Statistical Inference: An Integrated Bayesian/Likelihood Approach

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ISBN: 9781420093438 | 1420093436
Cover: Hardcover
Copyright: 6/2/2010

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This book sets out an integrated approach to statistical inference using the likelihood function as the primary measure of evidence for statistical model parameters, and for the statistical models themselves. The author provides both an alternative to standard Bayesian inference and the foundation for a course sequence in modern Bayesian theory at the graduate or advanced undergraduate level. The restriction of the book to evidence is deliberate: there are already many books on Bayesian and non-Bayesian decision theory, and the purpose of this one is less ambitious, but perhaps more relevant scientifically, in providing a detailed prescription for the assessment of statistical evidence.

Preface	p. xiii
Theories of Statistical Inference	p. 1
Example	p. 1
Statistical models	p. 1
The likelihood function	p. 2
Theories	p. 3
Pure likelihood theory	p. 3
Bayesian theory	p. 5
Likelihood-based repeated sampling theory	p. 10
"Model-guided" survey sampling theory	p. 12
Nonmodel-based repeated sampling	p. 18
Conclusion	p. 19
The Integrated Bayes/Likelihood Approach	p. 21
Introduction	p. 21
Probability	p. 21
Prior ignorance	p. 23
The importance of parametrization	p. 24
Inference about the Binomial N	p. 25
Profiling	p. 26
Conditioning	p. 26
The effect size	p. 31
The simple/simple hypothesis testing problem	p. 34
Bayes calibration	p. 35
Frequentist calibration (fixed sample size)	p. 36
Nonfixed error probabilities	p. 37
Frequentist test (sequential sampling)	p. 38
Bayesian interpretation of type I error probabilities	p. 39
The simple/composite hypothesis testing problem	p. 40
Fixed-sample frequentist analysis	p. 40
Sequential sampling approach	p. 41
Posterior likelihood approach	p. 41
Large-sample result	p. 43
Strength of support for the null hypothesis	p. 44
Credible intervals for the deviance difference	p. 44
Bayes factors	p. 45
Difficulties with the Bayes factor	p. 47
Conjugate prior difficulties	p. 50
Stone example	p. 51
The comparison of unrelated models	p. 53
Large-sample result	p. 53
Bayes factor	p. 54
Example	p. 55
Misleading conclusions from Bayes factors	p. 58
Modified Bayes factors	p. 63
Example - GHQ score and psychiatric diagnosis	p. 64
t-Tests and Normal Variance Tests	p. 69
One-sample t-test	p. 69
Credible interval	p. 69
Model comparisons	p. 70
Example	p. 71
Two samples: equal variances	p. 72
Credible interval	p. 72
Model comparisons	p. 73
Example	p. 74
The two-sample test	p. 76
Informative prior for the effect size	p. 76
Two samples: different variances	p. 77
Credible interval	p. 77
Model comparison	p. 78
Example	p. 79
The normal model variance	p. 81
Credible interval	p. 81
Model comparisons	p. 81
Example	p. 82
Marginal and full likelihoods	p. 83
Variance heterogeneity test	p. 85
Nonrobustness of variance tests	p. 87
Unified Analysis of Finite Populations	p. 91
Sample selection indicators	p. 92
The Bayesian bootstrap	p. 95
Multinomial model	p. 96
Dirichlet prior	p. 97
Confidence coverage of credible intervals	p. 98
Example - income population	p. 99
Simulation study	p. 105
Extensions of the Bayesian bootstrap	p. 107
Sampling without replacement	p. 107
Simulation study	p. 109
Regression models	p. 110
Design-based approach	p. 112
Bayesian bootstrap approach	p. 112
Simulation study	p. 113
Ancillary information	p. 116
Sampling without replacement	p. 116
Simulation study	p. 118
More general regression models	p. 118
The multinomial model for multiple populations	p. 119
Complex sample designs	p. 120
Stratified sampling	p. 120
Cluster sampling	p. 121
Shrinkage estimation	p. 125
A complex example	p. 127
Discussion	p. 130
Regression and Analysis of Variance	p. 133
Multiple regression	p. 133
Nonnested models	p. 137
Binomial and Multinomial Data	p. 143
Single binomial samples	p. 143
Bayes factor	p. 144
Example	p. 144
Single multinomial samples	p. 146
Two-way tables for correlated proportions	p. 147
Likelihood	p. 147
Bayes factor	p. 148
Posterior likelihood ratio	p. 148
Multiple binomial samples	p. 150
A social network table	p. 150
Network model	p. 150
Frequentist analysis	p. 152
Choice of alternative model prior	p. 152
Simulations	p. 153
Two-way tables for categorical responses - no fixed margins	p. 154
The ECMO study	p. 155
Bayes analysis	p. 155
Multinomial likelihoods	p. 156
Dirichlet prior	p. 156
Simulations	p. 157
Two-way tables for categorical responses - one fixed margin	p. 160
Multinomial "nonparametric" analysis	p. 164
Goodness of Fit and Model Diagnostics	p. 169
Frequentist model diagnostics	p. 169
Bayesian model diagnostics	p. 171
The posterior predictive distribution	p. 172
Marginalization and model diagnostics	p. 174
Multinomial deviance computation	p. 179
Model comparison through posterior deviances	p. 181
Examples	p. 182
Three binomial models	p. 182
Poisson model	p. 186
Simulation study	p. 188
Discussion	p. 194
Complex Models	p. 197
The data augmentation algorithm	p. 197
Two-level variance component models	p. 198
Two-level fixed effects model	p. 199
Two-level random effects model	p. 199
Posterior inference	p. 200
Likelihood	p. 200
Maximum likelihood estimates	p. 201
Posteriors	p. 201
Box-Tiao example	p. 202
Test for a zero variance component	p. 208
Alternative tests	p. 209
Generalized linear mixed models	p. 210
Finite mixtures	p. 210
Example - the galaxy velocity study	p. 211
Data examination	p. 211
Maximum likelihood estimates	p. 212
Bayes analysis	p. 213
Posterior likelihood analysis	p. 215
Simulation studies	p. 221
References	p. 223
Author Index	p. 229
Subject Index	p. 233
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Statistical Inference: An Integrated Bayesian/Likelihood Approach

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