Box’s M test is a statistical test used to check for the equality of covariance matrices coming from two or more different populations following the multivariate normal distribution. This test can only be applied under the assumption that the populations are normally distributed.

It is generally used to check for equality of covariances when applying the MANOVA (multivariate analysis of variance) procedure.

Suppose we are given ‘m’ different populations. The null hypothesis in this test is that the covariance matrices of all ‘m’ populations are equal whereas the alternative hypothesis is that the covariance matrices are not equal to each other.

**Procedure to carry out Box’s test for equality of covariance matrices**:

1) Formulate the null and alternative hypothesis as shown below:

2) Calculate the value of the box’s test statistic and compare it with the table values to decide whether to accept or reject the null hypothesis.

3) Since the calculations for the boxes test statistic are generally very complicated we use statistical software such as R to carry out the test. The biotools package in R software has a boxM() function for testing the equality of covariance matrices between groups. Click here to know how to carry out this test on R software.

**Tests used for checking equality of variances**:

Some of the tests which can be used to check for equality of variances are: