Box's M Test for Covariance Homogeneity

Many multivariate methods, linear discriminant analysis, MANOVA, assume that all groups being compared share not just similar averages but the same shape of variability: the same covariance matrix, meaning the same variances and the same correlations between variables. Box's M test checks that assumption directly, comparing each group's sample covariance matrix against a pooled covariance matrix computed by combining all the groups, under the null hypothesis that they're all really drawn from populations with identical covariance structure.

The test statistic compares the log-determinant of each group's own covariance matrix to the log-determinant of the pooled version, weighted by sample size, and converts the result into an approximate chi-squared statistic. A large M statistic (and correspondingly small p-value) means the groups' covariance structures differ enough that it's unlikely to be sampling noise, for example, one group's variables are far more tightly correlated with each other than another group's, even if the group means look similar.

The test is notoriously sensitive to non-normality and large sample sizes, flagging statistically significant differences even when they're economically tiny, so in practice it's used more as a diagnostic flag to inspect the covariance matrices directly than as a strict accept/reject rule, especially before running a MANOVA or LDA whose validity depends on the homogeneity assumption holding at least approximately.