Combining Effect Sizes Taking the Average

How to Combine (1) Study ES 1 2 .5 3 .3 Take the simple mean (add all ES, divide by number of ES) Study ES 1 2 .5 3 .3 M=(1+.5+.3)/3 M = 1.8/3 M=.6 Unbiased, consistent, but not efficient estimator. But see Bonnet for an argument for using unit wts

How to Combine (2) Study ES W 1 2 .5 3 .3 .9 W(ES) Take a weighted average Study ES W (weight) W(ES) 1 2 .5 3 .3 .9 M=(1+1+.9)/(1+2+3) M=(2.9)/6 M=.48 (cf .6 w/ unit wt) (Unit weights are special case where w=1.)

How to Combine (3) Choice of Weights (all are consistent, will give good estimates as the number of studies and sample size of studies increases) Unit Unbiased, inefficient Sample size Unbiased (maybe), efficient relative to unit Inverse variance – Reciprocal of sampling variance (or Ve+REVC) Biased (if parameter figures in sampling variance), most efficient Other – special weights depend on model, e.g., adjust for reliability (Schmidt & Hunter)

How to Combine (4) Inverse Variance Weights (fixed effects) are a function of the sample size, and sometimes also a parameter. For the mean: For r: For r transformed to z: Note that for two of these, the parameter is not part of the weight. But for r (not z transform), larger observed values will get more weight. Mean can be biased.