Multivariate normal inference for heterogeneous samples and an application to meta analysis

Abstract

Scope and Method of Study:

When extending likelihood inference in the case of the normal distribution to heterogeneous samples, one discovers that this is easily done in the univariate case but is prohibitive in the multivariate cases.

In the current work, the exact maximum likelihood estimates for the core mean and the covariance matrix are obtained for samples of different means but with core parameter vector and an unknown covariance matrix but a structured one. Then the celebrated Hotelling's T-square statistic is generalized to this case where the exact null distribution is derived. The approximate Chi-Square test is then obtained as well.

Next, we derive analogous results in the k-sample situation. The generalized Hotelling T- square statistic developed allows us to proceed to testing hypotheses in the one-way multivariate ANOVA when samples are heterogeneous. A cutting edge application of this work is its introduction to multivariate meta analysis approaches for multivariate heterogeneous data for the first time.

Finding and Conclusions:

The one-sample and multi-sample inferences for testing the core parameter vector(s) using likelihood ratio test approaches when the covariance matrix has compound symmetry are obtained. The null distribution of the LRT statistic for each case is derived exactly. Both of them follow a distribution of a powered function of two independent beta random variables but different forms. The exact distribution of the ML estimator of the intra-correlation is derived for one-sample case and it is distributed as a function of F random variable. As for the inference about the MLE of the common variance, it is exactly distributed as sum of two weighted chisquare random variables. An approximate Chi-square test is also derived for testing the core parameter vector(s) for each of one-sample and two-sample cases.

An application to multivariate meta analysis of the heterogeneous data for k independent studies is addressed. Both fixed and random effects models are investigated. A simulation study for inference of the overall core parameter vector is performed when data are heterogeneous for one-stage random effects model.