The log-ANOVA test was proposed by Box (1953). In this procedure, sub-samples are created out of the observations in the ith sample (where i=1,...,K). Each of the sub-samples hence contains separate observations where,
The statistic approximates an F-distribution with (K-1) and () degrees of freedom. However, where the values diverge (thus they are not equal), a generalisation developed by Scheffe (1959) can be used to obtain the statistic:
Hence it can also be demonstrated that
The statistic also approximates an F-distribution with (K-1) and () degrees of freedom. Other modifications of the Box method such as the Bargmann modification (Gartside, 1972) of the Box test have been proposed remove bias and to improve the approximation to the homogeneity condition of ANOVA.