Schmidt & Hunter Approach to r

1. Schmidt & Hunter Approach to r Bare Bones

2. Statistical Artifacts • Extraneous factors that influence observed effect • Sampling error* • Reliability • Range restriction • Computational error • Dichotomization of variables *addressed in the (bare-bones) analysis

3. Bare Bones r • Find weighted mean and variance: • Note sample size weight. • Note that for unit weights, the weighted variance estimator is the sample, not population, estimate.

4. Confidence Interval for Mean There are k studies, with Ni observations. This is not the only formula they use, but it’s the best one IMHO.

5. Estimated Sampling Error Variance • The variance of r Estimated variance for a study. Estimated sampling variance for a meta-analysis. Note mean r is constant. This is the variance of sampling error we expect if all the studies have a common effect estimated by r-bar.

6. Variance of Rho Classical Test Theory Sampling Error A definition

7. Estimated Variance of rho Note that the variance of rho will be called tau-squared by Hedges - To find the variance of infinite-sample correlations, find the variance of r in the meta-analysis and subtract expected sampling error variance. Schmidt would be quick to add that part of the estimated variance of infinite-sample correlations is artifactual.

8. Credibility Interval The credibility interval and the confidence interval are quite different things. The CI is a standard statistical estimate (intended to contain rho, or average of rho). The CR is intended to contain a percentage of the values of a random variable – infinite-sample effect sizes. The S&H value forgets that there is also uncertainty in the mean value; the two should be added. There are Bayesian programs that will do this; there is also an approximation called the prediction interval described in Borenstein et al.

9. Bare-Bones Example (1) <- Unit weighted mean

10. Bare-Bones Example (2)

11. BB Example (3) Recall unwighted or unit weighted mean = .30. Why are they different?

12. BB Example (4)

13. BB Example (5)

14. Interpretation • Schmidt says this is a random-effects meta-analysis. It uses a sample of studies to represent a larger population of studies. • People interpret the Credibility Interval, but typically do not recognize that it is poorly estimated.