Parametric and nonparametric bootstrap methods for general MANOVA

Abstract

a b s t r a c tWe develop parametric and nonparametric bootstrap methods for multi-factor multivariate data, without assuming normality, and allowing for covariance matrices that are heterogeneous between groups. The newly proposed, general procedure includes several situations as special cases, such as the multivariate Behrens-Fisher problem, the multivariate one-way layout, as well as crossed and hierarchically nested two-way layouts. We derive the asymptotic distribution of the bootstrap tests for general factorial designs and evaluate their performance in an extensive comparative simulation study. For moderate sample sizes, the bootstrap approach provides an improvement to existing methods in particular for situations with nonnormal data and heterogeneous covariance matrices in unbalanced designs. For balanced designs, less computationally intensive alternatives based on approximate sampling distributions of multivariate tests can be recommended.

Testing Mean Differences among Groups: Multivariate and Repeated Measures Analysis with Minimal Assumptions

“…The simulation results presented in the supplementary materials indicate that the methods proposed here are rather robust in this regard. Also, in simulations by Konietschke et al (2015, Figure 1), leptokurtic distributions had a larger effect on the power of classical tests without resampling than on tests involving parametric or nonparametric bootstrap.…”

“…The generalized inverse needs to be used in lieu of the regular matrix inverse since the latter may not exist, which would make the WTS Q N invalid. Konietschke et al (2015) have shown that, under the technical assumptions mentioned above in Section 2, and under H 0 : Tμ = 0, Q N (T) has asymptotically, as N → ∞, a central χ 2 -distribution with degrees of freedom equal to the rank of T. However, they only considered matrices T where the final component is the identity matrix I p (see sec. 4 of Konietschke et al, 2015).…”

“…Konietschke et al (2015) have shown that, under the technical assumptions mentioned above in Section 2, and under H 0 : Tμ = 0, Q N (T) has asymptotically, as N → ∞, a central χ 2 -distribution with degrees of freedom equal to the rank of T. However, they only considered matrices T where the final component is the identity matrix I p (see sec. 4 of Konietschke et al, 2015). This leads to the multivariate hypotheses discussed above in Section 2.1 A closer examination of their method of proof reveals that the established asymptotic results remain correct even when I p is replaced by other projection matrices, such as those mentioned in Section 2.2 Even the mathematical theory for the asymptotic model-based bootstrap, often referred to as parametric bootstrap, can be transferred to the more general situation without having to impose any additional assumptions.…”

“…However, the exclusive use of univariate techniques has in large part been driven by the fact that appropriate inference methods to analyze multivariate data have not existed. Indeed, classical multivariate analysis of variance (MANOVA) techniques (Bartlett, 1939;Dempster, 1958Dempster, , 1960Hotelling, 1947Hotelling, , 1951Lawley, 1938;Nanda, 1950;Pillai, 1955;Wilks, 1946) assume multivariate normal responses with equal covariance matrices across groups, and they are known to perform poorly when covariance matrices do in fact differ and the design is unbalanced (Konietschke, Bathke, Harrar, & Pauly, 2015;Vallejo & Ato, 2012). Unbalancedness and heteroscedasticity are however common real data features that one needs to properly address for valid analyses.…”

“…Apart from Konietschke et al (2015), the only other mean-based inference method using a multivariate factorial model without normality or equal covariance matrix assumption is that of Harrar and Bathke (2012). However, due to its design limitations, it cannot provide inferential answers to the research questions formulated above regarding the Alzheimer data.…”