# [R] How to test homogeneity of covariance matrices?

Date: Mon 19 Sep 2005 - 03:42:25 EST

Dear Group Members,

Forgive me if I am a little bit out of subject. I am looking for a good way to test the homogeneity of two variance-covariance matrices using R, prior to a Hotelling T² test. You’ll probably tell me that it is better to use a robust version of T², but I have no precise idea of the statistical behaviour of my variables, because they are parameters from the harmonics of Fourier series used to describe the outlines of specimens. I rather like to explore precisely these harmonics parameters.

It is known that Box’s M-test of homogeneity of variance-covariance matrices is oversensitive to heteroscedasticity and to deviation from multivariate normality and that it I not useful (Everitt, 2005 ; Seber, 1984 ; Layard, 1974). I have tried a “quick and dirty” intuitive comparison between two covariance matrices and I am seeking the opinion of professional statisticians about this stuff. The idea is to compare the two matrices using the absolute value of their difference, then to make a quadratic form using a unity vector and its transpose. One obtain a scalar that must be close to zero if the two covariance matrices are homogeneous :

Let S1 and S2 be two variance-covariance matrices of dimension n,

Let a be a vector of n ones : a <- rep(1, times = n)

b = a’ * |S1 – S2| * a, i.e. in R:

b <- a %*% abs(S1 – S2) %*% a

Is b distributed following a chi-square distribution? Is this idea total crap? Did someone tried this before and published something?

My data gave two 77 x 77 covariance matrices and b = 0.003243, a value close to 0, hence I expect my two covariance matrices are homogeneous. Am I right?

If this comparison is incorrect, could someone suggest a useful way to make this comparison using R?

Franck

Dr Franck BAMEUL

Le Clos d'Ornon
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