REPAIR: Random-effect prediction adjustment for inference in regression

Random-effect predictions from multilevel models are often used to relate individual differences in processes, such as growth or stress reactivity, to other variables. Yet, regressions using these predictions can distort regression coefficients and standard errors because the predictions combine shrinkage with sampling error. Here, we introduce and evaluate Random-Effect Prediction Adjustment for Inference in Regression (REPAIR), a two-step framework for linear regressions involving random effects. For linear Gaussian multilevel models, REPAIR obtains likelihood-only predictions by generalized least squares. A regularized method-of-moments estimator then uses estimated cluster-specific sampling-error covariances to correct structural moments. Four simulation studies examined random effects as predictors, outcomes, or both, including unequal cluster sizes. REPAIR generally showed small bias at medium to high predictor reliability and often had lower root mean squared error than alternative methods. Low predictor reliability remained challenging, particularly for confidence interval coverage in small samples. These findings support REPAIR as a practical framework for estimating structural relationships involving random effects while accounting for their measurement uncertainty.