A Note on the Existence of the Posteriors for One-way Random Effect Probit Models.

The existence of the posterior distribution for one-way random effect probit models has been investigated when the uniform prior is applied to the overall mean and a class of noninformative priors are applied to the variance parameter. The sufficient conditions to ensure the propriety of the posterior are given for the cases with replicates at some factor levels. It is shown that the posterior distribution is never proper if there is only one observation at each factor level. For this case, however, a class of proper priors for the variance parameter can provide the necessary and sufficient conditions for the propriety of the posterior.

A Bayesian hierarchical model of nontraumatic lower-extremity amputation rates.

A Bayesian hierarchical generalized linear model is used to estimate the risk of lower-extremity amputations (LEA) among diabetes patients from different counties in the state of Missouri. The model includes fixed age effects, fixed gender effect, random geographic effects, and spatial correlations between neighboring counties. The computation is done by Gibbs sampling using OPENBUGS. DIC (Deviance Information Criterion) is used as a criterion of goodness of fit to examine age effects, gender effect, and spatial correlations among counties in the risks of having LEAs. The Bayesian estimates are also shown to be quite robust in terms of choices of hyper-parameters.

Bayesian t tests for accepting and rejecting the null hypothesis.

Progress in science often comes from discovering invariances in relationships among variables; these invariances often correspond to null hypotheses. As is commonly known, it is not possible to state evidence for the null hypothesis in conventional significance testing. Here we highlight a Bayes factor alternative to the conventional t test that will allow researchers to express preference for either the null hypothesis or the alternative. The Bayes factor has a natural and straightforward interpretation, is based on reasonable assumptions, and has better properties than other methods of inference that have been advocated in the psychological literature. To facilitate use of the Bayes factor, we provide an easy-to-use, Web-based program that performs the necessary calculations.

A hierarchical process-dissociation model.

In fitting the process-dissociation model (L. L. Jacoby, 1991) to observed data, researchers aggregate outcomes across participant, items, or both. T. Curran and D. L. Hintzman (1995) demonstrated how biases from aggregation may lead to artifactual support for the model. The authors develop a hierarchical process-dissociation model that does not require aggregation for analysis. Most importantly, the Curran and Hintzman critique does not hold for this model. Model analysis provides for support of process dissociation--selective influence holds, and there is a dissociation in correlation patterns among participants and items. Items that are better recollected also elicit higher automatic activation. There is no correlation, however, across participants; that is, participants with higher recollection have no increased tendency toward automatic activation. The critique of aggregation is not limited to process dissociation. Aggregation distorts analysis in many nonlinear models, including signal detection, multinomial processing tree models, and strength models. Hierarchical modeling serves as a general solution for accurately fitting these psychological-processing models to data.

A Bayesian semi-parametric model for colorectal cancer incidences.

A Bayesian semi-parametric model is proposed to capture the interaction among demographic effects (age and gender), spatial effects (county) and temporal effects of colorectal cancer incidences simultaneously. In particular, an extension of multivariate conditionally autoregressive (CAR) processes to a partially informative Gaussian demographic spatial temporal CAR (DSTCAR) process for a spatial-temporal setting is proposed. The precision matrix of the Gaussian DSTCAR process is the Kronecker product of several components. The spatial component is modelled with a CAR prior. A pth order intrinsic autoregressive prior (IAR(p)) is implemented for the temporal component to estimate a smoothed and non-parametric temporal trend. The demographic component is modelled with a Wishart prior. Data analysis shows significant spatial correlation only exists in the age group of 50-59. Males and females in their 50s and 60s show fairly strong correlation. The hypothesis testing based on Bayes factor suggests that gender correlation cannot be ignored in this model.

A hierarchical model for estimating response time distributions.

We present a statistical model for inference with response time (RT) distributions. The model has the following features. First, it provides a means of estimating the shape, scale, and location (shift) of RT distributions. Second, it is hierarchical and models between-subjects and within-subjects variability simultaneously. Third, inference with the model is Bayesian and provides a principled and efficient means of pooling information across disparate data from different individuals. Because the model efficiently pools information across individuals, it is particularly well suited for those common cases in which the researcher collects a limited number of observations from several participants. Monte Carlo simulations reveal that the hierarchical Bayesian model provides more accurate estimates than several popular competitors do. We illustrate the model by providing an analysis of the symbolic distance effect in which participants can more quickly ascertain the relationship between nonadjacent digits than that between adjacent digits.

Hierarchical models for the probabilities of conception.

In the last thirty years, there has been considerable interest in finding better models to fit data for probabilities of conception. An important early model was proposed by Barrett and Marshall (1969) and extended by Schwartz, MacDonald and Heuchel (1980). Recently, researchers have further extended these models by adding covariates. However, the increasingly complicated models are challenging to analyze with frequentist methods such as the EM algorithm. Bayesian models are more feasible, and the computation can be done via Markov chain Monte Carlo (MCMC). We consider a Bayesian model with an effect for protected intercourse to analyze data from the California Women's Reproductive Health Study and assess the effects of water contaminants and hormones. There are two main contributions in the paper. (1) For protected intercourse, we propose modeling the ratios of daily conception probabilities with protected intercourse to corresponding daily conception probabilities with unprotected intercourse. Due to the small sample size of our data set, we assume the ratios are the same for each day but unknown. (2) We consider Bayesian analysis under a unimodality assumption where the probabilities of conception increase before ovulation and decrease after ovulation. Gibbs sampling is used for finding the Bayesian estimates. There is some evidence that the two covariates affect fecundability.

Bayesian inference on the patient population size given list mismatches.

In applying capture-recapture methods for closed populations to epidemiology, one needs to estimate the total number of people with a certain disease in a certain research area by using several lists with information of patients. Problems of lists error often arise due to mistyping or misinformation. Adopting the concept of tag-loss methodology in animal populations, Seber et al. (Biometrics 2000; 56:1227-1232) proposed solutions to a two-list problem. This article reports an interesting simulation study, where Bayesian point estimates based on improper constant and Jeffreys prior for unknown population size N could have smaller frequentist standard errors and MSEs compared to the estimates proposed in Seber et al. (2000). The Bayesian credible intervals based on the same priors also have super frequentist coverage probabilities while some of the frequentist confidence intervals procedures have drastically poor coverage. Seber's real data set on gestational diabetics is analysed with the proposed new methods.

Small-area incidence trends in breast cancer.

BACKGROUND: During the past 2 decades, the observed incidence of in situ and early-stage invasive breast cancer has increased substantially as a result of increased use of mammography. Geographic variability in the increase in breast cancer incidence has been observed among large areas. Examining the variability among small areas in the incidence over time will facilitate appropriate geographic allocation of resources aimed at increasing screening. METHODS: We examined county-specific increases in breast cancer incidence over time, specifically the variability and spatial correlation in the increase in breast cancer incidence. The analyses were based on county-level data (1973-1997) from the Iowa Surveillance, Epidemiology, and End Results program. A spatiotemporal hierarchical Bayesian model was used to examine variability in county-specific rates (intercepts, slopes, and spatial correlations) among white women at least 40 years of age. RESULTS: Posterior values indicate there was little variability among counties in the change in breast cancer incidence over time (slope) but substantial variation among intercepts. There was considerable spatial correlation among the county-specific intercepts but a lack of a spatial correlation among the county-specific slopes. There was no correlation between the county-specific intercept and slope. CONCLUSIONS: Breast cancer incidence increased over time, but county-specific rates increased independently relative to their neighboring counties or their initial rate.