A modified two-factor multivariate analysis of variance: asymptotics and small sample approximations

Abstract

In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods.

Multiple Contrast Tests for Multiple Endpoints in the Presence of Heteroscedasticity

“…The simulation results show that they yield conservative test decisions if the data follow distributions with a positive skew. In this case or for other non-normal distributions, non-parametric procedures -like those of Munzel and Brunner [22]; Bathke and Harrar [23]; Harrar and Bathke [24,25] -should be used instead. Note that these procedures do not provide multiplicity-adjusted p-values or SCIs for each contrast-endpoint combination as they use ANOVA-type χ 2 or F statistics, respectively.…”

A Modified Bartlett Test for Heteroscedastic Two-way MANOVA

“…[3] showed that for heteroscedastic data, the large-a asymptotics of the three tests are also equivalent. Based on some simulations, [3] further showed that their modified WLR, LHT and BNP tests also perform similarly for finite samples, with the modified LHT test, namely LHT m , slightly outperforming the BNP test.…”

“…Recently, [3] tried to solve this problem via modifying the WLR, LHT and BNP tests. Their main ideas focus on modifying the degrees of freedom of the random matrices involved in the test statistics so that the heteroscedasticity of the cell covariance matrices is taken into account and the WLR, LHT and BNP tests can still be used but with the degrees of freedom estimated from the data via matching the first two moments.…”

“…Their main ideas focus on modifying the degrees of freedom of the random matrices involved in the test statistics so that the heteroscedasticity of the cell covariance matrices is taken into account and the WLR, LHT and BNP tests can still be used but with the degrees of freedom estimated from the data via matching the first two moments. Although their approaches are simple to understand, these approaches admit the following three main drawbacks: (1) one needs to estimate the degrees of freedom of both the random matrices involved in the test statistics; (2) the estimated degrees of freedom, as given in Section 3 of [3], are complicated, case-sensitive, and not affine invariant; and (3) the null distributions of the WLR, LHT and BNP tests with known degrees of freedom are not immediately available; further approximations subsection, we aim to represent the main and interaction effects as linear combinations of the cell means which are estimable in the two-way MANOVA model (2) as described below.…”

“…We show that it admits several invariant properties. Simulation studies show that the MB test generally outperforms the classical LawleyHotelling trace (LHT) test and a modified LHT test of [3] under various parameter configurations in terms of size controlling and power. A real data example illustrates our method and the effect of heteroscedasticity.…”