Bayesian significance testing and multiple comparisons from MCMC outputs

This article proposes a Bayesian method to directly evaluate and test hypotheses in multiple comparisons. Transformation and integration over the coordinates relevant to the hypothesis are shown to enable us to directly test the hypotheses expressed as a linear equation of a parameter vector, given a linear constraint. When the conditional posterior distribution of the parameter vector we are interested in is the multivariate normal distribution, the proposed method can be applied to calculate the p-value of hypotheses pertaining to the parameters in any complex model such as generalized linear mixed effect models with latent variables, by using outputs from Markov chain Monte Carlo (MCMC) methods. Further, the proposed testing can be implemented without prior information. Some applications are presented, and the simulation results are provided to compare the powers of this method with those of other methods of conventional multiple comparisons. Simulation studies have shown that the proposed method is valid for multiple comparisons under nonequivalent variances and mean comparisons in latent variable modeling with categorical variables.