# Diagnostic plot for assessing homogeneity of variance-covariance matrices

Is there a handy plot for comparing the variance-covariance matrices of two (or perhaps more) groups? An alternative to looking at lots of marginal plots, especially in the multivariate Normal case?

• – Andy W May 19 '13 at 13:43
• Something smart like assessing multivariate Normality by putting squared Mahalanobis distances on a quantile-quantile plot against the chi-squared distribution. – Scortchi - Reinstate Monica May 19 '13 at 15:44
• Well, at least we are getting somewhere! Here are two questions (Q1 & Q2) that mention a Box M test that may be of interest. You may be able to decompose that test to provide the plot you want, not sure though (this is getting beyond anyway I can help though!) – Andy W May 20 '13 at 19:36
• Came across this old question of yours. For two covariance matrices $S_1$ and $S_2$, I am wondering if one can compute $S_1^{-1} S_2$ that should be close to $I$ if $S_1 \approx S_2$, so one can plot the eigen-spectrum of this product and visually check how close it is to the horizontal line. Perhaps one can compute and plot, for comparison, the expected eigen-spectrum under the null: it will not be a flat line due to the finite sample size. I don't know how to compute it analytically given $n_1$, $n_2$, and $p$ (starting with Marchenko-Pastur?), but it should be easy to estimate numerically. – amoeba says Reinstate Monica Sep 23 '15 at 23:27
• @amoeba: Thanks; that sounds like an interesting approach. For more than two matrices, perhaps a 2x2 grid of eigen-spectra. I've also got two papers on the subject mouldering in my desk pile; & I need to think about decomposing the Box M-test. – Scortchi - Reinstate Monica Sep 25 '15 at 9:15