Testing on the common mean of normal distributions using Bayesian methods

Meta-analysis is performing statistical analysis utilizing results from multiple sources. An application of meta-analysis is inference on a common mean when independent random samples are available from different normal populations with unknown and possibly unequal variances. This is an extension of the classical Behrens–Fisher problem. Chang and Pal [Testing on the common mean of several normal distributions, Comput. Statist. Data Anal. 53 (2008), pp. 321–333] described several methods that can be used to test hypotheses concerning this common mean. They proposed four methods which are the likelihood ratio test, two tests based on the Graybill-Deal estimator of the common mean, and a parametric bootstrap test based on the maximum-likelihood estimator. This paper proposes several Bayesian procedures to test the hypotheses of interest. Simulation studies investigating the size, power, and robustness of the new tests as well as the tests in Chang and Pal [1] are performed and discussed. Some of the results are in conflict with Chang and Pal [1] and some of the Bayesian tests appear to be attractive alternatives to tests that have been proposed in the past.