Random Effects Coefficient of Determination for Mixed and Meta-Analysis Models

The key feature of a mixed model is the presence of random effects. We have developed a coefficient, called the random effects coefficient of determination, R-r(2), that estimates the proportion of the conditional variance of the dependent variable explained by random effects. This coefficient takes values from 0 to 1 and indicates how strong the random effects are. The difference from the earlier suggested fixed effects coefficient of determination is emphasized. If R-r(2) is close to 0, there is weak support for random effects in the model because the reduction of the variance of the dependent variable due to random effects is small; consequently, random effects may be ignored and the model simplifies to standard linear regression. The value of R-r(2) apart from 0 indicates the evidence of the variance reduction in support of the mixed model. If random effects coefficient of determination is close to 1 the variance of random effects is very large and random effects turn into free fixed effects-the model can be estimated using the dummy variable approach. We derive explicit formulas for R-r(2) in three special cases: the random intercept model, growth curve model, and meta-analysis model. Theoretical results are illustrated with three mixed model examples: (1) travel time to the nearest cancer center for women with breast cancer in the U. S.; (2) cumulative time watching alcohol related scenes in movies among young U. S. teens, as a risk factor for early drinking onset; and (3) the classic example of the meta-analysis model for combination of 13 studies on tuberculosis vaccine.