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An accurate test for homogeneity of odds ratios based on Cochran's Q-statistic
Elena Kulinskaya1, Michael B Dollinger2
1School of Computing Sciences, University of East Anglia, Norwich, NR4 7TJ, UK. e.kulinskaya@uea.ac.uk.
Background:
A frequently used statistic for testing homogeneity in a meta-analysis of K independent studies is Cochran's Q. For a standard test of homogeneity the Q statistic is referred to a chi-square distribution with K-1 degrees of freedom. For the situation in which the effects of the studies are logarithms of odds ratios, the chi-square distribution is much too conservative for moderate size studies, although it may be asymptotically correct as the individual studies become large.
Methods:
Using a mixture of theoretical results and simulations, we provide formulas to estimate the shape and scale parameters of a gamma distribution to fit the distribution of Q.
Results:
Simulation studies show that the gamma distribution is a good approximation to the distribution for Q.
Conclusions:
Use of the gamma distribution instead of the chi-square distribution for Q should eliminate inaccurate inferences in assessing homogeneity in a meta-analysis. (A computer program for implementing this test is provided.) This hypothesis test is competitive with the Breslow-Day test both in accuracy of level and in power.
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The test statistic for a test of independence is similar to that of a goodness-of-fit test:

