2009/06/16 by Elena Kulinskaya, Kulinskaya, Elena, Michael Döllinger +3 · 2 citations
Decision Sciences · #FOS: Computer and information sciences #FOS: Mathematics #Meta-analysis and systematic reviews #Methodology (stat.ME) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.0906.2999
openalex publication_date 2009/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.