Box’s M test serves as a check for homogeneity of covariance matrices across groups within a multivariate analysis of variance (MANOVA) or discriminant function analysis. In the specific context of a 2×2 ANOVA, where there are two independent variables each with two levels, this test assesses whether the population covariance matrices for the four resulting groups (2×2 = 4) are equal. A significant result suggests that the assumption of equal covariance matrices is violated, which can impact the validity of the ANOVA results.
The importance of verifying this assumption stems from the potential for inflated Type I error rates if it is not met. When covariance matrices are unequal, the F-statistic used in ANOVA may not accurately reflect the true differences between group means, leading to incorrect conclusions about the effects of the independent variables. Historically, Box’s M test has been a standard procedure for assessing this assumption, although its sensitivity to departures from normality, particularly with larger sample sizes, has led to debates regarding its routine application.
