Estimation of comparable standardized mean differences in cluster randomized trials with covariate adjustment

Standardized mean differences (SMDs) are widely used to quantify treatment effects in cluster-randomized trials. However, covariate adjustment in hierarchical linear models reduces the residual variance components used for standardization, which artificially inflates effect size estimates and undermines comparability across studies. We propose a unified family of estimators that recover the unadjusted variance components by rescaling the covariate-adjusted variance components using pseudo- $R^2$ indices. This rescaling places effect size estimates on a common reference scale, thereby improving comparability across studies and model specifications under standard modeling assumptions. The framework accommodates three covariate adjustment scenarios including level-1, level-2, and simultaneous both-level adjustments. Furthermore, it introduces three estimator types spanning method of moments, maximum likelihood, and a t-statistic reformulation suitable for meta-analysis from published summaries, alongside delta-method variance approximations for each. An empirical example and a simulation study apply the proposed covariate-adjusted SMDs across these scenarios to illustrate their implementation and demonstrate the consequences of omitting the correction.