  Statistical power is often a concern for clustered RCTs due to variance
inflation from design effects and the high cost of adding study clusters (such
as hospitals, schools, or communities). While covariate pre-specification is
the preferred approach for improving power to estimate regression-adjusted
average treatment effects (ATEs), further precision gains can be achieved
through covariate selection once primary outcomes have been collected. This
article uses design-based methods underlying clustered RCTs to develop a
Lasso-OLS hybrid procedure for the post-hoc selection of covariates and ATE
estimation that avoids model overfitting and lack of transparency. In the first
stage, lasso estimation is conducted using cluster-level averages, where
asymptotic normality is proved using a new central limit theorem for finite
population regression estimators. In the second stage, ATEs and design-based
standard errors are estimated using weighted least squares with the first stage
lasso covariates. This nonparametric approach applies to continuous, binary,
and discrete outcomes. Simulation results indicate that Type 1 errors of the
second stage ATE estimates are near nominal values and standard errors are near
true ones, although somewhat conservative with small samples. The method is
demonstrated using data from a large, federally funded clustered RCT testing
the effects of school-based programs promoting behavioral health.
