1
Appendix 14: Box’s M Test
Box’s M statistic is used to test for homogeneity of covariance matrices. The jth set
of r dependent variables in the ith cell are ′=′+′
yx e
ij ij ij
B where eNw
ij r ij i
~,01−Σ
49
for ig=1, ,K and
j
ni
=1, ,K. The null hypothesis of the test for homogeneity of
covariance matrices is
H
og
:ΣΣ
1==L. Box (1949) derived a test statistic based
on the likelihood-ratio test. The test statistic is called Box’s M statistic. For
moderate to small sample sizes, an F approximation is used to compute its
significance.
Box’s M statistic is not designed to be used in a linear model context;1 therefore
the observed cell means are used in computing the statistic.
Notation
The following notation is used throughout this chapter, unless otherwise stated:
g Number of cells with non-singular covariance matrices.
ni Number of cases in the ith cell.
n Total sample size, nn n
g
=++
1L.
yij The jth set of dependent variables in the ith cell. A column vector of length r.
wij Regression weight associated with yij . It is assumed wij >0.
1 Although Anderson (1958, Section 10.2) mentioned that the population cell
means can be expressed as linear combinations of parameters, he assumed that the
combination coefficients are different for different cells, which is not the model
assumed for GLM .
