The asymptotic variance of the treatment effect estimator in PROCOVA is identical whether the prognostic score is known or estimated from historical data.
Then, the following holds: √ n { ˆβ emp n (ˆθ˜n) − ˆβ emp n (θ∗) } = √ n { ˆβn(ˆθ˜n) − ˆβn(θ∗) } +op(1)
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On the uncertainty from the first-stage estimation of prognostic covariate adjustment in randomized controlled trials
The asymptotic variance of the treatment effect estimator in PROCOVA is identical whether the prognostic score is known or estimated from historical data.