Box M test: Definition & Examples

The Box M test, usually called Box’s M test, tests whether several populations have the same covariance matrix. For a quant comparing groups of asset returns or model residuals, it asks whether their joint spread can plausibly share one covariance model. It does not test whether their average returns are equal.

For $g$ groups of $p$ variables, the null hypothesis is

H_0:\Sigma_1=\Sigma_2=\cdots=\Sigma_g.

The group means may differ under this null. Each group is centred on its own mean before its covariance is calculated. A test of mean vectors, such as MANOVA, addresses a different hypothesis. Friendly and Sigal, §§1.2–1.3, distinguish these questions.

Compare the pooled model with separate models

Assume independent groups, with independent observations from a multivariate normal distribution within each group. Write

n_i

for group size,

N=\sum_i n_i

, and use the usual sample covariance, with divisor

n_i-1

S_i=\frac{1}{n_i-1}\sum_{j=1}^{n_i}(\mathbf x_{ij}-\bar{\mathbf x}_i)(\mathbf x_{ij}-\bar{\mathbf x}_i)^\top.

The pooled within-group covariance is

S_P=\frac{\sum_{i=1}^{g}(n_i-1)S_i}{N-g}.

This is not the covariance obtained by ignoring group labels and centring every observation on one grand mean: that would also include between-group mean differences.

Define the uncorrected log-determinant statistic here as

M=(N-g)\ln\det S_P-\sum_{i=1}^{g}(n_i-1)\ln\det S_i.

The determinant summarises multivariate scatter. The comparison uses the determinant of the pooled matrix, not merely a check that the separate determinants match. Equal determinants can conceal different matrix shapes. Large

M

indicates greater disagreement with a common covariance model.

For the conventional corrected chi-squared approximation, set

C=\left(\sum_{i=1}^{g}\frac{1}{n_i-1}-\frac{1}{N-g}\right) \frac{2p^2+3p-1}{6(p+1)(g-1)}, \qquad X^2=(1-C)M,

with

\nu=\frac{(g-1)p(p+1)}{2}

degrees of freedom. Under the null and the approximation’s assumptions, compare $X^2$ with the upper tail of $\chi^2_\nu$ .

Check the reporting convention. Here

M

already means the displayed log-determinant difference; do not apply another

-2\ln

to it. Some software reports only the corrected

X^2

, and other presentations use an F approximation. The heplots implementation uses the correction above and labels its result “Chi-Sq (approx.)”.

Two variables, four times the variance

Consider fictional summaries from two independent normal groups, each with $20$ observations of two dimensionless variables. Suppose their sample covariances, using divisor $19$ , are

S_1=I_2,\qquad S_2=4I_2.

Both have zero sample cross-covariance; group 2 has four times each marginal sample variance. Pooling gives

S_P=\frac{19I_2+19(4I_2)}{38}=2.5I_2.

Thus $\det S_1=1$ , $\det S_2=16$ , and $\det S_P=6.25$ . The calculation is

\begin{aligned} M&=38\ln(6.25)-19\ln(16)\approx16.95891,\\ C&=\left(\frac{2}{19}-\frac{1}{38}\right)\frac{13}{18} \approx0.05701754,\\ X^2&=(1-C)M\approx15.99195,\qquad \nu=3. \end{aligned}

The chi-squared approximation gives a p-value of about

0.00114

. This is evidence against equal population covariances within the stipulated normal, independent-sample model; it is not an exact small-sample probability or a probability that the null is true.

Fictional two-variable example with 20 observations per group. Covariance contours for I and 4I have radii 1 and 2, while pooled 2.5I has radius square root 2.5. Pooled log determinant 1.83258 exceeds weighted group log determinant 1.38629. Multiplying the gap by 38 gives raw M 16.95891; the corrected statistic is 15.99195 with 3 degrees of freedom.

Original fictional covariance summaries. The contours are centred separately to compare spread; they are not confidence regions for group means. The lower panel shows the gap that produces raw M.

What a result cannot settle

Box’s M is sensitive to non-normality and outliers, and unbalanced group sizes can worsen its behaviour. A rejection in heavy-tailed return data may reflect poor calibration of the normal-theory reference distribution. Serial dependence also violates the independent-observation model. These concerns belong in the interpretation, alongside group covariance plots. See Friendly and Sigal, §§1.3–1.4, and the official heplots details.

The ordinary calculation requires positive-definite group covariance matrices. Having $n_i>p$ is necessary for a full-rank sample covariance, but redundant variables can still make it singular. Near-singularity also makes log determinants numerically fragile; a stable factorisation helps computation but does not repair the sampling model.

A non-rejection does not prove equality: statistical power depends on sample sizes, dimension and the alternatives. Nor is “passing Box’s M” a universal gate before using a robust method. Choose an analysis suited to the target question and data structure; a preliminary covariance test alone cannot establish its validity.

For the covariance quantities being compared, Expectation, Variance and Covariance develops the link between co-movement and portfolio variance. The public page is a preview; full access requires a free signed-in account.

Sources

Friendly, M. and Sigal, M., Visualizing Tests for Equality of Covariance Matrices (2018 preprint), §§1.2–1.4 and 2.1: covariance versus mean hypotheses, Box’s M, sensitivity and covariance contours.
heplots official documentation, Box’s M-test for Homogeneity of Covariance Matrices: chi-squared output, normality, imbalance and nonsingularity requirements.
heplots authors, R/boxM.R, pinned revision 0b906346: pooled covariance, raw log-determinant calculation, correction factor and upper-tail inference. The explicit raw- $M$ convention above follows this computation.

Compare the pooled model with separate models

Two variables, four times the variance

What a result cannot settle

Sources

Related terms

Chi-squared distribution

Determinant

MANOVA and Wilks lambda

Multivariate normal distribution

P-value

Sample covariance

Statistical power