Variables Represent Categories of Predictors
4.3.2 Example: Survey about Marijuana Use
4.3.3 ANOVA-Type Model Representation of Factors
4.3.4 Tests of Conditional Independence and of Homogeneity for Three-Way Contingency Tables
4.4 MULTIPLE LOGISTIC REGRESSION
4.4.1 Example: Horseshoe Crabs with Color and Width Predictors
4.4.2 Model Comparison to Check Whether a Term is Needed
4.4.3 Example: Treating Color as Quantitative or Binary
4.4.4 Allowing Interaction between Explanatory Variables
4.4.5 Effects Depend on Other Explanatory Variables in Model
4.5 SUMMARIZING EFFECTS IN LOGISTIC REGRESSION
4.5.1 Probability-Based Interpretations
4.5.2 Marginal Effects and Their Average
4.5.3 Standardized Interpretations
4.6 SUMMARIZING PREDICTIVE POWER: CLASSIFICATION TABLES, ROC CURVES, AND MULTIPLE CORRELATION
4.6.1 Summarizing Predictive Power: Classification Tables
4.6.2 Summarizing Predictive Power: ROC Curves
4.6.3 Summarizing Predictive Power: Multiple Correlation
5 Building and Applying Logistic Regression Models
5.1 STRATEGIES IN MODEL SELECTION
5.1.1 How Many Explanatory Variables Can the Model Handle?
5.1.2 Example: Horseshoe Crab Satellites Revisited
5.1.3 Stepwise Variable Selection Algorithms
5.1.4 Purposeful Selection of Explanatory Variables
5.1.5 Example: Variable Selection for Horseshoe Crabs
5.1.6 AIC and the Bias/Variance Tradeoff
5.2.1 Goodness of Fit: Model Comparison Using the Deviance
5.2.2 Example: Goodness of Fit for Marijuana Use Survey
5.2.3 Goodness of Fit: Grouped versus Ungrouped Data and Continuous Predictors
5.2.4 Residuals for Logistic Models with Categorical Predictors
5.2.5 Example: Graduate Admissions at University of Florida
5.2.6 Standardized versus Pearson and Deviance Residuals
5.2.7 Influence Diagnostics for Logistic Regression
5.2.8 Example: Heart Disease and Blood Pressure
5.3 INFINITE ESTIMATES IN LOGISTIC REGRESSION
5.3.1 Complete and Quasi-Complete Separation: Perfect Discrimination
5.3.2 Example: Infinite Estimate for Toy Example
5.3.3 Sparse Data and Infinite Effects with Categorical Predictors
5.3.4 Example: Risk Factors for Endometrial Cancer Grade
5.4 BAYESIAN INFERENCE, PENALIZED LIKELIHOOD, AND CONDITIONAL LIKELIHOOD FOR LOGISTIC REGRESSION
5.4.1 Bayesian Modeling: Specification of Prior Distributions
5.4.2 Example: Risk Factors for Endometrial Cancer Revisited
5.4.3 Penalized Likelihood Reduces Bias in Logistic Regression
5.4.4 Example: Risk Factors for Endometrial Cancer Revisited
5.4.5 Conditional Likelihood and Conditional Logistic Regression
5.4.6 Conditional Logistic Regression and Exact Tests for Contingency Tables
5.5 ALTERNATIVE LINK FUNCTIONS: LINEAR PROBABILITY AND PROBIT MODELS
5.5.1 Linear Probability Model
5.5.2 Example: Political Ideology and Belief in Evolution
5.5.3 Probit Model and Normal Latent Variable Model
5.5.4 Example: Snoring and Heart Disease Revisited
5.5.5 Latent Variable Models Imply Binary Regression Models
5.5.6 CDFs and Shapes of Curves for Binary Regression Models
5.6 SAMPLE SIZE AND POWER FOR LOGISTIC REGRESSION
5.6.1 Sample Size for Comparing Two Proportions
5.6.2 Sample Size in Logistic Regression Modeling
5.6.3 Example: Modeling the Probability of Heart Disease
6 Multicategory Logit Models
6.1 BASELINE-CATEGORY LOGIT MODELS FOR NOMINAL RESPONSES
6.1.1 Baseline-Category Logits
6.1.2 Example: What Do Alligators Eat?
6.1.3 Estimating Response Probabilities
6.1.4 Checking Multinomial Model Goodness of Fit
6.1.5 Example: Belief in Afterlife
6.1.6 Discrete Choice Models
6.1.7 Example: Shopping Destination Choice
6.2 CUMULATIVE LOGIT MODELS FOR ORDINAL RESPONSES
6.2.1 Cumulative Logit Models with Proportional Odds
6.2.2 Example: Political Ideology and Political Party Affiliation
6.2.3 Inference about Cumulative Logit Model Parameters
6.2.4 Increased Power for Ordinal Analyses
6.2.5 Example: Happiness and Family Income
6.2.6 Latent Variable Linear Models Imply Cumulative Link Models
6.2.7 Invariance to Choice of Response Categories
6.3 CUMULATIVE LINK MODELS: MODEL CHECKING AND EXTENSIONS
6.3.1 Checking Ordinal Model Goodness of Fit
6.3.2 Cumulative Logit Model without Proportional Odds
6.3.3 Simpler Interpretations Use Probabilities
6.3.4 Example: Modeling Mental Impairment
6.3.5 A Latent Variable Probability Comparison of Groups
6.3.6 Cumulative Probit Model
6.3.7 R2 Based on the Latent Variable Model
6.3.8 Bayesian Inference for Multinomial Models
6.3.9 Example: Modeling Mental Impairment Revisited
6.4 PAIRED-CATEGORY LOGIT MODELING OF ORDINAL RESPONSES
6.4.1 Adjacent-Categories Logits
6.4.2 Example: Political Ideology Revisited
6.4.4 Example: Tonsil Size and Streptococcus
7 Loglinear Models for Contingency Tables and Counts
7.1 LOGLINEAR MODELS FOR COUNTS IN CONTINGENCY TABLES
7.1.1 Loglinear Model of Independence for Two-Way Contingency Tables
7.1.2 Interpretation of Parameters in the Independence Model
7.1.3 Example: Happiness and Belief in Heaven
7.1.4 Saturated Model for Two-Way Contingency Tables
7.1.5 Loglinear Models for Three-Way Contingency Tables
7.1.6 Two-Factor Parameters Describe Conditional Associations
7.1.7 Example: Student Alcohol, Cigarette, and Marijuana Use
7.2 STATISTICAL INFERENCE FOR LOGLINEAR MODELS
7.2.1 Chi-Squared Goodness-of-Fit Tests
7.2.2 Cell Standardized Residuals for Loglinear Models
7.2.3 Significance Tests about Conditional Associations
7.2.4 Confidence Intervals for Conditional Odds Ratios
7.2.5 Bayesian Fitting of Loglinear Models
7.2.6 Loglinear Models for Higher-Dimensional Contingency Tables
7.2.7 Example: Automobile Accidents and Seat Belts
7.2.8 Interpreting Three-Factor Interaction Terms
7.2.9 Statistical Versus Practical Significance: Dissimilarity Index
7.3 THE LOGLINEAR – LOGISTIC MODEL CONNECTION
7.3.1 Using Logistic Models to Interpret Loglinear Models
7.3.2 Example: Auto Accident Data Revisited
7.3.3 Condition for Equivalent Loglinear and Logistic Models
7.3.4 Loglinear/Logistic Model Selection Issues
7.4 INDEPENDENCE GRAPHS AND COLLAPSIBILITY
7.4.1 Independence Graphs
7.4.2 Collapsibility Conditions for Contingency Tables
7.4.3 Example: Loglinear Model Building for Student Substance Use
7.4.4 Collapsibility and Logistic Models
7.5 MODELING ORDINAL ASSOCIATIONS IN CONTINGENCY TABLES
7.5.1 Linear-by-Linear Association Model
7.5.2 Example: Linear-by-Linear Association for Sex Opinions
7.5.3 Ordinal Significance Tests of Independence
7.6 LOGLINEAR MODELING OF COUNT RESPONSE VARIABLES
7.6.1 Count Regression Modeling of Rate Data
7.6.2 Example: Death Rates for Lung Cancer Patients
7.6.3 Negative Binomial Regression Models
7.6.4 Example: Female Horseshoe Crab Satellites Revisited
8 Models for Matched Pairs
8.1 COMPARING DEPENDENT PROPORTIONS FOR BINARY MATCHED PAIRS
8.1.1 McNemar Test Comparing Marginal Proportions
8.1.2 Estimating the Difference between Dependent Proportions
8.2 MARGINAL MODELS AND SUBJECT-SPECIFIC MODELS FOR MATCHED PAIRS
8.2.1 Marginal Models for Marginal Proportions
8.2.2 Example: Environmental Opinions Revisited
8.2.3 Subject-Specific and Population-Averaged Tables
8.2.4 Conditional Logistic Regression for Matched-Pairs
8.2.5 Logistic Regression for Matched Case-Control Studies
8.3 COMPARING PROPORTIONS FOR NOMINAL MATCHED-PAIRS RESPONSES
8.3.1 Marginal Homogeneity for Baseline-Category Logit Models
8.3.2 Example: Coffee Brand Market Share
8.3.3 Using the Cochran–Mantel–Haenszel Test to Test Marginal Homogeneity
8.3.4 Symmetry and Quasi-Symmetry Models for Square Contingency Tables
8.3.5 Example: Coffee Brand Market Share Revisited
8.4 COMPARING PROPORTIONS FOR ORDINAL MATCHED-PAIRS RESPONSES
8.4.1 Marginal Homogeneity and Cumulative Logit Marginal Model
8.4.2 Example: Recycle or Drive Less to Help the Environment?
8.4.3 An Ordinal Quasi-Symmetry Model
8.4.4 Example: Recycle or Drive Less Revisited?
8.5 ANALYZING RATER AGREEMENT
8.5.1 Example: Agreement on Carcinoma Diagnosis
8.5.2 Cell Residuals for Independence Model
8.5.3 Quasi-Independence Model
8.5.4 Quasi Independence and Odds Ratios Summarizing Agreement
8.5.5 Kappa Summary Measure of Agreement
8.6 BRADLEY–TERRY MODEL FOR PAIRED PREFERENCES
8.6.1 The Bradley–Terry Model and Quasi-Symmetry
8.6.2 Example: Ranking Men Tennis Players
9 Marginal Modeling of Correlated, Clustered Responses
9.1 MARGINAL MODELS VERSUS SUBJECT-SPECIFIC MODELS
9.1.1 Marginal Models for a Clustered Binary Response
9.1.2 Example: Repeated Responses on Similar Survey Questions
9.1.3 Subject-Specific Models for a Repeated Response
9.2 MARGINAL MODELING: THE GENERALIZED ESTIMATING EQUATIONS (GEE) APPROACH
9.2.1 Quasi-Likelihood Methods
9.2.2 Generalized Estimating Equation Methodology: Basic Ideas
9.2.3 Example: Opinion about Legalized Abortion Revisited
9.2.4 Limitations of GEE Compared to ML
9.3 MARGINAL MODELING FOR CLUSTERED MULTINOMIAL RESPONSES
9.3.1 Example: Insomnia Study
9.3.2 Alternative GEE Specification of Working Association
9.4 TRANSITIONAL MODELING, GIVEN THE PAST
9.4.1 Transitional Models with Explanatory Variables
9.4.2 Example: Respiratory Illness and Maternal Smoking
9.4.3 Group Comparisons Treating Initial Response as a Covariate
9.5 DEALING WITH MISSING DATA
9.5.1 Missing at Random: Impact on ML and GEE Methods
9.5.2 Multiple Imputation: Monte Carlo Prediction of Missing Data
10 Random Effects: Generalized Linear Mixed Models
10.1 RANDOM EFFECTS MODELING OF CLUSTERED CATEGORICAL DATA
10.1.1 The Generalized Linear Mixed Model (GLMM)
10.1.2 A Logistic GLMM for Binary Matched Pairs
10.1.3 Example: Environmental Opinions Revisited
10.1.4 Differing Effects in GLMMs and Marginal Models
10.1.5 Model Fitting for GLMMs
10.1.6 Inference for Model Parameters and Prediction
10.2 EXAMPLES: RANDOM EFFECTS MODELS FOR BINARY DATA
10.2.1 Small-Area Estimation of Binomial Probabilities
10.2.2 Example: Estimating Basketball Free Throw Success
10.2.3 Example: Opinions about Legalized Abortion Revisited
10.2.4 Item Response Models: The Rasch Model
10.2.5 Choice of Marginal Model or Random Effects Model
10.3 EXTENSIONS TO MULTINOMIAL RESPONSES AND MULTIPLE RANDOM EFFECT TERMS
10.3.1 Example: Insomnia Study Revisited
10.3.2 Meta-Analysis: Bivariate Random Effects for Association Heterogeneity
10.4 MULTILEVEL (HIERARCHICAL) MODELS
10.4.1 Example: Two-Level Model for Student Performance
10.4.2 Example: Smoking Prevention and Cessation Study
10.5.1 Independence Given a Latent Categorical Variable
10.5.2 Example: Latent Class Model for Rater Agreement
11 Classification and Smoothing
11.1 CLASSIFICATION: LINEAR DISCRIMINANT ANALYSIS
11.1.1 Classification with Fisher’s Linear Discriminant Function
11.1.2 Example: Horseshoe Crab Satellites Revisited
11.1.3 Discriminant Analysis Versus Logistic Regression
11.2 CLASSIFICATION: TREE-BASED PREDICTION
11.2.1 Classification Trees
11.2.2 Example: A Classification Tree for Horseshoe Crab Mating
11.2.3 How Does the Classification Tree Grow?
11.2.4 Pruning a Tree and Checking Prediction Accuracy
11.2.5 Classification Trees Versus Logistic Regression and Discriminant Analysis
11.3 CLUSTER ANALYSIS FOR CATEGORICAL RESPONSES
11.3.1 Measuring Dissimilarity Between Observations
11.3.2 Hierarchical Clustering Algorithm and Dendrograms
11.3.3 Example: Clustering States on Presidential Elections
11.4 SMOOTHING: GENERALIZED ADDITIVE MODELS
11.4.1 Generalized Additive Models
11.4.2 Example: GAMs for Horseshoe Crab Data
11.4.3 How Much Smoothing? The Bias/Variance Tradeoff
11.4.4 Example: Smoothing to Portray Probability of Kyphosis
11.5 REGULARIZATION FOR HIGH-DIMENSIONAL CATEGORICAL DATA (LARGE p)
11.5.1 Penalized-Likelihood Methods and Lq-Norm Smoothing
11.5.2 Implementing the Lasso
11.5.3 Example: Predicting Opinion on Abortion with Student Survey
11.5.4 Why Shrink ML Estimates Toward 0?
11.5.5 Issues in Variable Selection (Dimension Reduction)
11.5.6 Controlling the False Discovery Rate
11.5.7 Large p also Makes Bayesian Inference Challenging
12 A Historical Tour of Categorical Data Analysis
The Pearson–Yule Association Controversy
R.A. Fisher’s Contributions
Multiway Contingency Tables and Loglinear Models
Appendix: Software for Categorical Data Analysis
A.1 R FOR CATEGORICAL DATA ANALYSIS
A.2 SAS FOR CATEGORICAL DATA ANALYSIS
Chapters 1–2: Introduction and Contingency Tables
Chapters 3–5: Generalized Linear Models and Logistic Regression
Chapters 6–7: Multicategory Logit Models and Loglinear Models
Chapters 9–10: Marginal Models and Random Effects Models (GLMMs)
Chapter 11: Non-Model-Based Classification and Clustering
A.3 STATA FOR CATEGORICAL DATA ANALYSIS
Chapters 1–2: Introduction and Contingency Tables
Chapters 3–5: Generalized Linear Models and Logistic Regression
Chapters 6–7: Multicategory Logit Models and Loglinear Models
Chapters 8–11: Correlated Observations, Advanced Methods
A.4 SPSS FOR CATEGORICAL DATA ANALYSIS
Chapters 1–2: Introduction and Contingency Tables
Chapters 3–5: Generalized Linear Models and Logistic Regression
Chapters 6–7: Multicategory Logit Models and Loglinear Models
Chapters 8–11: Correlated Observations, Advanced Methods
Brief Solutions to Odd-Numbered Exercises
An Introduction to Categorical Data Analysis
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