Incorporating variability in estimates of heterogeneity in the random effects model in meta-analysis

When combining results from separate investigations in a meta-analysis, random effects methods enable the modelling of differences between studies by incorporating a heterogeneity parameter tau 2 that accounts explicitly for across-study variation. We develop a simple form for the variance of Cochran's homogeneity statistic Q, leading to interval estimation of tau 2 utilizing an approximating distribution for Q; this enables us to extend the point estimation of DerSimonian and Laird. We also develop asymptotic likelihood methods and compared them with this method. We then use these approximating distributions to give a new method of calculating the weight given to the individual studies' results when estimating the overall mean which takes into account variation in these point estimates of tau 2. Two examples illustrate the methods presented, where we show that the new weighting scheme is between the standard fixed and random effects models in down-weighting the results of large studies and up-weighting those of small studies.