On the uncertainty from the first-stage estimation of prognostic covariate adjustment in randomized controlled trials
Prognostic covariate adjustment (PROCOVA) is a two-sample two-stage estimation method for covariate adjustment in randomized controlled trials. In the first stage, a prognostic score, defined as the conditional expectation of an outcome given covariates under the control treatment, is estimated using historical data. In the second stage, analysis of covariance with the estimated prognostic score and treatment assignment as explanatory variables is performed, and the average treatment effect is estimated. Although the prognostic score is estimated in this procedure, the variance estimator, which treats the prognostic score as known, has been used. Furthermore, the difference in the asymptotic variance between cases where the prognostic score is known versus where it is estimated has not been previously clarified. In this study, we derived these two asymptotic variances and showed that they are equal. This result also holds when the prognostic score is estimated using machine learning with $L_2$ consistency. We also constructed two variance estimators: one that treats the prognostic score as known, and another that accounts for its estimation, and compared their performance through simulation studies and data applications. For PROCOVA, since both variance estimators are asymptotically valid, it is generally recommended to use a variance estimator that treats the prognostic score as known, as it is simpler to derive and implement. When historical data is small, a variance estimator that explicitly accounts for prognostic score estimation is recommended if conservative inference is preferred.