Ordinal Data Modeling

by Johnson, Valen E.; Albert, James H.

• ISBN13: 9780387987187
• ISBN10: 0387987185
• Format: Hardcover
• Copyright: 1999-04-01
• Publisher: Springer Verlag

Summary

Ordinal Data Modelingis a comprehensive treatment of ordinal data models from both likelihood and Bayesian perspectives. Written for graduate students and researchers in the statistical and social sciences, this book describes a coherent framework for understanding binary and ordinal regression models, item response models, graded response models, and ROC analyses, and for exposing the close connection between these models. A unique feature of this text is its emphasis on applications. All models developed in the book are motivated by real datasets, and considerable attention is devoted to the description of diagnostic plots and residual analyses. Software and datasets used for all analyses described in the text are available on websites listed in the preface.

Table of Contents

Preface	v
Review of Classical and Bayesian Inference	1	(32)
Learning about a binomial proportion	1	(17)
Sampling: The binomial distribution	3	(1)
The likelihood function	3	(1)
Maximum likelihood estimation	4	(2)
The sampling distribution of the MLE	6	(1)
Classical point and interval estimation for a proportion	6	(1)
Bayesian inference	7	(1)
The prior density	7	(2)
Updating one's prior beliefs	9	(1)
Posterior densities with alternative priors	10	(3)
Summarizing the posterior density	13	(4)
Prediction	17	(1)
Inference for a normal mean	18	(5)
A classical analysis	19	(2)
Bayesian analysis	21	(2)
Inference about a set of proportions	23	(4)
Further reading	27	(1)
Exercises	28	(5)
Review of Bayesian Computation	33	(42)
Integrals, integrals, integrals,	34	(1)
An example	35	(2)
Non-Simulation-Based Algorithms	37	(6)
The Multivariate normal approximation	37	(3)
Grid integration	40	(3)
Comments about the two computational methods	43	(1)
Direct Simulation	43	(10)
Simulating random variables	44	(2)
Inference based on simulated samples	46	(1)
Inference for a binomial proportion	46	(2)
Accuracy of posterior simulation computations	48	(1)
Direct simulation for a multiparameter posterior: The composition method	49	(1)
Inference for a normal mean	49	(1)
Direct simulation for a multiparameter posterior with independent components	49	(1)
Smoking example (continued)	50	(3)
Markov Chain Monte Carlo	53	(12)
Introduction	53	(1)
Metropolis-Hastings sampling	54	(4)
Gibbs sampling	58	(4)
Output analysis	62	(3)
A two-stage exchangeable model	65	(3)
Further reading	68	(1)
Appendix: Iterative implementation of Gauss-Hermite quadrature	68	(1)
Exercises	69	(6)
Regression Models for Binary Data	75	(51)
Basic modeling considerations	76	(6)
Link functions	77	(5)
Grouped data	82	(1)
Estimating binary regression coefficients	82	(8)
The likelihood function	82	(2)
Maximum likelihood estimation	84	(1)
Bayesian estimation and inference	85	(2)
An example	87	(3)
Latent variable interpretation of binary regression	90	(2)
Residual analysis and goodness of fit	92	(16)
Case analysis	93	(9)
Goodness of fit and model selection	102	(6)
An example	108	(7)
A note on retrospective sampling and logistic regression	115	(3)
Further reading	118	(1)
Appendix: iteratively reweighted least squares	118	(2)
Exercises	120	(6)
Regression Models for Ordinal Data	126	(32)
Ordinal data via latent variables	127	(4)
Cumulative probabilities and model interpretation	130	(1)
Parameter constraints and prior models	131	(2)
Noninformative priors	131	(1)
Informative priors	132	(1)
Estimation strategies	133	(4)
Maximum likelihood estimation	133	(1)
MCMC sampling	134	(3)
Residual analysis and goodness of fit	137	(2)
Examples	139	(9)
Grades in a statistics class	139	(9)
Prediction of essay scores from grammer attributes	148	(5)
Further reading	153	(1)
Appendix: iteratively reweighted least squares	153	(2)
Exercises	155	(3)
Analyzing Data from Multiple Raters	158	(24)
Essay scores from five raters	159	(2)
The multiple rater model	161	(6)
The likelihood function	161	(2)
The prior	163	(3)
Analysis of essay scores from five raters (without regression)	166	(1)
Incorporating regression functions into multirater data	167	(7)
Regression of essay grades obtained from five raters	171	(3)
ROC analysis	174	(6)
Further reading	180	(1)
Exercises	180	(2)
Item Response Modeling	182	(33)
Introduction	182	(1)
Modeling the probability of a correct response	183	(5)
Latent trait	183	(1)
Item response curve	184	(1)
Item characteristics	184	(4)
Modeling test results for a group of examinees	188	(1)
Data structure	188	(1)
Model assumptions	188	(1)
Classical estimation of item and ability parameters	189	(2)
Bayesian estimation of item parameters	191	(3)
Prior distributions on latent traits	191	(1)
Prior distributions on item parameters	192	(1)
Posterior distributions	193	(1)
Describing item response models (probit link)	193	(1)
Estimation of model parameters (probit link)	194	(3)
A Gibbs sampling algorithm	195	(2)
An example	197	(5)
Posterior inference	199	(3)
One-parameter (item response) models	202	(2)
The Rasch model	203	(1)
A Bayesian fit of the probit one-parameter model	203	(1)
Three-parameter item response models	204	(1)
Model checking	205	(2)
Bayesian residuals	205	(1)
Example	206	(1)
An exchangeable model	207	(4)
Prior belief of exchangeability	207	(2)
Application of a hierarchical prior model to the shyness data	209	(2)
Further reading	211	(1)
Exercises	211	(4)
Graded Response Models: A Case Study of Undergraduate Grade Data	215	(24)
Background	217	(3)
Previously proposed methods for grade adjustment	218	(2)
A Bayesian graded response model	220	(5)
Achievement indices and grade cutoffs	220	(2)
Prior distributions	222	(3)
Parameter estimation	225	(1)
Posterior simulation	225	(1)
Posterior optimization	226	(1)
Applications	226	(5)
Larkey and Caulkin data	227	(2)
A Class of Duke University undergraduates	229	(2)
Alternative models and sensitivity analysis	231	(4)
Discussion	235	(1)
Appendix: selected transcripts of Duke University undergraduates	236	(3)
Appendix: Software for Ordinal Data Modeling	239	(10)
References	249	(6)
Index	255