Robert K. Tsutakawa

Estimation in Covariance Components Models

Abstract Estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes. The estimation of fixed and random effects when the variances and covariances are known is presented in Bayesian terms, Point estimates of the unknown variances and covariances are computed using the EM algorithm for maximum likelihood estimation from incomplete data. The techniques are illustrated with data on law schools, field mice, and professional football teams.

The effect of uncertainty of item parameter estimation on ability estimates

The conventional method of measuring ability, which is based on items with assumed true parameter values obtained from a pretest, is compared to a Bayesian method that deals with the uncertainties of such items. Computational expressions are presented for approximating the posterior mean and variance of ability under the three-parameter logistic (3PL) model. A 1987 American College Testing Program (ACT) math test is used to demonstrate that the standard practice of using maximum likelihood or empirical Bayes techniques may seriously underestimate the uncertainty in estimated ability when the pretest sample is only moderately large.

Spatio-temporal interaction with disease mapping.

Markov chain Monte Carlo methods are used to estimate mortality rates under a Bayesian hierarchical model. Spatial correlations are introduced to examine spatial effects relative to both regional and regional changes over time by groups. A special feature of the models is the inclusion of longitudinal variables which will describe temporal trends in mortality or incidences for different population groups. Disease maps are used to illustrate the role of different parameters in the model and pinpointing areas of interesting patterns. The methods are demonstrated by male cancer mortality data from the state of Missouri during 1973-1992. Of special interest will be the geographic variations in the trend of lung cancer mortality over the recent past. Marginal posterior distributions are used to examine effects due to spatial correlations and age difference in temporal trends. Numerical results from the Missouri data show that although spatial correlations exist, they do not have a large effect on the estimated mortality rates.

A Bivariate Bayes Method for Improving the Estimates of Mortality Rates With a Twofold Conditional Autoregressive Model

A bivariate Bayes method is proposed for estimating the mortality rates of a single disease for a given population, using additional information from a second disease. The information on the two diseases is assumed to be from the same population groups or areas. The joint frequencies of deaths for the two diseases for given populations are assumed to have a bivariate Poisson distribution with joint means proportional to the population sizes. The relationship between the mortality rates of the two different diseases if formulated through the twofold conditional autoregressive (CAR) model, where spatial effects as well as indexes of spatial dependence are introduced to capture the structured clusterings among areas. This procedure is compared to a univariate hierarchical Bayes procedure that uses information from one disease only. Comparisons of two procedures are made by the optimal property, a Monte Carlo study, real data, and the Bayes factor. All of the methods that we consider demonstrate a substantial improvement in the bivariate over the univariate procedure. For analyzing male and female lung cancer data from the state of Missouri, Markov chain Monte Carlo methods are used to estimate mortality rates.

Parameter Estimation in Latent Trait Models.

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parametersθ=(θ1, …,θn) and item parametersβ=(β1, …,βk), whereβj may be vector valued. Withθ considered a random sample from a prior distribution with parameterφ, the estimation of (θ, β) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.

BAYESIAN ESTIMATION OF ITEM RESPONSE CURVES

Item response curves for a set of binary responses are studied from a Bayesian viewpoint of estimating the item parameters. For the two-parameter logistic model with normally distributed ability, restricted bivariate beta priors are used to illustrate the computation of the posterior mode via the EM algorithm. The procedure is illustrated by data from a mathematics test.

Design of Experiment for Bioassay

Abstract The one-parameter logistic distribution is used to illustrate certain numerical approximations for finding one- and two-stage bioassay designs which produce small posterior variances. The article discusses the use of two prior distributions, one for design and another for inference. Graphs are given for designing experiments when the prior distributions are normal. These graphs illustrate the importance of using additional dose levels when the variance of the prior distribution is large.

Approximation for Bayesian Ability Estimation.

An approximation is proposed for the posterior mean and standard deviation of the ability parameter in an item response model. The procedure assumes that approximations to the posterior mean and covariance matrix of item parameters are available. It is based on the posterior mean of a Taylor series approximation to the posterior mean conditional on the item parameters. The method is illustrated for the two-parameter logistic model using data from an ACT math test with 39 items. A numerical comparison with the empirical Bayes method using n = 400 examinees shows that the point estimates are very similar but the standard deviations under empirical Bayes are about 2% smaller than those under Bayes. Moreover, when the sample size is decreased to n = 100, the standard deviation under Bayes is shown to increase by 14% in some cases.

Estimation of Two-Parameter Logistic Item Response Curves.

Under the assumption that ability parameters are sampled from a normal distribution, the EM algorithm is used to derive maximum likelihood estimates for item parameters of the two-parameter logistic item response curves. The observed information matrix is then used to approximate the covariance matrix of these estimates. Responses to a questionnaire on general arthritis knowledge are used to illustrate the procedure and simulated data are used to compare the estimated and actual item parameters. The resulting estimates are found to be very close to those obtained from LOGIST. A computational note is included to facilitate the extensive numerical work required to implement the procedure.

Lognormal vs. Gamma: Extra Variations

Within a Bayesian framework of hierarchical modeling, the inclusion of extra variation effects becomes popular with Poisson and Binomial sampling processes. For the less populated areas, mortality rates are heterogeneous due to environmental effects or other socio-economic status. Thus, the extra variation in the frequency of deaths will usually exceed that expected from sampling distributions. In this paper, we propose a quasi-multiplicative spatio-temporal model (PGC) with gamma extra variation effects. Then we compare the performance of proposed model to loglinear model (PLC) with lognormal extra variation effects in Sun et al. (2000). Gibbs sampling is used to compute the posterior moments and marginal posterior densities. The numerical results based on Missouri male lung cancer data show that PGC and PLC models are almost interchangeable. The extra variation effects are important to predict the mortality rates adequately under both models.