A method for covariate adjustment in randomized controlled trials is prognostic covariate adjustment (PROCOVA). PROCOVA is a two-sample two-stage estimation method. In the first stage, the prognostic score, which is the conditional expectation of an outcome given covariates under control treatment, is estimated using historical data. In the second stage, ANCOVA 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 actually estimated in this procedure, the variance estimator, which treats the prognostic score as known, has been used. Furthermore, the difference in asymptotic variance between cases where the prognostic score is known and cases where it is estimated has not been clarified. In this study, we derived these two asymptotic variances and showed that they are equal. We also constructed the variance estimator, which treats the prognostic score as known, and the variance estimator, which accounts for the prognostic score estimation, and compared their performance through simulation studies and data application. Both variance estimators are asymptotically valid. When historical data is small, the variance estimator which explicitly accounts for the prognostic score estimation is recommended if one prefers conservative inference.

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