Schmidt & Hunter Approach to r. Statistical Artifacts. Extraneous factors that influence observed effect Sampling error Reliability Range restriction Computational error Dichotomization of variables. Bare Bones r. Find weighted mean and variance: Note sample size weight.
There are k studies, with Ni observations.
This is not the only formula they use, but it’s the best one IMHO.
Estimated variance for a study.
Estimated sampling variance for a meta-analysis. Note mean r is constant.
Classical Test Theory
To find the variance of infinite-sample correlations, find the variance of r in the meta-analysis and subtract sampling error variance. Schmidt would be quick to add that part of the estimated variance is artifactual.
The credibility interval and the confidence interval are quite different things. The CI is a standard statistical estimate (intended to contain rho). The CR is a Bayesian estimate (intended to contain a percentage of the values of a random variable).
Ignore the last 3 columns for now.
Recall unwighted or unit weighted mean = .30.
Why are they different?
Disattenuation for reliability
Correction for both
Correction for IV
Correction for DV
Suppose rxy = .30, rxx = ryy = .80. Then:
These are examples of direct RR.
Suppose rxy = .33, SD1=12, SD2 = 20. Then:
Can also invert by uX = 1/UX
Reliability of IV in restricted sample (job incumbents in I/O validation study).
Reliability of IV in unrestricted sample (job applicants in I/O validation study).
Ratio of SD of true scores; analogous to uX.
You will need rxxa for DIRECT range restriction correction.
You will need uT AND rxxi for INDIRECT range restriction correction.
Record data (N, r, artifact values rxx, etc.)
Compute the corrected correlation for each study:
If there is only 1 kind of artifact, disattenuation is simple:
Where a is the disattenuation factor.
Note ro is observed and rC is corrected.
If there is range restriction, things are tricky. If INDIRECT range restriction, then use Ut instead of Ux and disattenuate for reliability before adjusting for range restriction. Use reliabilities from the restricted group.
If DIRECT range restriction, adjust for ryy, then range restriction, then rxx, but rxxa, the reliability in the unrestricted group.
For each study, compute compound attenuation factor:
Compute sampling variance of uncorrected r:
Note this is sampling variance for one study.
Compute sampling variance of disattenuated r:
If there is range restriction, then do the following 2 steps.
Compute adjustment for range restriction:
Adjust sampling variance of disattenuated r:
Note A is the compound attenuation factor.
Compute the weighted mean:
Compute the weighted variance:
Compute average corrected r sampling error:
Compute variance of rho:
We’ve already done the bare-bones analysis of these data. Now we’ll analyze 3 ways: (1) just criterion reliability, (2) all artifacts with INDIRECT RR, (3) all artifacts DIRECT rr.
Suppose we only wish to correct for criterion unreliability.
Study 1 r = .20, rxx = .90, ryy = .80, Ux = 1.5
Disattenuation ryy : rC = .2/sqrt(.8) = .223607.
Compound attenuation factor A = .20/.223607 = .894.
Already know bare-bones mean.