A Bayesian Multiple Comparison Procedure for Order‐Restricted Mixed Models

Abstract

A Bayesian hierarchical mixed model is developed for multiple comparisons under a simple order restriction. The model facilitates inferences on the successive differences of the population means, for which we choose independent prior distributions that are mixtures of an exponential distribution and a discrete distribution with its entire mass at zero. We employ Markov Chain Monte Carlo (MCMC) techniques to obtain parameter estimates and estimates of the posterior probabilities that any two of the means are equal. The latter estimates allow one both to determine if any two means are significantly different and to test the homogeneity of all of the means. We investigate the performance of the model-based inferences with simulated data sets, focusing on parameter estimation and successive-mean comparisons using posterior probabilities. We then illustrate the utility of the model in an application based on data from a study designed to reduce lead blood concentrations in children with elevated levels. Our results show that the proposed hierarchical model can effectively unify parameter estimation, tests of hypotheses and multiple comparisons in one setting. Copyright (c) 2008 The Authors. Journal compilation (c) 2008 International Statistical Institute.

A Bayesian Multiple Comparison Procedure for Simple Order-Restricted Mixed Models with Missing Values

“…Prior to the continuity of the next derivation, we may comment on this prior. Due to the simple order restriction held between the means, i.e., µ 1 ≤ • • • ≤ µ k , the utility of a discrete-continuous mixture prior can effectively reflect this relationship and limit the difference between any two successive means to two possible values, zero or a positive number (see Gottardo and Raftery (2008), Shang et al (2008) , and Nashimoto and Wright (2008)). Therefore, from this point of view, the proposed mixture prior will play an important role in contributing to handling the problem of multiple comparisons for the simple order restricted means in the mixed modeling setting.…”

“…Let Pr{ k−1 i=1 δ i = 0 | Y } denote the joint posterior probability that all δ i 's are simultaneously zero. One could use the global test based on the joint posterior probability of the δ i 's, that is, the test that rejects H 0 if Pr Shang et al (2008)).…”

“…The data is provided and analyzed in Fitzmaurice et al (2004). This data has been utilized in Shang et al (2008) for demonstrating the performance of a similar hierarchical model in balanced mixed models. Here, we apply the proposed hierarchical model to the data for the illustration of its effectiveness.…”

Should we be concerned about multiple comparisons in hierarchical Bayesian models?

“…The potential utility of using formal decision theoretic or model selection approaches in place of a more traditional multiple comparisons procedure is just one of several potential solutions. Other solutions have been suggested in the context of Bayesian models (e.g., Li & Shang, 2015;Nashimoto & Wright, 2008;Shang et al, 2008;Westfall et al, 1997), but as noted previously, they introduce additional concerns. We emphasize that our goal is to lend insight into whether or not HB models are immune to multiplicity issues -they are notand to offer PPI as a tool for evaluating when alternative modelling approaches or adjustment procedures should be explored.…”

“…The pooling property essentially 'pulls' each θ k towards the global mean, reducing differences among the levels (Gelman & Hill, 2007;Gelman, Hwang & Vehtari 2014). Others, however, suggest modifications to HB models to account for multiplicity (e.g., Li & Shang, 2015;Nashimoto & Wright, 2008;Shang, Cavanaugh, & Wright, 2008;Westfall et al, 1997), but the approaches can greatly increase the complexity of the model, require specification of informative priors that may not be supported by existing information or require an a priori ranking of the group-level means, which may be difficult to determine in practice. Thus, the suggestion that partial pooling diminishes the need to adjust for multiplicity is attractive but has not be rigorously tested against the types of 'messy' data that ecologists routinely work with.…”