Testing homogeneity of covariance matrices: Box’s M Test

Consider a sample set {x11,…,x1n1} in Rm sampled from population Θ₁ and a sample set {x21,…,x2n2} in Rm sampled from population Θ₂. Assuming that the sample sizes n₁ and n₂ are sufficiently large, Box’s M Test tests the null hypothesis that the population covariance matrices are equal, i.e., H₀: Σ₁ = Σ₂. Let S₁ and S₂ be the sample covariance matrices from the populations Θ₁ and Θ₂, where each S_j is based on n_j independent observations, we define the pool variance S_pooled as follows:

and the value of M is given by:

Then, M(1 − c) has an approximate χdf2-distribution, where:

The null hypothesis H₀ is rejected when M(1-c)>χdf2(α) (or p-value < α).