We assume independent samples of size $n_1$, $n_2$, $\ldots$, $n_g$ from a $v$-multivariate normal distribution. The hypothesis of equality of covariances is: $$H_0= \Sigma_1 = \Sigma_2 = \ldots = \Sigma_g$$ To make the test we calculate: $$M=\frac{ |S_1|^{n_1-1} |S_2|^{n_2-1} \ldots |S_g|^{n_g-1} }{|S_{pl}|^{\sum_{i=1}^g n_i-1} }$$

where $S_i$ is the covariance matrix of group $i$ and $S_{pl}$ is the pooled sample covariance matrix. It is clear that we must have $n_i-1>v$;

otherwise $|S_i|=0$ for some $i$ and $M$ would be zero. The statistic $M$ is a modification of the likelihood ratio test and varies between 0 and 1 with values near 1 favouring $H_0$ and values near 0 leading to the rejection of $H_0$ (see Rencher (2002) p. 256 for further details). The quantity $-2 \ln M$ is approximately distributed as a $\chi^2$ distribution and is given in $\mbox{out.LR}$. The quantity $-2(1-c_1) \ln M$ (where $c_1$ is a small sample correction factor) is usually called correct Box test and is approximately distributed as a $\chi^2$ with $0.5 (g-1) v(v+1)$ degrees of freedom. We reject $H_0$ if $$-2(1-c_1) \ln M >\chi^2_{1-\alpha}$$ where $\chi^2_{1-\alpha}$ is the $1-\alpha$ quantile. The value of this test is contained in $\mbox{out.LRchi2approx}$ and the corresponding p-value of this test is given in $\mbox{out.LRchi2approx_pval}$.

Box also derived and F approximation to the test. This test is computed just if input option Fapprox is true. The value of this last test and the corresponding p-values are given in $\mbox{out.LRFapprox}$ and $\mbox{LRFapprox_pval}$.

WARNING: if the absolute value of the determinant of the covariance matrix of any group is less than 1e-40, a missing value for LR test is reported.