1. Hedges’ Approach

2. 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

3. 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).

4. 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

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

6. 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).

7. 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.

8. 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.

9. 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.

10. 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

11. 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

12. 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.

13. 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.

14. 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)