Finite-Sample Correction For Box’s M

Why Correct the Raw Statistic?

Box’s M compares several group covariance matrices with their pooled estimate. Its raw statistic is a log-determinant contrast, and under the null hypothesis it has an asymptotic chi-square distribution. In finite samples, however, the raw statistic tends to have a mean slightly larger than the degrees of freedom of that limiting distribution. The finite-sample correction reduces this discrepancy.

The correction depends on two features of the design: the number of variables, and each group’s residual degrees of freedom. It changes the scale used for chi-square calibration; it does not change the covariance matrices or turn the corrected statistic into a new likelihood ratio.

Raw Box’s M and Its Reference Distribution

Suppose there are \(g\) independent groups, with \(n_i\) observations in group \(i\), and \(p\) measured variables. Under multivariate normal sampling, the null hypothesis is that all groups have the same covariance matrix. Let \(S_i\) be the usual unbiased sample covariance matrix for group \(i\), and define

where \(N=\sum_i n_i\). Assume \(n_i>p\) so that each \(S_i\) is nonsingular with probability one under a nonsingular multivariate normal model. The pooled matrix \(S_p\) is then also nonsingular.

For fixed \(p\) and \(g\), as the group sizes grow under the null hypothesis,

The dimension \(p(p+1)/2\) is the number of distinct entries in a symmetric covariance matrix. The null imposes equality across \(g\) covariance matrices, leaving \(g-1\) sets of covariance restrictions. This gives the degrees of freedom \(d\). The chi-square reference is asymptotic, not an exact finite-sample distribution.

Deriving the Finite-Sample Factor

The standard Box correction is a Bartlett-type adjustment: it rescales \(M\) to reduce its leading finite-sample bias relative to a chi-square variable with mean \(d\). A convenient derivation uses the exact expectation of the log determinant of a Wishart matrix under the null.

Under the null, write \(W_i=a_iS_i\). For independent multivariate normal samples, \(W_i\) has a Wishart distribution with covariance \(\Sigma\) and \(a_i\) degrees of freedom, and \(W=\sum_i W_i\) has a Wishart distribution with covariance \(\Sigma\) and \(\nu\) degrees of freedom. For an integer \(m>p-1\), if \(W_m\sim W_p(\Sigma,m)\), then

where \(\psi\) is the digamma function. Define

Since \(S_p=W/\nu\), linearity of expectation gives the exact null expectation

To obtain its first finite-sample term, use the digamma expansion \(\psi(x)=\log x-\frac{1}{2x}-\frac{1}{12x^2}+O(x^{-4})\) as \(x\) grows. Expanding each term in \(h(m)\) in powers of \(1/m\) gives

For clarity, the coefficient of \(m^{-2}\) comes from summing \(1/6-j^2/2\) over \(j=1,\ldots,p\); using \(\sum_{j=1}^{p}j^2=p(p+1)(2p+1)/6\) yields the displayed expression. Substituting this expansion into the exact expectation gives the following result.

To see the leading terms directly, the \(1/m\) term in \(h(m)\) contributes \((g-1)p(p+1)/2=d\). The \(1/m^2\) term contributes the displayed group-size expression. Dividing its coefficient by \(d\) shows that the relative first-order inflation is

Thus the correction factor is one minus this relative inflation. Multiplying \(M\) by that factor cancels the leading excess in its null mean. In particular, the resulting statistic is calibrated against the same asymptotic chi-square degrees of freedom; the correction does not change \(d\).

This derivation explains the correction’s first-order purpose, not an exact distributional identity. The chi-square calibration, corrected or uncorrected, remains an approximation. The corrected statistic is not itself a likelihood-ratio statistic arising from a new pair of hypotheses. Indeed, its associated exponential, \(\exp(-CM/2)=\Lambda_B^C\), is simply a power of the raw determinant ratio.

Worked Calculations

What the Correction Does—and Does Not Do

The factor \(C\) depends only on the design \((g,p,n_1,\ldots,n_g)\), not on the observed covariance matrices. For a fixed number of groups and variables, the size term shrinks as each \(n_i\) grows, so \(C\) approaches one. A larger dimension generally increases the coefficient \(A\), reflecting the larger finite-sample effect of estimating more covariance entries.

A common mistake is to multiply \(M\) by \(C\) and then treat the result as an exact chi-square statistic, or to change its degrees of freedom because its numerical value is smaller. Neither step is justified. The correction is a first-order improvement to an approximation: use \(CM\) with \(d=(g-1)p(p+1)/2\), and retain the usual caveat that the approximation may be inadequate in small samples.

Another mistake is to confuse the raw and corrected statistics. The raw determinant ratio corresponds to \(M=-2\log\Lambda_B\); \(CM\) is a calibration adjustment. It is not the unmodified log likelihood ratio, and the value \(\exp(-CM/2)\) should not be reported as if it were the original likelihood ratio.