Simultaneous test procedures and the choice of a test statistic in MANOVA.

Abstract

Problems associated with the use of a preliminary multivariate analysis of variance (MANOVA) test in conjunction with univariate follow-up tests can be avoided if follow-up tests are carried out with a simultaneous test procedure (STP) derived from the MANOVA test statistic used for the overall test. It is argued that the choice of a MANOVA test statistic for such analyses should be based on the power and robustness of MANOVA STPS rather than on the properties of the corresponding overall tests. Monte Carlo data are presented which show that the STP based on the trace statistic V can be extremely conservative relative to the STP based on the largest root statistic R. The data suggest that the lack of robustness of the R statistic is unlikely to produce problems if the R STP is used to evaluate interpretable contrasts on linear combinations of variates of interest to the experimenter as opposed to contrasts for which coefficients referring to groups and variates are determined by the data. • When multivariate analysis of variance (MANOVA) is applied to data from a k group experiment (k > 2) with p dependent variables (p> 1), the analysis is usually regarded as a two-stage procedure. Initially an overall test of a general null hypothesis is carried out using one of the various MANOVA test statistics. If, and only if, this general hypothesis is rejected, follow-up tests are carried out in order to clarify the nature of the differences between groups. The properties of MANOVA test statistics and the properties of multiple comparison procedures for follow-up tests are usually treated as unrelated questions. Recent discussions of MANOVA test statistics, for example, make no reference to follow-up tests (Olson, 1974, 1976, 1979; Stevens, 1979). The commonly recommended practice of changing from a multivariate to a univariate test statistic for follow-up tests (Bock, 1975; Finn, 1974; Hummel & Sligo, 1971) is based on the assumptions that the only function of the initial overall test is to protect the second stage of analysis from inflation of the Type I error rate and that the efficiency of this protection does not depend in any way on the nature of the follow-up tests. As Ram