Chapter
1.4 Statistical Inference for Binomial Parameters
1.4.1 Tests About a Binomial Parameter
1.4.2 Confidence Intervals for a Binomial Parameter
1.4.3 Example: Estimating the Proportion of Vegetarians
1.4.4 Exact Small-Sample Inference and the Mid P- Value
1.5 Statistical Inference for Multinomial Parameters
1.5.1 Estimation of Multinomial Parameters
1.5.2 Pearson Chi-Squared Test of a Specified Multinomial
1.5.3 Likelihood-Ratio Chi-Squared Test of a Specified Multinomial
1.5.4 Example: Testing Mendel's Theories
1.5.5 Testing with Estimated Expected Frequencies
1.5.6 Example: Pneumonia Infections in Calves
1.5.7 Chi-Squared Theoretical Justification
1.6 Bayesian Inference for Binomial and Multinomial Parameters
1.6.1 The Bayesian Approach to Statistical Inference
1.6.2 Binomial Estimation: Beta and Logit-Normal Prior Distributions
1.6.3 Multinomial Estimation: Dirichlet Prior Distributions
1.6.4 Example: Estimating Vegetarianism Revisited
1.6.5 Binomial and Multinomial Estimation: Improper Priors
2 Describing Contingency Tables
2.1 Probability Structure for Contingency Tables
2.1.2 Joint/Marginal/Conditional Distributions for Contingency Tables
2.1.3 Example: Sensitivity and Specificity for Medical Diagnoses
2.1.4 Independence of Categorical Variables
2.1.5 Poisson, Binomial, and Multinomial Sampling
2.1.6 Example: Seat Belts and Auto Accident Injuries
2.1.7 Example: Case–Control Study of Cancer and Smoking
2.1.8 Types of Studies: Observational Versus Experimental
2.2 Comparing Two Proportions
2.2.1 Difference of Proportions
2.2.4 Properties of the Odds Ratio
2.2.5 Example: Association Between Heart Attacks and Aspirin Use
2.2.6 Case–Control Studies and the Odds Ratio
2.2.7 Relationship Between Odds Ratio and Relative Risk
2.3 Conditional Association in Stratified 2 × 2 Tables
2.3.2 Example: Racial Characteristics and the Death Penalty
2.3.3 Conditional and Marginal Odds Ratios
2.3.4 Marginal Independence Versus Conditional Independence
2.3.5 Homogeneous Association
2.3.6 Collapsibility: Identical Conditional and Marginal Associations
2.4 Measuring Association in I × J Tables
2.4.1 Odds Ratios in I x J Tables
2.4.2 Association Factors
2.4.3 Summary Measures of Association
2.4.4 Ordinal Trends: Concordant and Discordant Pairs
2.4.5 Ordinal Measure of Association: Gamma
2.4.6 Probabilistic Comparisons of Two Ordinal Distributions
2.4.7 Example: Comparing Pain Ratings After Surgery
2.4.8 Correlation for Underlying Normality
3 Inference for Two-Way Contingency Tables
3.1 Confidence Intervals for Association Parameters
3.1.1 Interval Estimation of the Odds Ratio
3.1.2 Example: Seat-Belt Use and Traffic Deaths
3.1.3 Interval Estimation of Difference of Proportions and Relative Risk
3.1.4 Example: Aspirin and Heart Attacks Revisited
3.1.5 Deriving Standard Errors with the Delta Method
3.1.6 Delta Method Applied to the Sample Logit
3.1.7 Delta Method for the Log Odds Ratio
3.1.8 Simultaneous Confidence Intervals for Multiple Comparisons
3.2 Testing Independence in Two-way Contingency Tables
3.2.1 Pearson and Likelihood-Ratio Chi-Squared Tests
3.2.2 Example: Education and Belief in God
3.2.3 Adequacy of Chi-Squared Approximations
3.2.4 Chi-Squared and Comparing Proportions in 2 x 2 Tables
3.2.5 Score Confidence Intervals Comparing Proportions
3.2.6 Profile Likelihood Confidence Intervals
3.3 Following-up Chi-Squared Tests
3.3.1 Pearson Residuals and Standardized Residuals
3.3.2 Example: Education and Belief in God Revisited
3.3.3 Partitioning Chi-Squared
3.3.4 Example: Origin of Schizophrenia
3.3.5 Rules for Partitioning
3.3.6 Summarizing the Association
3.3.7 Limitations of Chi-Squared Tests
3.3.8 Why Consider Independence If It's Unlikely to Be True?
3.4 Two-Way Tables with Ordered Classifications
3.4.1 Linear Trend Alternative to Independence
3.4.2 Example: Is Happiness Associated with Political Ideology?
3.4.3 Monotone Trend Alternatives to Independence
3.4.4 Extra Power with Ordinal Tests
3.4.5 Sensitivity to Choice of Scores
3.4.6 Example: Infant Birth Defects by Maternal Alcohol Consumption
3.4.7 Trend Tests for I x 2 and 2 x J Tables
3.4.8 Nominal-Ordinal Tables
3.5 Small-Sample Inference for Contingency Tables
3.5.1 Fisher's Exact Test for 2 x 2 Tables
3.5.2 Example: Fisher's Tea Drinker
3.5.3 Two-Sided P-Values for Fisher's Exact Test
3.5.4 Confidence Intervals Based on Conditional Likelihood
3.5.5 Discreteness and Conservatism Issues
3.5.6 Small-Sample Unconditional Tests of Independence
3.5.7 Conditional Versus Unconditional Tests
3.6 Bayesian Inference for Two-way Contingency Tables
3.6.1 Prior Distributions for Comparing Proportions in 2 x 2 Tables
3.6.2 Posterior Probabilities Comparing Proportions
3.6.3 Posterior Intervals for Association Parameters
3.6.4 Example: Urn Sampling Gives Highly Unbalanced Treatment Allocation
3.6.5 Highest Posterior Density Intervals
3.6.6 Testing Independence
3.6.7 Empirical Bayes and Hierarchical Bayesian Approaches
3.7 Extensions for Multiway Tables and Nontabulated Responses
3.7.1 Categorical Data Need Not Be Contingency Tables
4 Introduction to Generalized Linear Models
4.1 The Generalized Linear Model
4.1.1 Components of Generalized Linear Models
4.1.2 Binomial Logit Models for Binary Data
4.1.3 Poisson Loglinear Models for Count Data
4.1.4 Generalized Linear Models for Continuous Responses
4.1.6 Advantages of GLMs Versus Transforming the Data
4.2 Generalized Linear Models for Binary Data
4.2.1 Linear Probability Model
4.2.2 Example: Snoring and Heart Disease
4.2.3 Logistic Regression Model
4.2.4 Binomial GLM for 2 x 2 Contingency Tables
4.2.5 Probit and Inverse cdf Link Functions
4.2.6 Latent Tolerance Motivation for Binary Response Models
4.3 Generalized Linear Models for Counts and Rates
4.3.1 Poisson Loglinear Models
4.3.2 Example: Horseshoe Crab Mating
4.3.3 Overdispersion for Poisson GLMs
4.3.4 Negative Binomial GLMs
4.3.5 Poisson Regression for Rates Using Offsets
4.3.6 Example: Modeling Death Rates for Heart Valve Operations
4.3.7 Poisson GLM of Independence in Two-Way Contingency Tables
4.4 Moments and Likelihood for Generalized Linear Models
4.4.1 The Exponential Dispersion Family
4.4.2 Mean and Variance Functions for the Random Component
4.4.3 Mean and Variance Functions for Poisson and Binomial GLMs
4.4.4 Systematic Component and Link Function of a GLM
4.4.5 Likelihood Equations for a GLM
4.4.6 The Key Role of the Mean–Variance Relationship
4.4.7 Likelihood Equations for Binomial GLMs
4.4.8 Asymptotic Covariance Matrix of Model Parameter Estimators
4.4.9 Likelihood Equations and cov(β) for Poisson Loglinear Model
4.5 Inference and Model Checking for Generalized Linear Models
4.5.1 Deviance and Goodness of Fit
4.5.2 Deviance for Poisson GLMs
4.5.3 Deviance for Binomial GLMs: Grouped Versus Ungrouped Data
4.5.4 Likelihood-Ratio Model Comparison Using the Deviances
4.5.5 Score Tests for Goodness of Fit and for Model Comparison
4.5.7 Covariance Matrices for Fitted Values and Residuals
4.5.8 The Bayesian Approach for GLMs
4.6 Fitting Generalized Linear Models
4.6.1 Newton–Raphson Method
4.6.2 Fisher Scoring Method
4.6.3 Newton–Raphson and Fisher Scoring for Binary Data
4.6.4 ML as Iterative Reweighted Least Squares
4.6.5 Simplifications for Canonical Link Functions
4.7 Quasi-Likelihood and Generalized Linear Models
4.7.1 Mean–Variance Relationship Determines Quasi-likelihood Estimates
4.7.2 Overdispersion for Poisson GLMs and Quasi-likelihood
4.7.3 Overdispersion for Binomial GLMs and Quasi-likelihood
4.7.4 Example: Teratology Overdispersion
5.1 Interpreting Parameters in Logistic Regression
5.1.1 Interpreting β: Odds, Probabilities, and Linear Approximations
5.1.2 Looking at the Data
5.1.3 Example: Horseshoe Crab Mating Revisited
5.1.4 Logistic Regression with Retrospective Studies
5.1.5 Logistic Regression Is Implied by Normal Explanatory Variables
5.2 Inference for Logistic Regression
5.2.1 Inference About Model Parameters and Probabilities
5.2.2 Example: Inference for Horseshoe Crab Mating Data
5.2.3 Checking Goodness of Fit: Grouped and Ungrouped Data
5.2.4 Example: Model Goodness of Fit for Horseshoe Crab Data
5.2.5 Checking Goodness of Fit with Ungrouped Data by Grouping
5.2.6 Wald Inference Can Be Suboptimal
5.3 Logistic Models with Categorical Predictors
5.3.1 ANOVA-Type Representation of Factors
5.3.2 Indicator Variables Represent a Factor
5.3.3 Example: Alcohol and Infant Malformation Revisited
5.3.4 Linear Logit Model for I × 2 Contingency Tables
5.3.5 Cochran–Armitage Trend Test
5.3.6 Example: Alcohol and Infant Malformation Revisited
5.3.7 Using Directed Models Can Improve Inferential Power
5.3.8 Noncentral Chi-Squared Distribution and Power for Narrower Alternatives
5.3.9 Example: Skin Damage and Leprosy
5.3.10 Model Smoothing Improves Precision of Estimation
5.4 Multiple Logistic Regression
5.4.1 Logistic Models for Multiway Contingency Tables
5.4.2 Example: AIDS and AZT Use
5.4.3 Goodness of Fit as a Likelihood-Ratio Test
5.4.4 Model Comparison by Comparing Deviances
5.4.5 Example: Horseshoe Crab Satellites Revisited
5.4.6 Quantitative Treatment of Ordinal Predictor
5.4.7 Probability-Based and Standardized Interpretations
5.4.8 Estimating an Average Causal Effect
5.5 Fitting Logistic Regression Models
5.5.1 Likelihood Equations for Logistic Regression
5.5.2 Asymptotic Covariance Matrix of Parameter Estimators
5.5.3 Distribution of Probability Estimators
5.5.4 Newton–Raphson Method Applied to Logistic Regression
6 Building, Checking, and Applying Logistic Regression Models
6.1 Strategies in Model Selection
6.1.1 How Many Explanatory Variables Can Be in the Model?
6.1.2 Example: Horseshoe Crab Mating Data Revisited
6.1.3 Stepwise Procedures: Forward Selection and Backward Elimination
6.1.4 Example: Backward Elimination for Horseshoe Crab Data
6.1.5 Model Selection and the "Correct" Model
6.1.6 AIC: Minimizing Distance of the Fit from the Truth
6.1.7 Example: Using Causal Hypotheses to Guide Model Building
6.1.8 Alternative Strategies, Including Model Averaging
6.2 Logistic Regression Diagnostics
6.2.1 Residuals: Pearson, Deviance, and Standardized
6.2.2 Example: Heart Disease and Blood Pressure
6.2.3 Example: Admissions to Graduate School at Florida
6.2.4 Influence Diagnostics for Logistic Regression
6.3 Summarizing the Predictive Power of a Model
6.3.1 Summarizing Predictive Power: R and R-Squared Measures
6.3.2 Summarizing Predictive Power: Likelihood and Deviance Measures
6.3.3 Summarizing Predictive Power: Classification Tables
6.3.4 Summarizing Predictive Power: ROC Curves
6.3.5 Example: Evaluating Predictive Power for Horseshoe Crab Data
6.4 Mantel–Haenszel and Related Methods for Multiple 2 × 2 Tables
6.4.1 Using Logistic Models to Test Conditional Independence
6.4.2 Cochran–Mantel–Haenszel Test of Conditional Independence
6.4.3 Example: Multicenter Clinical Trial Revisited
6.4.4 CMH Test Is Advantageous for Sparse Data
6.4.5 Estimation of Common Odds Ratio
6.4.6 Meta-analyses for Summarizing Multiple 2 x 2 Tables
6.4.7 Meta-analyses for Multiple 2 x 2 Tables: Difference of Proportions
6.4.8 Collapsibility and Logistic Models for Contingency Tables
6.4.9 Testing Homogeneity of Odds Ratios
6.4.10 Summarizing Heterogeneity in Odds Ratios
6.4.11 Propensity Scores in Observational Studies
6.5 Detecting and Dealing with Infinite Estimates
6.5.1 Complete or Quasi-complete Separation
6.5.2 Example: Multicenter Clinical Trial with Few Successes
6.5.3 Remedies When at Least One ML Estimate Is Infinite
6.6 Sample Size and Power Considerations
6.6.1 Sample Size and Power for Comparing Two Proportions
6.6.2 Sample Size Determination in Logistic Regression
6.6.3 Sample Size in Multiple Logistic Regression
6.6.4 Power for Chi–Squared Tests in Contingency Tables
6.6.5 Power for Testing Conditional Independence
6.6.6 Effects of Sample Size on Model Selection and Inference
7 Alternative Modeling of Binary Response Data
7.1 Probit and Complementary Log-log Models
7.1.1 Probit Models: Three Latent Variable Motivations
7.1.2 Probit Models: Interpreting Effects
7.1.3 Probit Model Fitting
7.1.4 Example: Modeling Flour Beetle Mortality
7.1.5 Complementary Log–Log Link Models
7.1.6 Example: Beetle Mortality Revisited
7.2 Bayesian Inference for Binary Regression
7.2.1 Prior Specifications for Binary Regression Models
7.2.2 Example: Risk Factors for Endometrial Cancer Grade
7.2.3 Bayesian Logistic Regression for Retrospective Studies
7.2.4 Probability–Based Prior Specifications for Binary Regression Models
7.2.5 Example: Modeling the Probability a Trauma Patient Survives
7.2.6 Bayesian Fitting for Probit Models
7.2.7 Bayesian Model Checking for Binary Regression
7.3 Conditional Logistic Regression
7.3.1 Conditional Likelihood
7.3.2 Small-Sample Inference for a Logistic Regression Parameter
7.3.3 Small-Sample Conditional Inference for 2 x 2 Contingency Tables
7.3.4 Small-Sample Conditional Inference for Linear Logit Model
7.3.5 Small-Sample Tests of Conditional Independence in 2 x 2 x K Tables
7.3.6 Example: Promotion Discrimination
7.3.7 Discreteness Complications of Using Exact Conditional Inference
7.4 Smoothing: Kernels, Penalized Likelihood, Generalized Additive Models
7.4.1 How Much Smoothing? The Variance/Bias Trade-off
7.4.3 Example: Smoothing to Portray Probability of Kyphosis
7.4.4 Nearest Neighbors Smoothing
7.4.5 Smoothing Using Penalized Likelihood Estimation
7.4.6 Why Shrink Estimates Toward 0?
7.4.7 Firth's Penalized Likelihood for Logistic Regression
7.4.8 Example: Complete Separation but Finite Logistic Estimates
7.4.9 Generalized Additive Models
7.4.10 Example: GAMs for Horseshoe Crab Mating Data
7.4.11 Advantages/Disadvantages of Various Smoothing Methods
7.5 Issues in Analyzing High–Dimensional Categorical Data
7.5.1 Issues in Selecting Explanatory Variables
7.5.2 Adjusting for Multiplicity: The Bonferroni Method
7.5.3 Adjusting for Multiplicity: The False Discovery Rate
7.5.4 Other Variable Selection Methods with High–Dimensional Data
7.5.5 Examples: High–Dimensional Applications in Genomics
7.5.6 Example: Motif Discovery for Protein Sequences
7.5.7 Example: The Netflix Prize
7.5.8 Example: Credit Scoring
8 Models for Multinomial Responses
8.1 Nominal Responses: Baseline–Category Logit Models
8.1.1 Baseline–Category Logits
8.1.2 Example: Alligator Food Choice
8.1.3 Estimating Response Probabilities
8.1.4 Fitting Baseline–Category Logistic Models
8.1.5 Multicategory Logit Model as a Multivariate GLM
8.1.6 Multinomial Probit Models
8.1.7 Example: Effect of Menu Pricing
8.2 Ordinal Responses: Cumulative Logit Models
8.2.2 Proportional Odds Form of Cumulative Logit Model
8.2.3 Latent Variable Motivation for Proportional Odds Structure
8.2.4 Example: Happiness and Traumatic Events
8.2.5 Checking the Proportional Odds Assumption
8.3 Ordinal Responses: Alternative Models
8.3.1 Cumulative Link Models
8.3.2 Cumulative Probit and Log-Log Models
8.3.3 Example: Happiness Revisited with Cumulative Probits
8.3.4 Adjacent–Categories Logit Models
8.3.5 Example: Happiness Revisited
8.3.6 Continuation–Ratio Logit Models
8.3.7 Example: Developmental Toxicity Study with Pregnant Mice
8.3.8 Stochastic Ordering Location Effects Versus Dispersion Effects
8.3.9 Summarizing Predictive Power of Explanatory Variables
8.4 Testing Conditional Independence in I × J × K Tables
8.4.1 Testing Conditional Independence Using Multinomial Models
8.4.2 Example: Homosexual Marriage and Religious Fundamentalism
8.4.3 Generalized Cochran-Mantel–Haenszel Tests for I x J x K Tables
8.4.4 Example: Homosexual Marriage Revisited
8.4.5 Related Score Tests for Multinomial Logit Models
8.5 Discrete-Choice Models
8.5.1 Conditional Logits for Characteristics of the Choices
8.5.2 Multinomial Logit Model Expressed as Discrete-Choice Model
8.5.3 Example: Shopping Destination Choice
8.5.4 Multinomial Probit Discrete–Choice Models
8.5.5 Extensions: Nested Logit and Mixed Logit Models
8.5.6 Extensions: Discrete Choice with Ordered Categories
8.6 Bayesian Modeling of Multinomial Responses
8.6.1 Bayesian Fitting of Cumulative Link Models
8.6.2 Example: Cannabis Use and Mother's Age
8.6.3 Bayesian Fitting of Multinomial Logit and Probit Models
8.6.4 Example: Alligator Food Choice Revisited
9 Loglinear Models for Contingency Tables
9.1 Loglinear Models for Two-way Tables
9.1.1 Independence Model for a Two-Way Table
9.1.2 Interpretation of Loglinear Model Parameters
9.1.3 Saturated Model for a Two-Way Table
9.1.4 Alternative Parameter Constraints
9.1.5 Hierarchical Versus Nonhierarchical Models
9.1.6 Multinomial Models for Cell Probabilities
9.2 Loglinear Models for Independence and Interaction in Three-way Tables
9.2.1 Types of Independence
9.2.2 Homogeneous Association and Three-Factor Interaction
9.2.3 Interpretation of Loglinear Model Parameters
9.2.4 Example: Alcohol, Cigarette, and Marijuana Use
9.3 Inference for Loglinear Models
9.3.1 Chi-Squared Goodness-of-Fit Tests
9.3.2 Inference about Conditional Associations
9.4 Loglinear Models for Higher Dimensions
9.4.1 Models for Four–Way Contingency Tables
9.4.2 Example: Automobile Accidents and Seat-Belt Use
9.4.3 Large Samples and Statistical Versus Practical Significance
9.4.4 Dissimilarity Index
9.5 Loglinear—Logistic Model Connection
9.5.1 Using Logistic Models to Interpret Loglinear Models
9.5.2 Example: Auto Accidents and Seat-Belts Revisited
9.5.3 Equivalent Loglinear and Logistic Models
9.5.4 Example: Detecting Gene–Environment Interactions in Case–Control Studies
9.6 Loglinear Model Fitting: Likelihood Equations and Asymptotic Distributions
9.6.1 Minimal Sufficient Statistics
9.6.2 Likelihood Equations for Loglinear Models
9.6.3 Unique ML Estimates Match Data in Sufficient Marginal Tables
9.6.4 Direct Versus Iterative Calculation of Fitted Values
9.6.5 Decomposable Models
9.6.6 Chi-Squared Goodness-of-Fit Tests
9.6.7 Covariance Matrix of ML Parameter Estimators
9.6.8 Connection Between Multinomial and Poisson Loglinear Models
9.6.9 Distribution of Probability Estimators
9.6.10 Proof of Uniqueness of ML Estimates
9.6.11 Pseudo ML for Complex Sampling Designs
9.7 Loglinear Model Fitting: Iterative Methods and Their Application
9.7.1 Newton-Raphson Method
9.7.2 Iterative Proportional Fitting
9.7.3 Comparison of IPF and Newton–Raphson Iterative Methods
9.7.4 Raking a Table: Contingency Table Standardization
10 Building and Extending Loglinear Models
10.1 Conditional Independence Graphs and Collapsibility
10.1.1 Conditional Independence Graphs
10.1.2 Graphical Loglinear Models
10.1.3 Collapsibility in Three–Way Contingency Tables
10.1.4 Collapsibility for Multiway Tables
10.2 Model Selection and Comparison
10.2.1 Considerations in Model Selection
10.2.2 Example: Model Building for Student Survey
10.2.3 Loglinear Model Comparison Statistics
10.2.4 Partitioning Chi-Squared with Model Comparisons
10.2.5 Identical Marginal and Conditional Tests of Independence
10.3 Residuals for Detecting Cell-Specific Lack of Fit
10.3.1 Residuals for Loglinear Models
10.3.2 Example: Student Survey Revisited
10.3.3 Identical Loglinear and Logistic Standardized Residuals
10.4 Modeling Ordinal Associations
10.4.1 Linear-by-Linear Association Model for Two-Way Tables
10.4.2 Corresponding Logistic Model for Adjacent Responses
10.4.3 Likelihood Equations and Model Fitting
10.4.4 Example: Sex and Birth Control Opinions Revisited
10.4.5 Directed Ordinal Test of Independence
10.4.6 Row Effects and Column Effects Association Models
10.4.7 Example: Estimating Category Scores for Premarital Sex
10.4.8 Ordinal Variables in Models for Multiway Tables
10.5 Generalized Loglinear and Association Models, Correlation Models, and Correspondence Analysis
10.5.1 Generalized Loglinear Model
10.5.2 Multiplicative Row and Column Effects Model
10.5.3 Example: Mental Health and Parents' SES
10.5.4 Correlation Models
10.5.5 Correspondence Analysis
10.5.6 Model Selection and Score Choice for Ordinal Variables
10.6 Empty Cells and Sparseness in Modeling Contingency Tables
10.6.1 Empty Cells: Sampling Versus Structural Zeros
10.6.2 Existence of Estimates in Loglinear Models
10.6.3 Effects of Sparseness on X2, G2, and Model-Based Tests
10.6.4 Alternative Sparse Data Asymptotics
10.6.5 Adding Constants to Cells of a Contingency Table
10.7 Bayesian Loglinear Modeling
10.7.1 Estimating Loglinear Model Parameters in Two-Way Tables
10.7.2 Example: Polarized Opinions by Political Party
10.7.3 Bayesian Loglinear Modeling of Multidimensional Tables
10.7.4 Graphical Conditional Independence Models
11 Models for Matched Pairs
11.1 Comparing Dependent Proportions
11.1.1 Confidence Intervals Comparing Dependent Proportions
11.1.2 McNemar Test Comparing Dependent Proportions
11.1.3 Example: Changes in Presidential Election Voting
11.1.4 Increased Precision with Dependent Samples
11.1.5 Small-Sample Test Comparing Dependent Proportions
11.1.6 Connection Between McNemar and Cochran-Mantel–Haenszel Tests
11.1.7 Subject-Specific and Population–Averaged (Marginal) Tables
11.2 Conditional Logistic Regression for Binary Matched Pairs
11.2.1 Subject–Specific Versus Marginal Models for Matched Pairs
11.2.2 Logistic Models with Subject-Specific Probabilities
11.2.3 Conditional ML Inference for Binary Matched Pairs
11.2.4 Random Effects in Binary Matched-Pairs Model
11.2.5 Conditional Logistic Regression for Matched Case–Control Studies
11.2.6 Conditional Logistic Regression for Matched Pairs with Multiple Predictors
11.2.7 Marginal Models and Subject-Specific Models: Extensions
11.3 Marginal Models for Square Contingency Tables
11.3.1 Marginal Models for Nominal Classifications
11.3.2 Example: Regional Migration
11.3.3 Marginal Models for Ordinal Classifications
11.3.4 Example: Opinions on Premarital and Extramarital Sex
11.4 Symmetry, Quasi-Symmetry, and Quasi-Independence
11.4.1 Symmetry as Logistic and Loglinear Models
11.4.3 Marginal Homogeneity and Quasi-symmetry
11.4.4 Quasi–independence
11.4.5 Example: Migration Revisited
11.4.6 Ordinal Quasi-symmetry
11.4.7 Example: Premarital and Extramarital Sex Revisited
11.5 Measuring Agreement Between Observers
11.5.1 Agreement: Departures from Independence
11.5.2 Using Quasi–independence to Analyze Agreement
11.5.3 Quasi-symmetry and Agreement Modeling
11.5.4 Kappa: A Summary Measure of Agreement
11.5.5 Weighted Kappa: Quantifying Disagreement
11.5.6 Extensions to Multiple Observers
11.6 Bradley-Terry Model for Paired Preferences
11.6.1 Bradley-Terry Model
11.6.2 Example: Major League Baseball Rankings
11.6.3 Example: Home Team Advantage in Baseball
11.6.4 Bradley-Terry Model and Quasi-symmetry
11.6.5 Extensions to Ties and Ordinal Pairwise Evaluations
11.7 Marginal Models and Quasi-Symmetry Models for Matched Sets
11.7.1 Marginal Homogeneity, Complete Symmetry, and Quasi-symmetry
11.7.2 Types of Marginal Symmetry
11.7.3 Comparing Binary Marginal Distributions in Multiway Tables
11.7.4 Example: Attitudes Toward Legalized Abortion
11.7.5 Marginal Homogeneity for a Multicategory Response
11.7.6 Wald and Generalized CMH Score Tests of Marginal Homogeneity
12 Clustered Categorical Data: Marginal and Transitional Models
12.1 Marginal Modeling: Maximum Likelihood Approach
12.1.1 Example: Longitudinal Study of Mental Depression
12.1.2 Modeling a Repeated Multinomial Response
12.1.3 Example: Insomnia Clinical Trial
12.1.4 ML Fitting of Marginal Logistic Models: Constraints on Cell Probabilities
12.1.5 ML Fitting of Marginal Logistic Models: Other Methods
12.2 Marginal Modeling: Generalized Estimating Equations (GEEs) Approach
12.2.1 Generalized Estimating Equations Methodology: Basic Ideas
12.2.2 Example: Longitudinal Mental Depression Revisited
12.2.3 Example: Multinomial GEE Approach for Insomnia Trial
12.3 Quasi-Likelihood and Its GEE Multivariate Extension: Details
12.3.1 The Univariate Quasi-likelihood Method
12.3.2 Properties of Quasi–likelihood Estimators
12.3.3 Sandwich Covariance Adjustment for Variance Misspecification
12.3.4 GEE Multivariate Methodology: Technical Details
12.3.5 Working Associations Characterized by Odds Ratios
12.3.6 GEE Approach: Multinomial Responses
12.3.7 Dealing with Missing Data
12.4 Transitional Models: Markov Chain and Time Series Models
12.4.2 Example: Changes in Evapotranspiration Rates
12.4.3 Transitional Models with Explanatory Variables
12.4.4 Example: Child's Respiratory Illness and Maternal Smoking
12.4.5 Example: Initial Response in Matched Pair as a Covariate
12.4.6 Transitional Models and Loglinear Conditional Models
13 Clustered Categorical Data: Random Effects Models
13.1 Random Effects Modeling of Clustered Categorical Data
13.1.1 Generalized Linear Mixed Model
13.1.2 Logistic GLMM with Random Intercept for Binary Matched Pairs
13.1.3 Example: Changes in Presidential Voting Revisited
13.1.4 Extension: Rasch Model and Item Response Models
13.1.5 Random Effects Versus Conditional ML Approaches
13.2 Binary Responses: Logistic-Normal Model
13.2.1 Shared Random Effect Implies Nonnegative Marginal Correlations
13.2.2 Interpreting Heterogeneity in Logistic-Normal Models
13.2.3 Connections Between Random Effects Models and Marginal Models
13.2.4 Comments About GLMMs Versus Marginal Models
13.3 Examples of Random Effects Models for Binary Data
13.3.1 Example: Small–Area Estimation of Binomial Proportions
13.3.2 Modeling Repeated Binary Responses: Attitudes About Abortion
13.3.3 Example: Longitudinal Mental Depression Study Revisited
13.3.4 Example: Capture–Recapture Prediction of Population Size
13.3.5 Example: Heterogeneity Among Multicenter Clinical Trials
13.3.6 Meta-analysis Using a Random Effects Approach
13.3.7 Alternative Formulations of Random Effects Models
13.3.8 Example: Matched Pairs with a Bivariate Binary Response
13.3.9 Time Series Models Using Autocorrelated Random Effects
13.3.10 Example: Oxford and Cambridge Annual Boat Race
13.4 Random Effects Models for Multinomial Data
13.4.1 Cumulative Logit Model with Random Intercept
13.4.2 Example: Insomnia Study Revisited
13.4.3 Example: Combining Measures on Ordinal Items
13.4.4 Example: Cluster Sampling
13.4.5 Baseline-Category Logit Models with Random Effects
13.4.6 Example: Effectiveness of Housing Program
13.5.1 Hierarchical Random Terms: Partitioning Variability
13.5.2 Example: Children's Care for an Unmarried Mother
13.6 GLMM Fitting, Inference, and Prediction
13.6.1 Marginal Likelihood and Maximum Likelihood Fitting
13.6.2 Gauss–Hermite Quadrature Methods for ML Fitting
13.6.3 Monte Carlo and EM Methods for ML Fitting
13.6.4 Laplace and Penalized Quasi-likelihood Approximations to ML
13.6.5 Inference for GLMM Parameters
13.6.6 Prediction Using Random Effects
13.7 Bayesian Multivariate Categorical Modeling
13.7.1 Marginal Homogeneity Analyses for Matched Pairs
13.7.2 Bayesian Approaches to Meta-analysis and Multicenter Trials
13.7.3 Example: Bayesian Analyses for a Multicenter Trial
13.7.4 Bayesian GLMMs and Marginal Models
14 Other Mixture Models for Discrete Data
14.1.1 Independence Given a Latent Categorical Variable
14.1.2 Fitting Latent Class Models
14.1.3 Example: Latent Class Model for Rater Agreement
14.1.4 Example: Latent Class Models for Capture-Recapture
14.1.5 Example: Latent Class Transitional Models
14.2 Nonparametric Random Effects Models
14.2.1 Logistic Models with Unspecified Random Effects Distribution
14.2.2 Example: Attitudes About Legalized Abortion
14.2.3 Example: Nonparametric Mixing of Logistic Regressions
14.2.4 Is Misspecification of Random Effects a Serious Problem?
14.2.5 Rasch Mixture Model
14.2.6 Example: Modeling Rater Agreement Revisited
14.2.7 Nonparametric Mixtures and Quasi-symmetry
14.2.8 Example: Attitudes About Legalized Abortion Revisited
14.3 Beta-Binomial Models
14.3.1 Beta-Binomial Distribution
14.3.2 Models Using the Beta-Binomial Distribution
14.3.3 Quasi-likelihood with Beta-Binomial Type Variance
14.3.4 Example: Teratology Overdispersion Revisited
14.3.5 Conjugate Mixture Models
14.4 Negative Binomial Regression
14.4.1 Gamma Mixture of Poissons Is Negative Binomial
14.4.2 Negative Binomial Regression Modeling
14.4.3 Example: Frequency of Knowing Homicide Victims
14.5 Poisson Regression with Random Effects
14.5.2 Marginal Model Implied by Poisson GLMM
14.5.3 Example: Homicide Victim Frequency Revisited
14.5.4 Negative Binomial Models versus Poisson GLMMs
15 Non-Model-Based Classification and Clustering
15.1 Classification: Linear Discriminant Analysis
15.1.1 Classification with Normally Distributed Predictors
15.1.2 Example: Horseshoe Crab Satellites Revisited
15.1.3 Multicategory Classification and Other Versions of Discriminant Analysis
15.1.4 Classification Methods for High Dimensions
15.1.5 Discriminant Analysis Versus Logistic Regression
15.2 Classification: Tree-Structured Prediction
15.2.1 Classification Trees
15.2.2 Example: Classification Tree for a Health Care Application
15.2.3 How Does the Classification Tree Grow?
15.2.4 Pruning a Tree and Checking Prediction Accuracy
15.2.5 Classification Trees Versus Logistic Regression
15.2.6 Support Vector Machines for Classification
15.3 Cluster Analysis for Categorical Data
15.3.1 Supervised Versus Unsupervised Learning
15.3.2 Measuring Dissimilarity Between Observations
15.3.3 Clustering Algorithms: Partitions and Hierarchies
15.3.4 Example: Clustering States on Election Results
16 Large- and Small-Sample Theory for Multinomial Models
16.1.1 O, o Rates of Convergence
16.1.2 Delta Method for a Function of a Random Variable
16.1.3 Delta Method for a Function of a Random Vector
16.1.4 Asymptotic Normality of Functions of Multinomial Counts
16.1.5 Delta Method for a Vector Function of a Random Vector
16.1.6 Joint Asymptotic Normality of Log Odds Ratios
16.2 Asymptotic Distributions of Estimators of Model Parameters and Cell Probabilities
16.2.1 Asymptotic Distribution of Model Parameter Estimator
16.2.2 Asymptotic Distribution of Cell Probability Estimators
16.2.3 Model Smoothing Is Beneficial
16.3 Asymptotic Distributions of Residuals and Goodness-of-fit Statistics
16.3.1 Joint Asymptotic Normality of p and π
16.3.2 Asymptotic Distribution of Pearson and Standardized Residuals
16.3.3 Asymptotic Distribution of Pearson X2 Statistic
16.3.4 Asymptotic Distribution of Likelihood-Ratio Statistic
16.3.5 Asymptotic Noncentral Distributions
16.4 Asymptotic Distributions for Logit/Loglinear Models
16.4.1 Asymptotic Covariance Matrices
16.4.2 Connection with Poisson Loglinear Models
16.5 Small-Sample Significance Tests for Contingency Tables
16.5.1 Exact Conditional Distribution for I x J Tables Under Independence
16.5.2 Exact Tests of Independence for I x J Tables
16.5.3 Example: Sexual Orientation and Party ID
16.6 Small-Sample Confidence Intervals for Categorical Data
16.6.1 Small-Sample CIs for a Binomial Parameter
16.6.2 CIs Based on Tests Using the Mid P- Value
16.6.3 Example: Proportion of Vegetarians Revisited
16.6.4 Small-Sample CIs for Odds Ratios
16.6.5 Example: Fisher's Tea Taster Revisited
16.6.6 Small-Sample CIs for Logistic Regression Parameters
16.6.7 Example: Diarrhea and an Antibiotic
16.6.8 Unconditional Small-Sample CIs for Difference of Proportions
16.7 Alternative Estimation Theory for Parametric Models
16.7.1 Weighted Least Squares for Categorical Data
16.7.2 Inference Using the WLS Approach to Model Fitting
16.7.3 Scope of WLS Versus ML Estimation
16.7.4 Minimum Chi-Squared Estimators
16.7.5 Minimum Discrimination Information
17 Historical Tour of Categorical Data Analysis
17.1 Pearson-Yule Association Controversy
17.2 R. A. Fisher's Contributions
17.4 Multiway Contingency Tables and Loglinear Models
17.5 Bayesian Methods for Categorical Data
17.6 A Look Forward, and Backward
Appendix A Statistical Software for Categorical Data Analysis
Appendix B Chi-Squared Distribution Values
Categorical Data Analysis
( Wiley Series in Probability and Statistics )
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