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

Abstract

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, , 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 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 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 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.