## What is Box’s M Test?

**Box’s M test** (also called Box’s Test for Equivalence of Covariance Matrices) is a parametric test used to compare variation in multivariate samples. More specifically, it tests if two or more covariance matrices are equal (homogeneous).

## Null Hypothesis

The null hypothesis for this test is that the observed covariance matrices for the dependent variables are equal across groups. In other words, a non-significant test result (i.e. one with a large p-value) indicates that the covariance matrices are equal. The generated test statistic is called Box’s M statistic.

## Disadvantages

Box’s M Test is extremely sensitive to departures from normality; the fundamental test assumption is that your data is multivariate normally distributed. Therefore, if your samples don’t meet the assumption of normality, you shouldn’t use this test.

In addition, Box’s M has very little power (Cohen, 2008) for **small sample sizes**; if your small-sample result is not significant, it doesn’t necessarily indicate that the covariance matrices are equal. The test has also been criticized for being overly sensitive for **large sample sizes**. In other words, it will report a statistically significant result when one doesn’t actually exist. To address this particular issue, a smaller alpha level (e.g. .001) is recommended (Hahs-Vaughn, 2016).

## Similar Tests

Bartlett’s test is test for homogeneity of variances for normally distributed samples. Box’s M Test is available in SPSS; SAS uses Bartlett’s test instead.

Levene’s test is another similar test, although it’s best for non-normal samples. While Box’s M determines whether the covariance matrices are similar, Levene’s assesses whether the *variances *are similar.

## References

Box, G. E. P., 1949. A general distribution theory for a class of likelihood criteria. Biometrika, 36: 317–346.

Cohen, B. (2008). Explaining Psychological Statistics. John Wiley & Sons.

Hahs-Vaughn, D. (2016). Applied Multivariate Statistical Concepts. Taylor & Francis.

Layard, M. (1974). A Monte Carlo Comparison tests for equality of covariance matrices. Biometrika. 16, 461-465.

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