Multiple comparison techniques are to compare more than one
pair of treatment means by employing various effective methodologies. In some applications, researchers may believe that before the data are collected, the underlying parameters satisfy an order restriction. But some researchers believe otherwise for certain applications. For instance, in quality control experiments, researchers study the effect of different setting of significant factors, such as the temperature, the pressure, and the different types of machines. The researchers believe that in order to find complete significant results, all treatments should be mixed up and comparisons should be applied to each pair of them. To facilitate multiple comparisons, certain methodologies are proposed and applied. Generally, the frequentist and the Bayesian methodologies are used to conduct multiple comparisons.
In this dissertation, we are interested in the Bayesian approach. We propose a hierarchical model in developing and applying multiple comparisons without any restriction in mixed models. The model facilitates inferences in parameterizing the successive differences of the population means, and for them we choose independent prior distributions that are mixtures of a normal distribution and a discrete distribution with its entire mass at zero. For the other parameters, we choose conjugate or non-informative priors, and we derive the full conditional posterior distributions for the parameters in the mixed models. A simulation study is performed to investigate the effectiveness of the proposed hierarchical model. In the simulations, a sequence of different simulated data sets is utilized. To determine the optimal priors for the parameters or hyper-parameters, we do comparisons of the different priors for the purpose of making a decision. In the applications, two real data sets are analyzed to compare the population means for illustrating the performance of the proposed in the hierarchical model. The Type I error, Family-wise Error rate (FWER), and the test power are also studied in the simulations. The simulation results exhibit that the proposed hierarchical model can effectively do multiple comparisons while keeping Type I error relative low and the test power reasonable large in our simulation sets.
Gibbs sampler, a special procedure of Markov chain Monte Carlo (MCMC), is used in the simulations to compute the parameters estimates and the posterior probabilities that two population means are equal. The iteration procedure allows one both to determine if any two means are significantly different and to test the homogeneity of all of the means. Hypothesis testing procedure is introduced and the simulations are based on it. The proposed hierarchical model is applied in two real data sets for illustrating the performance of the model. The computing procedure of the proposed hierarchical model can effectively unify parameters estimation, tests of hypotheses, and multiple comparisons in one setting.