Visualizing Tests for Equality of Covariance Matrices

Abstract

This paper explores a variety of topics related to the question of testing the equality of covariance matrices in multivariate linear models, particularly in the MANOVA setting. The main focus is on graphical methods that can be used to address the evaluation of this assumption. We introduce some extensions of data ellipsoids, hypothesis-error (HE) plots and canonical discriminant plots and demonstrate how they can be applied to the testing of equality of covariance matrices. Further, a simple plot of the components of Box's M test is proposed that shows how groups differ in covariance and also suggests other visualizations and alternative test statistics. These methods are implemented and freely available in the heplots and candisc packages for R. Examples from the paper are available in supplementary materials.

Impact of Color Matching Primaries on Observer Matching: Part II – Observer Variability

“…The inter-observer SD-ellipses, centered at the origin to better illustrate differences in variability (Friendly and Sigal 2018), are plotted in u'v' coordinates in Fig. 2 (calculated using 4 CMF sets and grouping per primary set).…”

Homogeneity Test of Several High-Dimensional Covariance Matrices for Stationary Processes under Non-normality

“…Let p ≥ c(n max ) 3r/r−2 . Under the original hypothesis H 0 in (1) with Σ > 0, for any p-dimensional vector y j = 0 defined in (10), we have that for every j (1 ≤ j ≤ m), the test statistic L kj in (11) converges in distribution to χ 2 k−1 as n min → ∞. Remark 3.2.…”

“…p j ×p j and S (i) p j ×p j are all (p j − p j−1 ) × (p j − p j−1 ) sub-matrices for 1 ≤ i ≤ k and 1 ≤ j ≤ m − 1, and Σ Let y p 1 = (1 1 , ..., 1 p 1 ), y p 2 = (1 p 1 +1 , ..., 1 p 2 ), ..., y pm = (1 p m−1 +1 , ..., 1 p ) and y 1 = (y p 1 , 0, ..., 0), y 2 = (0, ..., 0, y p 2 , 0, ..., 0), ..., y m = (0, ..., 0, y pm ), (10) be m p-dimensional vectors. It follows that…”

Making Meaning out of MANOVA: The Need for Multivariate Post Hoc Testing in Gifted Education Research

“…However, visualization is again strongly recommended to further evaluate the extent of the heterogeneity between the groups. Friendly and Sigal (2017, 2018) provide explanation and code for several visualizations of multivariate linear models. Figure 2 depicts one of these, a plot of the log determinants with surrounding asymptotic confidence intervals (see Cai, Liang, & Zhou, 2015) for each group matrix and the pooled covariance matrix.…”