  Meta-analysis seeks to combine the results of several experiments in order to
improve the accuracy of decisions. It is common to use a test for homogeneity
to determine if the results of the several experiments are sufficiently similar
to warrant their combination into an overall result. Cochran's Q statistic is
frequently used for this homogeneity test. It is often assumed that Q follows a
chi-square distribution under the null hypothesis of homogeneity, but it has
long been known that this asymptotic distribution for Q is not accurate for
moderate sample sizes. Here we present formulas for the mean and variance of Q
under the null hypothesis which represent O(1/n) corrections to the
corresponding chi-square moments in the one parameter case. The formulas are
fairly complicated, and so we provide a program (available at
http://www.imperial.ac.uk/stathelp/researchprojects/metaanalysis) for making
the necessary calculations. We apply the results to the standardized mean
difference (Cohen's d-statistic) and consider two approximations: a gamma
distribution with estimated shape and scale parameters and the chi-square
distribution with fractional degrees of freedom equal to the estimated mean of
Q. We recommend the latter distribution as an approximate distribution for Q to
use for testing the null hypothesis.
