Ordinal Data Modeling

Transcription

1 Valen E. Johnson James H. Albert Ordinal Data Modeling With 73 illustrations I ". Springer

2 Contents Preface v 1 Review of Classical and Bayesian Inference Learning about a binomial proportion Sampling: The binomial distribution The likelihood function Maximum likelihood estimation The sampling distribution of the MLE Classical point and interval estimation for a proportion Bayesian inference The prior density Updating one's prior beliefs Posterior densities with alternative priors Summarizing the posterior density Prediction ~> Inference for a normal mean A classical analysis Bayesian analysis Inference about a set of proportions Further reading Exercises 28 2 Review of Bayesian Computation Integrals, integrals, integrals, 34

3 viii Contents 2.2 An example Non-Simulation-Based Algorithms The Multivariate normal approximation Grid integration, Comments about the two computational methods Direct Simulation Simulating random variables Inference based on simulated samples Inference for a binomial proportion Accuracy of posterior simulation computations Direct simulation for a multiparameter posterior: The composition method. > Inference for a normal mean Direct simulation for a multiparameter posterior with independent components Smoking example (continued) Markov Chain Monte Carlo Introduction Metropolis-Hastings sampling Gibbs sampling Output analysis A two-stage exchangeable model Further reading Appendix: Iterative implementation of Gauss-Hermite quadrature Exercises 69 3 Regression Models for Binary Data Basic modeling considerations Link functions Grouped data Estimating binary regression coefficients The likelihood function Maximum likelihood estimation Bayesian estimation and inference An example Latent variable interpretation of binary regression Residual analysis and goodness of fit Case analysis Goodness of fit and model selection An example A note on retrospective sampling and logistic regression Further reading Appendix: iteratively reweighted least squares Exercises 120

4 Contents ix Regression Models for Ordinal Data Ordinal data via latent variables Cumulative probabilities and model interpretation Parameter constraints and prior models Noninformative priors Informative priors Estimation strategies Maximum likelihood estimation MCMC sampling Residual analysis and goodness of fit Examples Grades in a statistics class Prediction of essay scores from grammar attributes Further reading : Appendix: iteratively reweighted least squares Exercises 155 Analyzing Data from Multiple Raters Essay scores from five raters The multiple rater model The likelihood function The prior Analysis of essay scores from five raters (without regression) Incorporating regression functions into multirater data Regression of essay grades obtained from five raters ROC analysis Further reading Exercises 180 Item Response Modeling Introduction Modeling the probability of a correct response Latent trait Item response curve Item characteristics Modeling test results for a group of examinees Data structure Model assumptions Classical estimation of item and ability parameters Bayesian estimation of item parameters Prior distributions on latent traits Prior distributions on item parameters Posterior distributions Describing item response models (probit link) 193

5 x Contents 6.6 Estimation of model parameters (probit link) A Gibbs sampling algorithm An example Posterior inference One-parameter (item response) models The Rasch model A Bayesian fit of the probit one-parameter model Three-parameter item response models Model checking Bayesian residuals Example An exchangeable model Prior belief of exchangeability Application of a hierarchical prior model to the shyness data Further reading Exercises Graded Response Models: A Case Study of Undergraduate Grade Data Background Previously proposed methods for grade adjustment A Bayesian graded response model Achievement indices and grade cutoffs.' Prior distributions Parameter estimation Posterior simulation Posterior optimization Applications Larkey and Caulkin data A Class of Duke University undergraduates Alternative models and sensitivity analysis Discussion Appendix: selected transcripts of Duke University undergraduates 236 Appendix: Software for Ordinal Data Modeling 239 References 249 Index 255