MANOVA for Nested Designs with Unequal Cell Sizes and Unequal Cell Covariance Matrices

Li-Wen Xu

doi:10.1155/2014/649202

Abstract

We propose and study parametric bootstrap (PB) tests for heteroscedastic two-factor MANOVA with nested designs. For the problem of testing “main effects” of both factors, we develop a flexible test based on a parametric bootstrap approach. The PB test is shown to be invariant under affine-transformations. Moreover, the PB test does not depend on the chosen weights used to define the parameters uniquely. The proposed test is compared with the approximate HotellingT2(AHT) test by the simulations. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations and generally outperforms the AHT test in terms of controlling the nominal size. For the heteroscedastic cases, the PB test outperforms the AHT test in terms of power. In addition, the PB test does not lose too much power when the homogeneity assumption is actually valid.

Highlights

There are many situations where we record more than one response variable from each sampling or experimental unit and where these units are allocated to or occur in treatment groups
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 Wilks likelihood ratio (WLR), Lawley-Hotelling trace (LHT), and BNP tests can still be used but with the degrees of freedom estimated from the data via matching the first two moments; see some details in Section 2.2 of Zhang and Xiao [7]
Similar to Zhang [8], we proposed a test referred as the approximate Hotelling T2 (AHT) test, which is based on the test statistic (d − q + 1) T

Summary

Introduction

There are many situations where we record more than one response variable from each sampling or experimental unit and where these units are allocated to or occur in treatment groups. Some simulation studies conducted in Zhang [8] showed that the AHT test outperforms the modified LHT test of Harrar and Bathke [6]. Another useful two-factor design is the nested MANOVA [9]. Some technical proofs of the main results are given in the Appendix

Tests for Nested Effects

Tests for the Nesting Effects

Monte Carlo Studies

Concluding Remarks