Definition • Moderator - A third variable that conditions the relations of two other variables • SAT-Q and math grades in school • Correlation for females greater than for males • Sex is a moderator
Types of Moderators • Categorical or nominal • Analogous to factors in ANOVA • Sex, race, study type (published vs. disssertation vs. not) • Analyzed by analog to ANOVA • Continuous • Time between test & retest, age of participants, number of visits or duration of therapy • Analyzed by weighted regression
Fixed model The moderator(s) are expected to account for all systematic variance in effect sizes. One analysis only. Mixed model The moderators(s) account for some, but not all, variance in effect sizes. Some REVC left over. Recalculate weights for residual REVC and re-estimate parameters. Types of Analysis
Hypothetical SAT data What is the correlation overall? Is it different for males and females?
Test of Moderator The test of the moderator is the test of QB. The test has df = (Number groups –1). Here df=1, QB=18.99, p<.05.
Mixed Model In the previous example, the moderators accounted for all the variance in the effect sizes (excepting sampling error). Suppose there was remaining variance, e.g., QW = 6.
Mixed Model (2) In this example, it makes little difference whether fixed or mixed. Sometimes it matters.
Analog to ANOVA Review the Excel and SAS analog to ANOVA programs posted online.
Weighted Regression • OLS regression • Assume equal error variances (homoscedasticity) • Estimate magnitude of error, minimize SSE • Weighted regression • Error variances assumed known • Error variances are unequal • In meta-analysis, we know the sampling (error) variances, so can use weighted regression • Minimize weighted SSE
Weighted Regression Defined OLS WLS Assume uncorrelated errors not all equal to one another. p is the number of IVs If cjj is S.E.2, the jth diagonal element of If cjj is the jth diagonal element of
Hypothetical SAT-Q and pct quant courses in GPA Does the percentage of quantitative courses influence the size of the correlation between SAT-Q and GPA?
SAT-Q Matrices (1) V X’
SAT-Q Matrices (2) Intercept Slope V-1
SAT-Q (3) S. E. S.E. = sqrt(cjj) t t (or z) = b/S.E. Intercept Slope
Mixed Model In the event that homogeneity analysis reveals a large Q value for the residual, you can use the Q value to estimate the residual REVC. The REVC can be used to recalculate the weights, and the regression can be recomputed as a mixed model.
Weighted Regression Review the weighted regression programs posted online.