Hedges’ Approach

Two main camps in MA Schmidt & Hunter Hedges et al. Hedges & Olkin Hedges & Vevea Differ in Weights and Data Transformation Others – HLM, Rosenthal, Bayesian, not as common

Weights Defined SH use N, NA2 for weights Hedges uses inverse variance weights. Sampling variances and inverses: To analyze correlations, Hedges will use z and (N-3).

InV Weights Pros & Cons Precise definition of weight; math justification (min var estimator) Parameters often involved in weight, resulting in bias Sum of weights gives basis for confidence interval Parameter has to be estimated. Provides test of homogeneity of effect sizes

Data Transformation r .10 .20 .30 .40 .50 .60 .70 .80 .90 z .10 .20 .31 .42 .55 .69 .87 1.10 1.47

Confidence Interval Because w=N-3, this basically means that the confidence interval is the mean plus or minus 2 times the root of 1/(Total N).

Homogeneity Test When the null (homogeneous rho) is true, Q is distributed as chi-square with (k-1) df, where k is the number of studies. This is a test of whether Random Effects Variance Component is zero.

Estimating the REVC If REVC estimate is less than zero, set to zero. REVC is SH Var(rho), but in the metric of z, not r. Method due to DerSimonian & Laird. This method works well, even though the estimator gives an approximation. In metafor, this estimator is applied if method= “DL”. Iterative numerical analysis is needed for an exact solution. In metafor, this estimator is the default, method=“REML”, for restricted maximum likelihood.

Random-Effects Weights Inverse variance weights give weight to each study depending on the uncertainty for the true value of that study. For fixed-effects, there is only sampling error. For random-effects, there is also uncertainty about where in the distribution the study came from, so 2 sources of error.* The InV weight is, therefore: *we will go into this in greater detail next week. Try to accept this for now and to understand it next week.

Numerical Illustration (1) Study Ni r z w zw 1 200 .20 .203 197 39.94 2 100 97 19.67 3 150 .40 .424 147 62.28 4 80 77 32.62 Sum 518 154.50

Numerical Illustration (2) Study Ni z w w2 1 200 .203 197 1.80 38809 2 100 97 .89 9409 3 150 .424 147 2.31 21609 4 80 77 1.21 5929 Sum 518 6.20 75756

Numerical Illustration (3) Fixed-effects mean and CI: Parameter z r Lower B .212 .209 Mean .298 .289 Upper B .384 .366 But, generally best to use RE, even if Q is n.s.

Numerical Illustration (4) Study Ni z w w* zw* 1 200 .203 197 73.01 14.80 2 100 97 52.83 10.71 3 150 .424 147 64.84 27.47 4 80 77 46.28 19.61 Sum 518 236.95 72.58 Not prediction interval.

Numerical Illustration (5) Comparison of Results S & H Hedges FE Hedges RE Mean .287 .289 .297 CI low .190 .209 .177 CI high .384 .367 .408 CR low .172 .124 CR high .401 .453 (not a prediction interval)