Categorical data analysis

Contents ... 9
Preface ... 15
CHAPTER  1 Introduction:  Distributions and Inference for  Categorical Data ... 19
	1.1  CATEGORICAL RESPONSE DATA ... 19
		1.1.1  Response-Explanatory Variable Distinction ... 20
		1.1.2  Binary-Nominal-Ordinal Scale Distinction ... 20
		1.1.3  Discrete-Continuous Variable Distinction ... 21
		1.1.4  Quantitative-Qualitative Variable Distinction ... 21
		1.1.S  Organization  of Book  and  Online Computing Appendix ... 22
	1.2  DISTRIBUTIONS FOR CATEGORICAL DATA ... 23
		1.2.1  Binomial Distribution ... 23
		1.2.2  Multinomial Distribution ... 24
		1.2.3  Poisson Distribution ... 24
		1.2.4  Overdispersion ... 25
		1.2.5  Connection Between Poisson and Multinomial Distributions ... 25
		1.2.6  The Chi-Squared Distribution ... 26
	1.3  STATISTICAL INFERENCE FOR CATEGORICAL DATA ... 26
		1.3.1  Likelihood Functions and Maximum Likelihood Estimation ... 27
		1.3.2  Likelihood Function and ML Estimate for Binomial Parameter ... 27
		1.3.3  Wald-Likelihood  Ratio Score Test Triad ... 28
		1.3.4  Constructing Confi dence Intervals by Inverting Tests ... 30
	1.4  STATISTICAL INFERENCE FOR  BINOMIAL PARAME TERS ... 31
		1.4.1  Tests About a Binomial Parameter ... 31
		1.4.2  Confidence Intervals for a Binomial Parameter ... 32
		1.4.3  Example:  Estimating the  Proportion of Vegetarians ... 33
		1.4.4  Exact  Small-Sample Inference and  the Mid P-Value ... 34
	1.5  STATISTICAL INFERENCE FOR MULTINOMIAL PARAMETERS ... 35
		1.5.1  Estimation of Multinomial Parameters ... 35
		1.5.2  Pearson Chi-Squared Test of a Specifi ed Multinomial ... 36
		1.5.3  Likelihood-Ratio Chi-Squared Test of a Specifi ed Multinomial ... 36
		1.5.4  Example: Testing Mendel\'s Theories ... 37
		1.5.5  Testing with Estimated Expected Frequencies ... 38
		1.5.6  Example: Pneumonia Infections in Calves ... 38
		1.5.7  Chi-Squared Theoretical Justifi cation ... 40
	1.6  BAYESIAN INFERENCE FOR  BINOMIAL  AND MULTINOMIAL  PARAMETERS ... 40
		1.6.1  The Bayesian Approach to Statistical Inference ... 40
		1.6.2  Binomial Estimation:  Beta and  Logit-Normal Prior Distributions ... 42
		1.6.3  Multinomial Estimation: Dirichlet Prior Distributions ... 43
		1.6.4  Example:  Estimating Vegetarianism Revisited ... 44
		1.6.5  Binomial and Multinomial Estimation:  Improper Priors ... 44
	NOTES ... 45
	EXERCISES ... 46
CHAPTER  2 Describing Contingency  Tables ... 55
	2.1  PROBABILITY STRUCTURE FOR  CONTINGENCY TABLES ... 55
		2.1.1  Contingency Tables ... 55
		2.1.2  Joint/Marginal/Conditional Distributions for Contingency Tables ... 56
		2.1.3  Example:  Sensitivity and Specifi city  for Medical Diagnoses ... 57
		2.1.4  Independence of Categorical Variables ... 58
		2.1.5  Poisson, Binomial, and Multinomial Sampling ... 58
		2.1.6  Example:  Seat  Belts and Auto Accident  Injuries ... 59
		2.1.7  Example:  Case-Control Study of Cancer and Smoking ... 60
		2.1.8  Ty pes of Studies:  Observational  Versus Experimental ... 61
	2.2  COMPARING TWO PROPORTIONS ... 61
		2.2.1  Difference of Proportions ... 62
		2.2.2  Relative Risk ... 62
		2.2.3  Odds Ratio ... 62
		2.2.4  Properties of the Odds Ratio ... 63
		2.2.S  Example: Association Between Heart Attacks and Aspirin Use ... 64
		2.2.6  Case-Control Studies and the Odds Ratio ... 64
		2.2. 7  Relationship B etween Odds Ratio and Relative Risk ... 65
	2.3  CONDITIONAL ASSOCIATION IN STRATIFIED 2x2 TABLES ... 65
		2.3.1  Partial Tables ... 66
		2.3.2  Example:  Racial Characteristics and  the Death Penalty ... 66
		2.3.3  Conditional  and  Marginal  Odds Ratios ... 68
		2.3.4  Marginal Independence Versus Conditional Independence ... 69
		2.3.5  Homogeneous  Association ... 71
		2.3.6  Collapsibility:  Identical  Conditional  and  Marginal  Associations ... 71
	2.4  MEASURING ASSOCIATION IN IxJ  TABLES ... 72
		2.4.1  Odds Ratios in IxJ  Tables ... 72
		2.4.2  Association Factors ... 73
		2.4.3  Summary Measures of Association ... 74
		2.4.4  Ordinal Trends:  Concordant and Discordant Pairs ... 74
		2.4.5  Ordinal Measure of Association:  Gamma ... 75
		2.4.6  Probabilistic Comparisons of Tw o Ordinal Distributions ... 76
		2.4.7  Example:  Comparing Pain Ratings After Surgery ... 77
		2.4.8  Correlation for Underlying Normality ... 77
	NOTES ... 78
	EXERCISES ... 78
CHAPTER  3 Inference for Two-Way Contingency  Tables ... 87
	3.1  CONFIDENCE INTERVALS FOR ASSOCIATION PARAMETERS ... 87
		3.1.1  Interval Estimation of the Odds Ratio ... 87
		3.1.2  Example: Seat-Belt Use and Traffic  Deaths ... 88
		3.1.3  Interval Estimation of Diff erence of Proportions and Relative Risk ... 89
		3.1.4  Example:  Aspirin and Heart Attacks Revisited ... 89
		3.1.5  Deriving  Standard  Errors with the Delta Method ... 90
		3.1.6  Delta Method Applied to the Sample Logit ... 91
		3.1.7  Delta Method for the Log Odds Ratio ... 91
		3.1.8  Simultaneous  Confi dence  Intervals for Multiple Comparisons ... 93
	3.2  TESTING  INDEPENDENCE  IN  TWO-WAY  CONTINGENCY  TABLES ... 93
		3.2.1  Pearson and  Likelihood-Ratio  Chi-Squared  Tests ... 93
		3.2.2  Example: Education and Belief in God ... 95
		3.2.3  Adequacy of Chi-Squared Approximations ... 95
		3.2.4  Chi-Squared and Comparing Proportions in 2x2 Tables ... 96
		3.2.5  Score Confi dence Intervals Comparing Proportions ... 96
		3.2.6  Profi le Likelihood Confi dence Intervals ... 97
	3.3  FOLLOWING-UP CHI-SQUARED TESTS ... 98
		3.3.1  Pearson Residuals and Standardized Residuals ... 98
		3.3.2  Example: Education and Belief in God Revisited ... 99
		3.3.3  Partitioning Chi-Squared ... 99
		3.3.4  Example: Origin of Schizophrenia ... 101
		3.3.S  Rules for  Partitioning ... 102
		3.3.6  Summarizing  the Association ... 102
		3.3.7  Limitations  of Chi-Squared  Tests ... 102
		3.3.8  Why Consider Independence If It\'s Unlikely to Be True? ... 103
	3.4  TWO-WAY TABLES WITH ORDERED CLASSIFICATIONS ... 104
		3.4.1  Linear Trend Alternative to Independence ... 104
		3.4.2  Example:  Is  Happiness  Associated  with  Political  Ideology? ... 105
		3.4.3  Monotone Trend Alternatives to Independence ... 105
		3.4.4  Extra Power with Ordinal Tests ... 106
		3.4.5  Sensitivity to Choice of Scores ... 106
		3.4.6  Example: Infant Birth Defects by Maternal Alcohol Consumption ... 107
		3.4.7 Trend Tests for  Ix2  and 2xJ  Tables ... 108
		3.4.8  Nominal-Ordinal Tables ... 108
	3.5  SMALL-SAMPLE INFERENCE FOR CONTINGENCY TABLES ... 108
		3.5.1  Fisher\'s Exact Test for 2x2 Tables ... 108
		3.5.2  Example:  Fisher\'s Tea Drinker ... 109
		3.5.3  Two-Sided P-Values for  Fisher\'s  Exact Test ... 110
		3.5.4  Confidence  Intervals Based on  Conditional Likelihood ... 110
		3.5.5  Discreteness and Conservatism Issues ... 111
		3.5.6  Small-Sample Unconditional Tests of Independence ... 111
		3.5.7  Conditional Versus Unconditional Tests ... 112
	3.6  BAYESIAN INFERENCE  FOR TWO-WAY CONTINGENCY TABLES ... 114
		3.6.1  Prior Distributions for Comparing Proportions in 2x2  Tables ... 114
		3.6.2  Posterior Probabilities Comparing Proportions ... 115
		3.6.3  Posterior Intervals for Association Parameters ... 115
		3.6.4  Example: Urn Sampling Gives Highly Unbalanced Treatment Allocation ... 116
		3.6.5  Highest Posterior Density Intervals ... 116
		3.6.6  Testing Independence ... 117
		3.6.7  Empirical Bayes  and  Hierarchical Bayesian  Approaches ... 118
	3.7  EXTENSIONS  FOR  MULTIWAY TABLES  AND NONTABULATED RESPONSES ... 118
		3.7.1  Categorical Data Need  Not Be  Contingency Tables ... 118
	NOTES ... 119
	EXERCISES ... 121
CHAPTER  4 Introduction to Generalized Linear Models ... 131
	4.1  THE  GENERALIZED  LINEAR  MODEL ... 131
		4.1.1  Components of Generalized Linear Models ... 132
		4.1.2  Binomial Logit Models for Binary Data ... 132
		4.1.3  Poisson Loglinear Models for Count Data ... 133
		4.1.4  Generalized Linear Models for Continuous Responses ... 133
		4.1.5  Deviance of a GLM ... 133
		4.1.6  Advantages of GLMs Versus Transforming the Data ... 134
	4.2  GENERALIZED  LINEAR MODELS FOR BINARY DATA ... 135
		4.2.1  Linear Probability Model ... 135
		4.2.2  Example:  Snoring and  Heart Disease ... 136
		4.2.3  Logistic Regression Model ... 137
		4.2.4  Binomial GLM for 2x2  Contingency  Tables ... 138
		4.2.5  Probit and Inverse cdf Link Functions ... 139Black,notBold,notItalic,open,TopLeftZoom,358,2,0.0
		4.2.6  Latent  Tolerance Motivation for Binary Response Models ... 140
	4.3  GENERALIZED LINEAR  MODELS FOR  COUNTS  AND RATES ... 140
		4.3.1  Poisson Loglinear  Models ... 141
		4.3.2  Example:  Horseshoe  Crab  Mating ... 141
		4.3.3  Overdispersion for Poisson GLMs ... 144
		4.3.4  Negative Binomial  GLMs ... 145
		4.3.5  Poisson Regression for  Rates Using Offsets ... 146
		4.3.6  Example:  Modeling Death  Rates for  Heart Valve  Operations ... 146
		4.3.7  Poisson  GLM  of Independence in Two-Way  Contingency  Tables ... 148
	4.4  MOMENTS  AND  LIKELIHOOD FOR GENERALIZED  LINEAR MODELS ... 148
		4.4.1  The Exponential Dispersion Family ... 148
		4.4.2  Mean and Variance Functions for the Random Component ... 149
		4.4.3  Mean and Variance Functions for Poisson and Binomial GLMs ... 150
		4.4.4  Systematic Component and Link Function of a GLM ... 150
		4.4.S  Likelihood  Equations for a GLM ... 151
		4.4.6  The Key Role of the Mean-Variance Relationship ... 152
		4.4.7  Likelihood Equations for Binomial GLMs ... 152
		4.4.8  Asymptotic  Covariance  Matrix  of Model Parameter Estimators ... 153
		4.4.9  Likelihood  Equations and cov (p) for Poisson  Loglinear Model ... 154
	4.5  INFERENCE AND MODEL  CHECKING FOR GENERALIZED LINEAR MODELS ... 154
		4.5.1  Deviance and  Goodness of Fit ... 154
		4.5.2  Deviance for Poisson GLMs ... 155
		4.5.3  Deviance for Binomial GLMs: Grouped Versus Ungrouped Data ... 155
		4.5.4  Likelihood-Ratio Model Comparison Using the Deviances ... 156
		4.5.S  Score Tests for Goodness of Fit and for Model Comparison ... 157
		4.5.6  Residuals for GLMs ... 158
		4.5.7  Covariance Matrices for Fitted Values and Residuals ... 160
		4.5.8  The Bayesian Approach for GLMs ... 160
	4.6  FITTING GENERALIZED LINEAR MODELS ... 161
		4.6.1  Newton-Raphson Method ... 161
		4.6.2  Fisher Scoring Method ... 162
		4.6.3  Newton-Raphson and Fisher Scoring for Binary Data ... 163
		4.6.4  ML as Iterative Reweighted Least Squares ... 164Black,notBold,notItalic,open,TopLeftZoom,284,2,0.0
		4.6.5  Simplifi cations for  Canonical Link Functions ... 165
	4.7 QUASI-LIKELIHOOD  AND  GENERALIZED  LINEAR  MODELS ... 167
		4.7.1  Mean-Variance Relationship Determines Quasi-likelihood Estimates ... 167
		4.7.2  Overdispersion  for Poisson  GLMs and Quasi-likelihood ... 167
		4.7.3  Overdispersion for Binomial GLMs and Quasi-likelihood ... 168
		4.7.4  Example:  Teratology  Overdispersion ... 169
		NOTES ... 170
		EXERCISES ... 171
CHAPTER  5 Logistic  Regression ... 181
	5.1  INTERPRETING PARAMETERS IN LOGISTIC REGRESSION ... 181
		5.1.1  Interpreting p:  Odds, Probabilities, and Linear Approximations ... 182
		5.1.2  Looking at the  Data ... 183
		5.1.3  Example: Horseshoe Crab Mating Revisited ... 184
		5.1.4  Logistic Regression with Retrospective Studies ... 186
		5.1.5  Logistic Regression  Is Implied by  Normal Explanatory  Variables ... 187
	5.2  INFERENCE FOR LOGISTIC REGRESSION ... 187
		5.2.1  Inference About Model  Parameters and Probabilities ... 187
		5.2.2  Example: Inference for Horseshoe Crab Mating Data ... 188
		5.2.3  Checking Goodness of Fit: Grouped and Ungrouped Data ... 189
		5.2.4  Example: Model Goodness of Fit for Horseshoe Crab Data ... 190
		5.2.5  Checking Goodness of Fit with Ungrouped Data by Grouping ... 190
		5.2.6  Wald Inference Can Be Suboptimal ... 192
	5.3  LOGISTIC MODELS WITH CATEGORICAL PREDICTORS ... 193
		5.3.1  ANOVA-Type Representation of Factors ... 193
		5.3.2  Indicator Variables Represent  a Factor ... 193
		5.3.3  Example:  Alcohol and  Infant Malformation Revisited ... 194
		5.3.4  Linear Logit Model for Ix2 Contingency Tables ... 195
		5.3.5  Cochran-Armitage Trend Test ... 195
		5.3.6  Example:  Alcohol and  Infant  Malformation  Revisited ... 197
		5.3.7  Using Directed  Models Can Improve Inferential Power ... 197
		5.3.8  Noncentral Chi-Squared Distribution and Power for Narrower Alternatives ... 198
		5.3.9  Example:  Skin Damage and Leprosy ... 199
		5.3.10  Model Smoothing Improves Precision of Estimation ... 200
	5.4  MULTIPLE LOGISTIC REGRESSION ... 200
		5.4.1  Logistic Models for Multiway Contingency Tables ... 201
		5.4.2  Example:  AIDS and AZT Use ... 202
		5.4.3  Goodness of Fit as a Likelihood-Ratio Test ... 204
		5.4.4  Model  Comparison by Comparing Deviances ... 205
		5.4.5  Example:  Horseshoe Crab Satellites  Revisited ... 205
		5.4.6  Quantitative Treatment of Ordinal Predictor ... 207Black,notBold,notItalic,open,TopLeftZoom,292,2,0.0
		5.4.7  Probability-Based and Standardized Interpretations ... 208
		5.4.8  Estimating an Average  Causal Eff ect ... 209
	5.5  FITTING LOGISTIC REGRESSION MODELS ... 210
		5.5.1  Likelihood Equations for Logistic Regression ... 210
		5.5.2  Asymptotic  Covariance  Matrix of Parameter  Estimators ... 211
		5.5.3  Distribution of Probability Estimators ... 212
		5.5.4  Newton-Raphson Method Applied to Logistic Regression ... 212
	NOTES ... 213
	EXERCISES ... 214
CHAPTER  6 Building,  Checking,  and Applying Logistic Regression Models ... 225
	6.1  STRATEGIES IN MODEL SELECTION ... 225
		6.1.1  How Many Explanatory Variables Can Be in the Model? ... 226
		6.1.2  Example: Horseshoe Crab Mating Data Revisited ... 226
		6.1.3  Stepwise Procedures:  Forward Selection and Backward Elimination ... 227
		6.1.4  Example: Backward Elimination for Horseshoe Crab Data ... 228
		6.1.5  Model Selection and the \"Correct\" Model ... 229
		6.1.6  AIC:  Minimizing Distance of the Fit from the Tr uth ... 230
		6.1.7  Example:  Using Causal Hypotheses to  Guide Model Building ... 231
		6.1.8  Alternative Strategies,  Including  Model Averaging ... 233
	6.2  LOGISTIC REGRESSION DIAGNOSTICS ... 233
		6.2.1  Residuals: Pearson, Deviance, and Standardized ... 233
		6.2.2  Example:  Heart Disease and Blood Pressure ... 234
		6.2.3  Example:  Admissions  to Graduate School at Florida ... 236
		6.2.4  Infl uence Diagnostics for  Logistic Regression ... 238
	6.3  SUMMARIZING  THE  PREDICTIVE POWER OF A MODEL ... 239
		6.3.1  Summarizing  Predictive  Power:  Rand  R-Squared  Measures ... 239
		6.3.2  Summarizing Predictive Power: Likelihood and Deviance Measures ... 240
		6.3.3  Summarizing Predictive Power: Classifi cation Tables ... 241
		6.3.4  Summarizing Predictive Power: ROC Curves ... 242
		6.3.S  Example:  Evaluating Predictive Power for  Horseshoe Crab Data ... 242
	6.4  MANTEL-HAENSZEL  AND RELATED METHODS FOR MULTIPLE 2x2  TABLES ... 243
		6.4.1  Using Logistic Models to Test Conditional Independence ... 244
		6.4.2  Cochran-Mantel-Haenszel  Test of Conditional  Independence ... 245
		6.4.3  Example: Multicenter Clinical Trial Revisited ... 246
		6.4.4  CMH Test Is Advantageous for Sparse Data ... 246
		6.4.S  Estimation of Common Odds Ratio ... 247
		6.4.6  Meta-analyses for Summarizing Multiple 2x2 Tables ... 248
		6.4. 7  Meta-analyses for  Multiple 2x2 Tables:  Diff erence of Proportions ... 249
		6.4.8  Collapsibility and Logistic Models for  Contingency Tables ... 250
		6.4.9  Testing Homogeneity of Odds Ratios ... 250
		6.4.10  Summarizing Heterogeneity  in  Odds Ratios ... 251
		6.4.11  Propensity Scores in  Observational Studies ... 251
	6.5  DETECTING A ND DEALING  WITH INFINITE ESTIMATES ... 251
		6.5.1  Complete or Quasi-complete Separation ... 252
		6.5.2  Example:  Multicenter  Clinical  Trial with Few Successes ... 253
		6.5.3  Remedies When at Least One ML Estimate Is Infi nite ... 254
	6.6  SAMPLE  SIZE AND POWER CONSIDERATIONS ... 255
		6.6.1  Sample  Size  and Power for Comparing Two Proportions ... 255
		6.6.2  Sample  Size Determination in Logistic Regression ... 256
		6.6.3  Sample Size in Multiple Logistic Regression ... 257
		6.6.4  Power for Chi-Squared Tests in Contingency Tables ... 257
		6.6.5  Power for Testing Conditional Independence ... 258
		6.6.6  Effects  of Sample Size on Model Selection and Inference ... 259
	NOTES ... 259
	EXERCISES ... 261
CHAPTER  7 Alternative Modeling of Binary Response Data ... 269
	7.1  PROBIT AND COMPLEMENTARY LOG-LOG MODELS ... 269
		7.1.l  Probit Models:  Three Latent Variable Motivations ... 270
		7.1.2  Probit Models:  Interpreting Eff ects ... 270
		7.1.3  Probit  Model  Fitting ... 271
		7.1.4  Example:  Modeling Flour Beetle  Mortality ... 272
		7.1.5  Complementary Log-Log Link Models ... 273
		7.1.6  Example:  Beetle  Mortality  Revisited ... 275
	7.2  BAYESIAN INFERENCE  FOR BINARY REGRESSION ... 275
		7.2.1  Prior  Specifi cations  for  Binary  Regression  Models ... 275
		7.2.2  Example: Risk Factors for Endometrial Cancer Grade ... 276
		7 .2.3  Bayesian Logistic Regression for Retrospective Studies ... 278
		7.2.4  Probability-Based Prior Specifi cations for Binary Regression Models ... 278
		7.2.5  Example:  Modeling the Probability a Trauma Patient Survives ... 279
		7.2.6  Bayesian  Fitting for Probit  Models ... 281
		7.2.7  Bayesian Model Checking for Binary Regression ... 283
	7.3  CONDITIONAL LOGISTIC REGRESSION ... 283
		7.3.1  Conditional Likelihood ... 283
		7.3.2  Small-Sample Inference for a Logistic Regression Parameter ... 285
		7.3.3  Small-Sample Conditional Inference for 2x2 Contingency Tables ... 285
		7.3.4  Small-Sample Conditional Inference for Linear Logit Model ... 286
		7.3.5  Small-Sample Tests of Conditional Independence in 2x2  x  K Tables ... 287
		7.3.6  Example: Promotion Discrimination ... 287
		7.3.7  Discreteness Complications of Using Exact Conditional Inference ... 288
	7.4  SMOOTHING:  KERNELS, PENALIZED LIKELIHOOD, GENERALIZED  ADDITIVE MODELS ... 288
		7.4.l  How Much Smoothing The Variance -Bias  Trade-off ... 288
		7.4.2  Kernel Smoothing ... 289
		7.4.3  Example: Smoothing to Portray Probability of Kyphosis ... 290
		7.4.4  Nearest Neighbors Smoothing ... 290
		7.4.5  Smoothing Using Penalized Likelihood Estimation ... 291
		7.4.6  Why  Shrink Estimates Toward 0? ... 293
		7.4.7  Firth\'s Penalized  Likelihood  for Logistic Regression ... 293
		7.4.8  Example:  Complete  Separation but  Finite  Logistic Estimates ... 293
		7.4.9  Generalized Additive Models ... 294
		7.4.10  Example:  GAMs for Horseshoe  Crab Mating Data ... 295
		7.4.11  Advantages -Disadvantages of Various  Smoothing  Methods ... 295
	7.5  ISSUES IN ANALYZING HIGH-DIMENSIONAL CATEGORICAL DATA ... 296
		7.5.l  Issues in Selecting Explanatory Variables ... 296
		7.5.2  Adjusting for Multiplicity:  The Bonferroni  Method ... 297
		7.5.3  Adjusting for  Multiplicity:  The False Discovery Rate ... 298
		7.5.4  Other  Variable  Selection Methods with High-Dimensional Data ... 299
		7.5.S  Examples: High-Dimensional Applications in Genomics ... 300
		7.5.6  Example: Motif Discovery for Protein Sequences ... 301
		7.5.7  Example:  The Netfl ix Prize ... 302
		7.5.8  Example:  Credit Scoring ... 303
	NOTES ... 303
	EXERCISES ... 305
CHAPTER  8 Models for Multinomial Responses ... 311
	8.1  NOMINAL RESPONSES:  BASELINE-CATEGORY LOGIT MODELS ... 311
		8.1.1  Baseline-Category Logits ... 311
		8.1.2  Example:  Alligator Food Choice ... 312
		8.1.3  Estimating Response Probabilities ... 314
		8.1.4  Fitting Baseline-Category Logistic Models ... 315
		8.1.5  Multicategory Logit Model as a Multivariate GLM ... 317
		8.1.6  Multinomial Probit Models ... 317
		8.1.7  Example:  Eff ect of Menu Pricing ... 318
	8.2  ORDINAL RESPONSES:  CUMULATIVE LOGIT MODELS ... 319
		8.2.1  Cumulative  Logits ... 319
		8.2.2  Proportional Odds Form  of Cumulative  Logit Model ... 319
		8.2.3  Latent Variable Motivation for Proportional Odds Structure ... 321
		8.2.4  Example:  Happiness and Traumatic Events ... 322
		8.2.S  Checking the Proportional Odds Assumption ... 324
	8.3  ORDINAL RESPONSES:  ALTERNATIVE  MODELS ... 326
		8.3.1  Cumulative Link  Models ... 326
		8.3.2  Cumulative Probit and Log-Log Models ... 326
		8.3.3  Example:  Happiness  Revisited  with  Cumulative  Probits ... 327
		8.3.4  Adjacent-Categories Logit Models ... 327
		8.3.5  Example:  Happiness Revisited ... 328
		8.3.6  Continuation-Ratio Logit Models ... 329
		8.3.7  Example:  Developmental  Toxicity  Study with  Pregnant  Mice ... 330
		8.3.8  Stochastic Ordering Location  Eff ects Versus Dispersion Eff ects ... 331
		8.3.9  Summarizing Predictive Power of Explanatory Variables ... 332
	8.4  TESTING  CONDITIONAL INDEPENDENCE IN IxJ  x  K  TABLES ... 332
		8.4.1  Testing Conditional Independence Using Multinomial Models ... 332
		8.4.2  Example:  Homosexual  Marriage and Religious  Fundamentalism ... 334
		8.4.3  Generalized  Cochran-Mantel-Haenszel Tests for IxJ  x  K  Tables ... 335
		8.4.4  Example:  Homosexual  Marriage  Revisited ... 337
		8.4.5  Related Score Tests for Multinomial Logit Models ... 337
	8.5  DISCRETE-CHOICE MODELS ... 338
		8.5.1  Conditional  Logits for Characteristics of the Choices ... 338
		8.5.2  Multinomial Logit Model Expressed as Discrete-Choice Model ... 339
		8.5.3  Example:  Shopping Destination Choice ... 339
		8.5.4  Multinomial Probit Discrete-Choice Models ... 339
		8.5.5  Extensions:  Nested Logit and Mixed Logit Models ... 340
		8.5.6  Extensions:  Discrete Choice with Ordered Categories ... 340
	8.6  BAYESIAN MODELING OF MULTINOMIAL RESPONSES ... 341
		8.6.1  Bayesian Fitting of Cumulative  Link  Models ... 341
		8.6.2  Example:  Cannabis Use and Mother\'s Age ... 342
		8.6.3  Bayesian Fitting of Multinomial Logit and Probit Models ... 343
		8.6.4  Example:  Alligator Food Choice Revisited ... 344
		NOTES ... 344
	EXERCISES ... 347
CHAPTER  9 Loglinear Models  for Contingency Tables ... 357
	9.1  LOGLINEAR  MODELS  FOR  TWO-WAY TABLES ... 357
		9.1.1  Independence Model  for  a  Two-Way  Table ... 357
		9.1.2  Interpretation of Loglinear Model Parameters ... 358
		9.1.3  Saturated Model for a Two-Way Table ... 358
		9.1.4  Alternative Parameter Constraints ... 359
		9.1.5  Hierarchical Versus Nonhierarchical Models ... 359
		9.1.6  Multinomial Models for  Cell Probabilities ... 360
	9.2  LOGLINEAR MODELS FOR INDEPENDENCE  AND INTERACTION IN THREE-WAY TABLES ... 360
		9.2.1  Types of Independence ... 360
		9.2.2  Homogeneous Association and Three-Factor Interaction ... 362
		9.2.3  Interpretation of Loglinear Model  Parameters ... 363
		9.2.4  Example:  Alcohol,  Cigarette, and Marijuana Use ... 364
	9.3  INFERENCE FOR LOGLINEAR MODELS ... 366
		9.3.1  Chi-Squared  Goodness-of-Fit Tests ... 366
		9.3.2  Inference about Conditional Associations ... 366
	9.4  LOGLINEAR MODELS FOR HIGHER DIMENSIONS ... 368
		9.4.1  Models for  Four-Way Contingency Tables ... 368
		9.4.2  Example:  Automobile Accidents and Seat-Belt  Use ... 368
		9.4.3  Large Samples and Statistical Versus Practical Signifi cance ... 370
		9.4.4  Dissimilarity Index ... 370
	9.5  LOGLINEAR-LOGISTIC MODEL CONNECTION ... 371
		9.5.1  Using Logistic Models to Interpret Loglinear Models ... 371
		9.5.2  Example: Auto Accidents and Seat-Belts Revisited ... 372
		9.5.3  Equivalent Loglinear and Logistic Models ... 372
		9.5.4  Example:  Detecting Gene-Environment Interactions in Case-Control Studies ... 373
	9.6  LOG LINEAR MODEL FITTING:  LIKELIHOOD  EQUATIONS  AND ASYMPTOTIC DISTRIBUTIONS ... 374
		9.6.1  Minimal Suffi cient  Statistics ... 374
		9.6.2  Likelihood Equations for Loglinear Models ... 375
		9.6.3  Unique ML Estimates Match Data in Suffi cient  Marginal Tables ... 376
		9.6.4  Direct Versus Iterative Calculation of Fitted Values ... 376
		9.6.S  Decomposable Models ... 377
		9.6.6  Chi-Squared Goodness-of-Fit Tests ... 377
		9.6.7  Covariance Matrix of ML Parameter Estimators ... 378
		9.6.8  Connection Between Multinomial and Poisson Loglinear Models ... 379
		9.6.9  Distribution of Probability Estimators ... 380
		9.6.10  Proof of Uniqueness of ML Estimates ... 381
		9.6.11  Pseudo ML for Complex Sampling Designs ... 381
	9.7  LOGLINEA R MODEL FITTING: ITERATIVE METHODS AND THEIR APPLICATION ... 382
		9.7.1  Newton-Raphson Method ... 382
		9.7.2  Iterative Proportional Fitting ... 383
		9.7.3  Comparison ofIPF and Newton-Raphson Iterative Methods ... 384
		9.7.4  Raking a Table: Contingency Table Standardization ... 385
	NOTES ... 386
	EXERCISES ... 387Black,notBold,notItalic,open,TopLeftZoom,165,2,0.0
CHAPTER  10 Building and Extending Loglinear Models ... 395
	IO.I  CONDITIONAL INDEPENDENCE GRAPHS AND COLLAPSIBILITY ... 395
		IO.I.I  Conditional Independence Graphs ... 395
		10.1.2  Graphical Loglinear Models ... 396
		10.1.3  Collapsibility in Three-Way Contingency Tables ... 397
		10.1.4  Collapsibility for Multiway Tables ... 398
	10.2  MODEL SELECTION  AND COMPARISON ... 398
		10.2.1  Considerations  in Model Selection ... 398
		10.2.2  Example: Model Building for Student Survey ... 399
		10.2.3  Loglinear Model Comparison Statistics ... 401
		10.2.4  Partitioning Chi-Squared with Model Comparisons ... 402
		10.2.5  Identical Marginal and Conditional Tests of Independence ... 402
	10.3  RESIDUALS  FOR DETECTING  CELL-SPECIFIC LACK OF  FIT ... 403
		10.3.1  Residuals for Loglinear  Models ... 403
		10.3.2  Example:  Student  Survey Revisited ... 403
		10.3.3  Identical Loglinear and Logistic Standardized Residuals ... 404
	10.4  MODELING ORDINAL ASSOCIATIONS ... 404
		10.4.1  Linear-by-Linear Association Model for Two-Way Tables ... 405
		10.4.2  Corresponding Logistic Model for  Adjacent Responses ... 406
		10.4.3  Likelihood Equations and Model Fitting ... 407
		10.4.4  Example: Sex and Birth  Control  Opinions Revisited ... 407
		10.4.5  Directed Ordinal Test of Independence ... 409
		10.4.6  Row Effects and Column Effects Association Models ... 409
		10.4.7  Example:  Estimating  Category  Scores for  Premarital  Sex ... 410
		10.4.8  Ordinal  Variables in  Models for Multiway Tables ... 410
	10.S  GENERALIZED LOGLINEAR AND ASSOCIATION MODELS, CORRELATION MODELS, AND CORRESPONDENCE ANALYSIS ... 411
		10.S.1  Generalized Loglinear Model ... 411
		10.5.2  Multiplicative Row and Column Eff ects Model ... 412
		10.5.3  Example:  Mental Health and Parents\' SES ... 413
		10.5.4  Correlation Models ... 413
		10.5.5  Correspondence Analysis ... 414
		10.5.6  Model Selection and Score Choice for Ordinal Variables ... 416
	10.6  EMPTY CELLS  AND SPARSENESS IN MODELING CONTINGENCY TABLES ... 416
		10.6.1  Empty Cells:  Sampling Versus Structural Zeros ... 416
		10.6.2  Existence of Estimates in Loglinear Models ... 416
		10.6.3  Eff ects of Sparseness on X2,  G2, and Model-Based Tests ... 418
		10.6.4  Alternative Sparse Data Asymptotics ... 419
		10.6.5  Adding  Constants to  Cells of a  Contingency Table ... 419
	10.7  BAYESIAN LOGLINEAR  MODELING ... 419
		10.7.1  Estimating Loglinear Model Parameters in Two-Way Tables ... 420
		10.7.2  Example:  Polarized Opinions by Political Party ... 420
		10.7.3  Bayesian Loglinear Modeling of Multidimensional Tables ... 421
		10.7.4  Graphical  Conditional Independence Models ... 422
	NOTES ... 422
	EXERCISES ... 425
CHAPTER  11 Models  for Matched Pairs ... 431
	11.1  COMPARING DEPENDENT PROPORTIONS ... 432
		11.1.2  McNemar Test Comparing Dependent Proportions ... 433
		11.1.3  Example:  Changes in  Presidential Election Voting ... 433
		11.1.4  Increased  Precision  with  Dependent  Samples ... 434
		11.1.5  Small-Sample  Test  Comparing  Dependent  Proportions ... 434
		11.1.6  Connection  Between McNemar and  Cochran-Mantel-Haenszel Tests ... 435
		11.1.7  Subject-Specifi c and Population- Averaged (Marginal) Tables ... 436
	11.2  CONDITIONAL LOGISTIC  REGRESSION FOR BINARY  MATCHED  PA IRS ... 436
		11.2.1  Subject-Specific Versus  Marginal Models  for Matched Pairs ... 436
		11.2.2  Logistic Models with Subject-Specific  Probabilities ... 437
		11.2.3  Conditional  ML Inference for  Binary  Matched Pairs ... 438
		11.2.4  Random Effects in Binary Matched-Pairs Model ... 439
		11.2.S  Conditional Logistic Regression for Matched Case-Control Studies ... 439
		11.2.6  Conditional Logistic Regression for Matched Pairs with Multiple Predictors ... 440
		11.2.7  Marginal Models and Subject-Specifi c Models: Extensions ... 441
	11.3  MARGINAL  MODELS FOR SQUARE  CONTINGENCY  TABLES ... 442
		11.3.1  Marginal Models for Nominal Classifi cations ... 442
		11.3.2  Example:  Regional Migration ... 443
		11.3.3  Marginal Models for Ordinal Classifi cations ... 443
		11.3.4  Example: Opinions on Premarital and Extramarital Sex ... 444
	11.4  SYMMETRY, QUASI-SYMMETRY, AND QUASI-INDEPENDENCE ... 444
		11.4.1  Symmetry as Logistic and Loglinear Models ... 445
		11.4.2  Quasi-symmetry ... 445
		11.4.3  Marginal Homogeneity and Quasi-symmetry ... 447
		11.4.4  Quasi-independence ... 447
		11.4.5  Example:  Migration  Revisited ... 448
		11.4.6  Ordinal Quasi-symmetry ... 449
		11.4.7  Example:  Premarital and Extramarital Sex Revisited ... 450
	11.5  MEASURING AGREEMENT BETWEEN OBSERVERS ... 450
		11.5.1  Agreement:  Departures from Independence ... 451
		11.5.2  Using Quasi-independence to Analyze Agreement ... 451
		11.5.3  Quasi-symmetry and Agreement Modeling ... 452
		11.5.4  Kappa: A Summary Measure of Agreement ... 452
		11.5.5  Weighted Kappa: Quantifying Disagreement ... 453
		11.S.6  Extensions to Multiple Observers ... 453
	11.6  BRADLEY-TERRY  MODEL FOR PAIRED PREFERENCES ... 454
		11.6.1  Bradley-Terry  Model ... 454
		11.6.2  Example:  Major League Baseball Rankings ... 454
		11.6.3  Example:  Home  Team Advantage in Baseball ... 455
		11.6.4  Bradley-Terry Model and Quasi-symmetry ... 456
		11.6.S  Extensions to  Ties and Ordinal Pairwise Evaluations ... 457
	11.7  MARGINAL MODELS AND  QUASI-SYMMETRY MODELS FOR  MATCHED SETS ... 457
		11.7.1  Marginal Homogeneity,  Complete  Symmetry, and Quasi-symmetry ... 457
		11.7.2  Types of Marginal Symmetry ... 458
		11.7.3  Comparing Binary Marginal Distributions in Multiway Tables ... 458
		11.7.4  Example:  Attitudes Toward  Legalized  Abortion ... 459
		11.7.S  Marginal Homogeneity for a Multicategory Response ... 460
		11.7.6  Wald and Generalized CMH Score  Tests of Marginal Homogeneity ... 460
	NOTES ... 461
	EXERCISES ... 463
CHAPTER  12 Clustered Categorical Data: Marginal and Transitional Models ... 473
	12.1  MARGINAL MODELING: MAXIMUM LIKELIHOOD APPROACH ... 474
		12.1.1  Example:  Longitudinal Study of Mental Depression ... 474
		12.1.2  Modeling a Repeated Multinomial Response ... 476
		12.1.3  Example:  Insomnia Clinical Trial ... 476
		12.1.4  ML Fitting of Marginal Logistic Models:  Constraints on Cell Probabilities ... 477
		12.1.5  ML Fitting of Marginal Logistic Models: Other Methods ... 479
	12.2  MARGINAL  MODELING:  GENERALIZED ESTIMATING EQUATIONS (GEEs) APPROACH ... 480
		12.2.1  Generalized Estimating Equations Methodology:  Basic Ideas ... 480
		12.2.2  Example:  Longitudinal Mental Depression Revisited ... 481
		12.2.3  Example: Multinomial GEE Approach for Insomnia Trial ... 482
	12.3  QUASI-LIKELIHOOD  A ND ITS GEE MULTIVARIATE EXTENSION:  DETAILS ... 483
		12.3.1  The Univariate Quasi-likelihood Method ... 483
		12.3.2  Properties of Quasi-likelihood Estimators ... 484
		12.3.3  Sandwich Covariance Adjustment for Variance Misspecifi cation ... 485
		12.3.4  GEE Multivariate Methodology:  Technical Details ... 486
		12.3.S  Working Associations  Characterized by Odds Ratios ... 488
		12.3.6  GEE Approach: Multinomial Responses ... 488
		12.3. 7  Dealing with Missing Data ... 489
	12.4  TRANSITIONAL MODELS: MARKOV CHAIN AND TIME SERIES MODELS ... 491
		12.4.1  Markov Chains ... 491
		12.4.2  Example:  Changes  in  Evapotranspiration Rates ... 492
		12.4.3  Transitional Models with Explanatory  Variables ... 493
		12.4.4  Example: Child\'s Respiratory Illness and Maternal Smoking ... 494
		12.4.5  Example:  Initial Response in Matched Pair as a Covariate ... 495
		12.4.6  Transitional Models and Loglinear  Conditional Models ... 496
	NOTES ... 496
	EXERCISES ... 497
CHAPTER  13 Clustered  Categorical Data: Random Effects Models ... 507
	13.1  RANDOM EFFECTS MODELING OF  CLUSTERED  CATEGORICAL  DATA ... 507
		13.1.1  Generalized Linear Mixed Model ... 508
		13.1.2  Logistic GLMM with Random Intercept for Binary Matched Pairs ... 509
		13.1.3  Example:  Changes in Presidential  Voting Revisited ... 510
		13.1.4  Extension:  Rasch Model  and Item Response Models ... 510
		13.1.S  Random Eff ects Versus Conditional ML Approaches ... 511
	13.2  BINARY  RESPONSES:  LOGISTIC-NORMAL MODEL ... 512
		13.2.1  Shared Random Eff ect Implies Nonnegative Marginal Correlations ... 512
		13.2.2  Interpreting Heterogeneity in Logistic-Normal Models ... 512
		13.2.3  Connections Between Random Eff ects Models and Marginal Models ... 513
		13.2.4  Comments About GLMMs Versus Marginal Models ... 515
	13.3  EXAMPLES OF RANDOM EFFECTS MODELS FOR BINARY DATA ... 516
		13.3.1  Example:  Small-Area Estimation of Binomial Proportions ... 516
		13.3.2  Modeling Repeated Binary Responses: Attitudes About Abortion ... 518
		13.3.3  Example: Longitudinal Mental Depression Study Revisited ... 520
		13.3.4  Example:  Capture-Recapture Prediction of Population  Size ... 521
		13.3.S  Example:  Heterogeneity Among Multicenter Clinical  Trials ... 523
		13.3.6  Meta-analysis Using a Random Effects Approach ... 525
		13.3.7  Alternative Formulations of Random Effects Models ... 525
		13.3.8  Example:  Matched  Pairs with a Bivariate  Binary Response ... 526
		13.3.9  Time Series Models Using Autocorrelated Random Eff ects ... 527
		13.3.10  Example:  Oxford and Cambridge Annual Boat Race ... 528
	13.4  RANDOM EFFECT S  MODELS  FOR MULTINOMIAL DATA ... 529
		13.4.1  Cumulative  Logit Model with Random Intercept ... 529
		13.4.2  Example:  Insomnia  Study  Revisited ... 529
		13.4.3  Example: Combining Measures on Ordinal Items ... 530
		13.4.4  Example:  Cluster Sampling ... 531
		13.4.S  Baseline-Category Logit  Models with Random Eff ects ... 532
		13.4.6  Example: Eff ectiveness of Housing Program ... 532
	13.5  MULTILEVEL MODELING ... 533
		13.5.1  Hierarchical Random Terms:  Partitioning Variability ... 534
		13.5.2  Example:  Children\'s Care for an Unmarried  Mother ... 534
	13.6  GLMM FITTING, INFERENCE, AND PREDICTION ... 537
		13.6.1  Marginal Likelihood and Maximum Likelihood Fitting ... 537
		13.6.2  Gauss-Hermite Quadrature Methods for ML Fitting ... 538
		13.6.3  Monte Carlo and EM Methods for ML Fitting ... 538
		13.6.4  Laplace and Penalized Quasi-likelihood Approximations to ML ... 539
		13.6.5  Inference for GLMM  Parameters ... 540
		13.6.6  Prediction Using Random Effects ... 540
	13.7  BAYESIAN  MULTIVARIATE  CATEGORICAL  MODELING ... 541
		13.7.1  Marginal Homogeneity  Analyses for  Matched Pairs ... 541
		13.7.2  Bayesian  Approaches  to Meta-analysis and  Multicenter  Trials ... 541
		13.7.3  Example:  Bayesian Analyses for a Multicenter Trial ... 542
		13.7.4  Bayesian  GLMMs and Marginal Models ... 542
	NOTES ... 543
	EXERCISES ... 545
CHAPTER  14 Other Mixture Models  for Discrete Data ... 553
	14.1  LATENT CLASS MODELS ... 553
		14.1.1  Independence Given a Latent Categorical Variable ... 554
		14.1.2  Fitting Latent  Class Models ... 555
		14.1.3  Example: Latent Class Model for Rater Agreement ... 556
		14.1.4  Example: Latent Class Models for Capture-Recapture ... 558
		14.1.5  Example: Latent Class Tr ansitional Models ... 559
	14.2  NONPARAMETRIC RANDOM EFFECTS  MODELS ... 560
		14.2.1  Logistic  Models  with  Unspecifi ed Random Eff ects Distribution ... 560
		14.2.2  Example:  Attitudes  About Legalized  Abortion ... 560
		14.2.3  Example:  Nonparametric  Mixing of Logistic  Regressions ... 561
		14.2.4  Is  Misspecifi cation  of Random Eff ects a Serious Problem? ... 561
		14.2.5  Rasch  Mixture  Model ... 563
		14.2.6  Example:  Modeling  Rater  Agreement  Revisited ... 563
		14.2.7  Nonparametric Mixtures and Quasi-symmetry ... 564
		14.2.8  Example: Attitudes About Legalized Abortion Revisited ... 565
	14.3  BETA-BINOMIAL MODELS ... 566
		14.3.1  Beta-Binomial Distribution ... 566
		14.3.2  Models Using the  Beta-Binomial  Distribution ... 567
		14.3.3  Quasi-likelihood with  Beta-Binomial  Ty pe  Variance ... 567
		14.3.4  Example: Teratology Overdispersion Revisited ... 568
		14.3.5  Conjugate  Mixture  Models ... 570
	14.4  NEGATIVE BINOMIAL REGRESSION ... 570
		14.4.1  Gamma Mixture of Poissons Is  Negative Binomial ... 571
		14.4.2  Negative Binomial Regression Modeling ... 571
		14.4.3  Example: Frequency of Knowing Homicide Victims ... 572
	14.5  POISSON REGRESSION WITH RANDOM EFFECTS ... 573
		14.5.1  A Poisson GLMM ... 574
		14.5.2  Marginal Model Implied by Poisson GLMM ... 574
		14.5.3  Example:  Homicide Victim Frequency  Revisited ... 575
		14.5.4  Negative Binomial Models  versus  Poisson GLMMs ... 575
	NOTES ... 575
	EXERCISES ... 576
CHAPTER  15 Non-Model-Based Classification and Clustering ... 583
	15.1  CLASSIFICATION: LINEAR DISCRIMINANT ANALYSIS ... 583
		15.1.1  Classifi cation with Normally Distributed Predictors ... 583
		15.1.2  Example: Horseshoe Crab Satellites Revisited ... 585
		15.1.3  Multicategory Classifi cation and Other Versions of Discriminant Analysis ... 586
		15.1.4  Classifi cation Methods for High  Dimensions ... 587
		15.1.5  Discriminant Analysis Versus Logistic Regression ... 587
	15.2  CLASSIFICATION:  TREE-STRUCTURED PREDICTION ... 588
		15.2.1  Classifi cation Trees ... 588
		15.2.2  Example:  Classifi cation Tree for a Health  Care Application ... 589
		15.2.3  How Does the Classifi cation Tree Grow? ... 590
		15.2.4  Pruning a Tree and Checking Prediction Accuracy ... 591
		15.2.5  Classifi cation Trees Versus Logistic Regression ... 592
		15.2.6  Support Vector Machines for Classifi cation ... 593
	15.3  CLUSTER ANALYSIS FOR CATEGORICAL DATA ... 594
		15.3.1  Supervised Versus Unsupervised Learning ... 595
		15.3.2  Measuring Dissimilarity Between Observations ... 595
		15.3.3  Clustering Algorithms: Partitions and Hierarchies ... 596
		15.3.4  Example:  Clustering States on  Election  Results ... 597
	NOTES ... 599
	EXERCISES ... 600
CHAPTER  16 Large- and Small-Sample Theory for Multinomial Models ... 605
	16.1  DELTA METHOD ... 605
		16.1.1  0,  o  Rates of Convergence ... 606
		16.1.2  Delta Method for a Function of a Random Variable ... 606
		16.1.3  Delta Method for a Function of a Random Vector ... 607
		16.1.4  Asymptotic Normality of Functions of Multinomial Counts ... 607
		16.1.S  Delta Method for a Vector Function of a Random Vector ... 609
		16.1.6  Joint Asymptotic Normality of Log Odds Ratios ... 609
	16.2  ASYMPTOTIC DISTRIBUTIONS  OF ESTIMATORS  OF MODEL PARA METERS AND CELL PROBABILITIES ... 610Black,notBold,notItalic,closed,TopLeftZoom,2,2,0.0
		16.2.1  A symptotic Distribution  of Model  Parameter  Estimator ... 610Black,notBold,notItalic,open,TopLeftZoom,845,2,0.0
		16.2.2  Asymptotic Distribution of Cell Probability Estimators ... 611
		16.2.3  Model Smoothing Is Benefi cial ... 612
	16.3  ASYMPTOTIC DISTRIBUTIONS  OF RESIDUALS  AND GOODNESS-OF-FIT STATISTICS ... 612
		16.3.1  Joint  Asymptotic Normality of p and ii: ... 612
		16.3.2  Asymptotic Distribution of Pearson and Standardized Residuals ... 613
		16,3,3  Asymptotic Distribution of Pearson X2  Statistic ... 614
		16.3.4  Asymptotic Distribution of Likelihood-Ratio Statistic ... 615
		16.3.5  Asymptotic  Noncentral Distributions ... 616
	16.4  ASYMPTOTIC DISTRIBUTIONS FOR LOGIT - LOGLINEAR MODELS ... 617
		16.4.1  Asymptotic  Covariance Matrices ... 617
		16.4.2  Connection with Poisson Loglinear Models ... 618
	16.5  SMALL-SAMPLE SIGNIFICANCE TESTS FOR CONTINGENCY TABLES ... 619
		16.5.1  Exa ct Conditional  Distribution for  IxJ Tables Under  Independence ... 619
		16.5.2  Exact Tests of Independence for IxJ  Tables ... 620
		16.5.3  Example: Sexual Orientation and Party ID ... 620
	16.6  SMALL-SAMPLE CONFIDENCE INTERVALS  FOR CATEGORICAL DATA ... 621
		16.6.1  Small-Sample Cis for a Binomial Parameter ... 621
		16.6.2  Cls Based on Tests Using the Mid P-Value ... 623
		16.6.3  Example: Proportion of Vegetarians Revisited ... 623
		16.6.4  Small-Sample Cls for Odds Ratios ... 624
		16.6.5  Example: Fisher\'s Tea Taster Revisited ... 625
		16.6.6  Small-Sample Cls for Logistic Regression Parameters ... 625
		16.6.7  Example:  Diarrhea and an  Antibiotic ... 626
		16.6.8  Unconditional Small-Sample  Cls for Difference  of Proportions ... 627
	16.7  ALTERNATIVE ESTIMATION  THEORY FOR  PARAMETRIC MODELS ... 628
		16.7.1  Weighted Least Squares for Categorical Data ... 628
		16.7.2  Inference Using the WLS Approach to Model Fitting ... 629
		16.7.3  Scope of WLS Versus ML Estimation ... 630
		16.7.4  Minimum Chi-Squared Estimators ... 631
		16.7.S  Minimum Discrimination Information ... 632
	NOTES ... 633
	EXERCISES ... 634
CHAPTER  17 Historical Tour of Categorical Data Analysis ... 641
	17.1  PEARSON-YULE ASSOCIATION CONTROVERSY ... 641
	17.2  R. A. FISHER\'S CONTRIBUTIONS ... 643
	17.3  LOGISTIC REGRESSION ... 645
	17.4  MULTIWAY CONTINGENC Y  TABLES  AND  LOGLINEAR  MODEL S ... 647
	17.5  BAYESIAN METHODS FOR  CATEGORICAL DATA ... 651
	17.6  A LOOK FORWARD, AND BACKWARD ... 652
APPENDIX  A Statistical  Software for Categorical Data Analysis ... 655
References ... 661
Author Index ... 707
Subject Index ... 723