# Box’s M Test: Definition

## 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.

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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