Dongchu Sun

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.

THE FORMAL DEFINITION OF REFERENCE PRIORS

Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically.

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.

Bayesian analysis for the Poly-Weibull distribution

Abstract In this article Bayesian analysis for a Poly-Weibull distribution using informative priors is discussed. This distribution typically arises when the data is the minimum of several Weibull failure times from competing risks. To perform the Bayesian computations, simulation using the Gibbs sampler is suggested. This can be used to find posterior moments, the marginal posterior probability density function, and the predictive risk or reliability.

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 hierarchical bayesian statistical framework for response time distributions

This paper provides a statistical framework for estimating higher-order characteristics of the response time distribution, such as the scale (variability) and shape. Consideration of these higher order characteristics often provides for more rigorous theory development in cognitive and perceptual psychology (e.g., Luce, 1986). RT distribution for a single participant depends on certain participant characteristics, which in turn can be thought of as arising from a distribution of latent variables. The present work focuses on the three-parameter Weibull distribution, with parameters for shape, scale, and shift (initial value). Bayesian estimation in a hierarchical framework is conceptually straightforward. Parameter estimates, both for participant quantities and population parameters, are obtained through Markov Chain Monte Carlo methods. The methods are illustrated with an application to response time data in an absolute identification task. The behavior of the Bayes estimates are compared to maximum likelihood (ML) estimates through Monte Carlo simulations. For small sample size, there is an occasional tendency for the ML estimates to be unreasonably extreme. In contrast, by borrowing strength across participants, Bayes estimation “shrinks” extreme estimates. The results are that the Bayes estimators are more accurate than the corresponding ML estimators.

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.

Objective priors for the bivariate normal model

Summary Objective Bayesian inference for the multivariate normal distribution is illustrated, using dieren t types of formal objective priors (Jereys, invariant, reference and matching), dieren t modes of inference (Bayesian and frequentist), and dieren t criteria involved in selecting optimal objective priors (ease of computation, frequentist performance, marginalization paradoxes, and decision-theoretic evaluation). In the course of the investigation of the bivariate normal model in Berger and Sun (2006), a variety of surprising results were found, including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters, such as the correlation coecien t. Certain of these results are generalized to the multivariate normal situation. The prior that most frequently yields exact frequentist inference is the rightHaar prior, which unfortunately is not unique. Two natural proposals are studied for dealing with this non-uniqueness: rst, mixing over the right-Haar priors; second, choosing the ‘empirical Bayes’ right-Haar prior, that which maximizes the marginal likelihood of the data. Quite surprisingly, we show that neither of these possibilities yields a good solution. This is disturbing and sobering. It is yet another indication that improper priors do not behave as do proper priors, and that it can be dangerous to apply ‘understandings’ from the world of proper priors to the world of improper priors.

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.

Noninformative priors and frequentist risks of bayesian estimators of vector-autoregressive models

In this study, we examine posterior properties and frequentist risks of Bayesian estimators based on several noninformative priors in vector autoregressive (VAR) models. We prove existence of the posterior distributions and posterior moments under a general class of priors. Using a variety of priors in this class we conduct numerical simulations of posteriors. We find that in most examples Bayesian estimators with a shrinkage prior on the VAR coefficients and the reference prior of Yang and Berger (Ann. Statist. 22 (1994) 1195) on the VAR covariance matrix dominate MLE, Bayesian estimators with the diffuse prior, and Bayesian estimators with the prior used in RATS. We also examine the informative Minnesota prior and find that its performance depends on the nature of the data sample and on the tightness of the Minnesota prior. A tightly set Minnesota prior is better when the data generating processes are similar to random walks, but the shrinkage prior or constant prior can be better otherwise.