MANOVA Simultaneous Test Procedures: The Power and Robustness of Restricted Multivariate Contrasts

Abstract

A Monte Carlo study was conducted using SAS-IML to compare the multivariate analysis of variance (MANOVA) simultaneous test procedures (STPs) of 0, 'T, A, and V in terms of power and Type I error under various conditions including violations of MANOVA assumptions. The Type I error rates of moderately restricted contrasts in STPs were robust to violations of MANOVA assumptions such that the actual alpha remained below the nominal alpha. However, the power of the STPs was unacceptably low in moderately restricted contrasts under most conditions. For partially restricted contrasts, the power of the STPs was adequate for θ when the noncentrality structure was concentrated. When the noncentrality structure was diffuse, again θ had the highest power but decreased to a minimal level as the number of groups or variables increased.

A Monte Carlo Comparison of Robust MANOVA Test Statistics

“…These results for T 2 are similar to those reported in studies of the performance of Pillai's Trace, Wilk's Lambda, Hotelling-Lawley's Trace and Roy's Greatest Root when there are violations in the assumption of equality of covariance matrices (Finch, 2005;Olson, 1974;Sheehan-Holt, 1998). In these studies, when the smaller group had the larger variance the Type I error rates were inflated, whereas when the larger group had the larger elemental covariance elements, there was a reduction in power.…”

“…In the three group case, for condition (b) groups 1 and 2 were half the size of group 3, and for condition (c) groups 1 and 2 were twice the size of group 3. Thus, for example, in the n=60 case, there were 30 simulees per group in condition a, 20 in group 1 and 40 in group 2 in condition b, and 40 in group 1 and 20 in group 2 in condition c. The combination of unequal group sizes with unequal group covariance matrices has been shown to influence both Type I error and power rates (Sheehan-Holt, 1998;Stevens, 2001;Hakstian, Roed & Lind, 1979) and these particular ratios employed have been used in prior studies (e.g., Christensen & Rencher, 1997, Hakstian et al,1979). As noted above, when the smaller group has the larger covariance matrix elements the Type I error rate will be inflated; when the larger group has the larger elements power will be diminished.…”

“…Equality of the covariance matrices across groups is a vital assumption for the test statistics associated with MANOVA, and differences in these matrices can influence the performance of these tests (Fouladi & Yockey, 2002;Sheehan-Holt, 1998;Korin, 1972). Two unequal covariance conditions were simulated (a) the larger group had the larger elemental values and (b) the smaller group had the larger elemental values.…”

Glucose Intolerance in Patients With Cystic Fibrosis: Sex-Based Differences in Clinical Score, Pulmonary Function, Radiograph Score, and 6-Minute Walk Test

“…MANOVA takes advantage of this by combining the dependent variables in a way that makes the differences among the groups as large as possible. 30 We also reported a MANOVA for analysis of the same dependent and independent variables, but with age as the covariate.…”

Robust tests for multivariate factorial designs under heteroscedasticity

“…Several studies have been conducted to investigate the extent to which MANOVA tests are robust to violations of underlying assumptions (Finch & Davenport, 2009;Hakstian, Roed, & Lind, 1979;Ito, 1980;Mardia, 1971;Olson, 1974;Sheehan-Holt, 1998). The general conclusion to be drawn from these studies is that for equal group sizes, violations of normality and equality of covariance matrices produce slightly distorted error rates.…”