Within-Trial Prognostic Adjustment

• Within-trial prognostic adjustment is a variance-reduction strategy that uses baseline prognostic variables to improve precision in randomized trials without altering the treatment comparison.
• It employs methods such as augmented estimators, standardization, TMLE, and machine-learning derived scores to enhance power and reduce required sample sizes.
• The approach bridges trial design and analysis by combining stratification with advanced covariate adjustment, thereby yielding more robust and efficient inference.

Searching arXiv for papers on within-trial prognostic adjustment and closely related covariate-adjustment methods in randomized clinical trials. Within-trial prognostic adjustment is the use of baseline prognostic information in the analysis of a randomized clinical trial to improve precision without altering the randomized treatment comparison. Across recent work, the core rationale is consistent: treatment assignment is randomized, so baseline covariates or scores predictive of outcome can be used to reduce unexplained variability and thereby improve efficiency, power, or sample-size requirements. The contemporary literature frames this adjustment through several closely related devices—augmented estimators, standardization or G-computation, prognostic scores learned from historical control data, TMLE, and stratified designs—and emphasizes a recurring distinction between preserving a marginal trial estimand and inadvertently shifting to a conditional estimand in nonlinear models (Zhang, 2024, Liu et al., 2023, Lancker et al., 2024, Højbjerre-Frandsen et al., 31 Jul 2025, Wuethrich et al., 22 May 2026).

1. Definition and statistical objective

Within-trial prognostic adjustment means using baseline variables or a baseline-derived score that predicts the outcome under control or standard care to improve precision in the analysis of a randomized trial. The adjustment targets outcome variation, not treatment assignment, and therefore differs from confounding adjustment in observational studies (Siegfried et al., 2021). In randomized settings, the benefit is efficiency: unexplained residual variation is reduced, which can improve precision and power or lower the required sample size (Siegfried et al., 2021, Lancker et al., 2024).

Several papers formulate the target as a marginal treatment effect. For binary outcomes, one example is the unconditional risk difference

$$\mathrm{RD} = E\!\left(Y^{(1)}\right) - E\!\left(Y^{(0)}\right) = \Pr\!\left(Y^{(1)}=1\right)-\Pr\!\left(Y^{(0)}=1\right),$$

estimated by standardization over the observed baseline covariate distribution (Liu et al., 2023). In generalized linear model plug-in analyses, the target is written as

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$

which covers marginal causal effects such as the average treatment effect, risk ratio, or odds ratio (Jeppesen et al., 15 Oct 2025, Højbjerre-Frandsen et al., 28 Mar 2025). For continuous outcomes in linear models, the target is often the average treatment effect $\tau = \mu_1-\mu_0$ (Schuler et al., 2020), while other work studies marginal treatment effects embedded directly in transformation models so that adjustment preserves marginal interpretability in continuous, binary, ordinal, and time-to-event settings (Wuethrich et al., 22 May 2026).

A recurrent practical formulation uses a prognostic score: a one-dimensional summary of baseline covariates intended to predict the control outcome. In one notation this score is

$\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$

where $D$ indicates membership in the new trial, so the score is estimated on historical control data and then used as a covariate in the trial analysis (Jeppesen et al., 15 Oct 2025). Another paper writes the historical prediction target as

$m(X)\approx \mathbb E[Y_0\mid X],$

with trial-level score $M_i = m(X_i)$ (Schuler et al., 2020). This suggests a unifying perspective: within-trial prognostic adjustment is a variance-reduction strategy that compresses baseline prognostic structure into an analysis-stage adjustment object.

2. General efficiency theory and augmentation

A central modern account places within-trial prognostic adjustment in a general framework of regular, asymptotically linear estimators expressed as augmented estimators (Zhang, 2024). For a treatment effect $\delta=g(\mu_1)-g(\mu_0)$, the usual unadjusted estimator is written with influence function

$$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$

A broad class of estimators can then be represented as

$\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$

where $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$0 is any square-integrable function of baseline covariates (Zhang, 2024).

Under simple randomization, the asymptotic variance is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$1

and the optimal augmentation function is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$2

with

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$3

The paper’s interpretation is that efficient covariate adjustment is equivalent to approximating $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$4 (Zhang, 2024).

The same work gives a geometric identity,

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$5

which implies that efficiency loss is proportional to the squared $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$6 distance from the optimal augmentation function. This formulation explains why regression-based adjustment, standardization, and machine-learning-based predictors can all be seen as approximation strategies for the same efficiency target (Zhang, 2024). A plausible implication is that “within-trial prognostic adjustment” names a family of procedures whose common function is to move the estimator closer to the efficient influence-function-based augmentation.

Related efficient-estimation papers make the same point using AIPW or influence-function structure. One data-adaptive framework writes the adjusted arm-specific estimating functions as

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$7

and attributes robustness to misspecification to this orthogonalized form (Lancker et al., 2024). In GLM plug-in analyses, the influence-function-based arm-specific term is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$8

and marginal effects are obtained by delta-method combination (Jeppesen et al., 15 Oct 2025, Højbjerre-Frandsen et al., 28 Mar 2025). These formulations are analytically distinct but conceptually aligned: efficient prognostic adjustment operates by replacing unadjusted arm means with outcome predictions that are then debiased by augmentation terms.

3. Prognostic scores and external historical information

A large part of the literature emphasizes scores learned from historical control data. In an idealized two-arm $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$9 normal-outcome setting, one paper parameterizes the utility of prognostic adjustment by two quantities: $\tau = \mu_1-\mu_0$0, the explained variance of the true prognostic component on historical data, and $\tau = \mu_1-\mu_0$1, the correlation between the estimated and true prognostic scores (Siegfried et al., 2021). The data-generating model is

$\tau = \mu_1-\mu_0$2

with

$\tau = \mu_1-\mu_0$3

and adjusted residual variance

$\tau = \mu_1-\mu_0$4

The resulting residual-variance ratio is

$\tau = \mu_1-\mu_0$5

so the planned total sample size $\tau = \mu_1-\mu_0$6 for an unadjusted analysis reduces to

$\tau = \mu_1-\mu_0$7

under prognostic-score adjustment (Siegfried et al., 2021). The same paper states that substantial gains require both meaningful prognostic signal and accurate score estimation, and gives rule-of-thumb thresholds such as “more than 20% sample-size reduction” being plausible when $\tau = \mu_1-\mu_0$8 and $\tau = \mu_1-\mu_0$9 is around $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$0 or more (Siegfried et al., 2021).

In the linear-regression PROCOVA framework, historical data are used only to train the prognostic model, not to contribute outcomes directly to treatment-effect estimation. One formulation proceeds by fitting any learner $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$1 on historical data, generating $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$2, and then analyzing the trial with ordinary least squares including treatment, raw baseline covariates, the prognostic score, and, in the preferred specification, treatment interactions with both $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$3 and $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$4 (Schuler et al., 2020). Under constant treatment effects and exact control regression $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$5, oracle prognostic covariate adjustment is stated to be semiparametrically efficient; under $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$6 convergence of the learned score and comparable growth of historical and trial samples, the feasible estimator is asymptotically equivalent to the oracle one (Schuler et al., 2020).

That paper also gives a simplified bound for asymptotic variance,

$\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$7

and in the special case of $\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$8 randomization and common variance/correlation assumptions,

$\rho(W) \coloneqq \mathbb{E}[Y \,|\, W,A=0, D=0],$9

It reports that sample size reductions between $D$0 and $D$1 are attainable when prognostic models explain a clinically realistic percentage of outcome variance, and in a DHA Alzheimer’s disease example under $D$2 randomization found $D$3 versus $D$4 for 80% power under prognostic adjustment, described as about a $D$5 reduction in enrollment (Schuler et al., 2020).

Subsequent work extends the external-score idea to binary and non-Gaussian settings. PROCOVA-LR uses a prognostic score $D$6 from historical control data in logistic regression for binary endpoints,

$D$7

and derives power and sample-size calculations for the Wald test for the conditional odds ratio (Li et al., 2024). GLM plug-in extensions use the historical score on the link scale as an extra covariate and state that, under an additive treatment effect on the link scale, the resulting estimator is locally semiparametrically efficient (Højbjerre-Frandsen et al., 28 Mar 2025, Jeppesen et al., 15 Oct 2025).

A separate Bayesian literature keeps the basic prognostic covariate adjustment structure but supplements it with historical-data priors. One Bayesian prognostic covariate adjustment model writes

$D$8

with priors on $D$9 controlling trust in prognostic-model bias (Walsh et al., 2020). Another Bayesian PROCOVA paper uses a digital twin generator, prognostic score

$m(X)\approx \mathbb E[Y_0\mid X],$0

and an additive mixture prior combining informative and weakly informative components (Vanderbeek et al., 2023). These Bayesian developments are not purely within-trial in the narrow sense, since they explicitly leverage external history, but they are part of the same prognostic-adjustment tradition.

4. Trial-only construction, TMLE, and data-adaptive adjustment

A distinct line of work asks whether the prognostic score can be constructed from the trial data itself. One paper states that once this is done, “within-trial” prognostic score adjustment is nothing more than a form of TMLE (Højbjerre-Frandsen et al., 31 Jul 2025). The setup is a randomized trial with i.i.d. observations

$m(X)\approx \mathbb E[Y_0\mid X],$1

targeting

$m(X)\approx \mathbb E[Y_0\mid X],$2

The standard plug-in uses $m(X)\approx \mathbb E[Y_0\mid X],$3 to estimate

$m(X)\approx \mathbb E[Y_0\mid X],$4

while the within-trial prognostic score is taken to be

$m(X)\approx \mathbb E[Y_0\mid X],$5

The paper then shows that linear ANCOVA adjustment on this in-trial score corresponds to a TMLE fluctuation or update, and that within-trial prognostic adjustment and TMLE had very similar performance in simulation (Højbjerre-Frandsen et al., 31 Jul 2025). The reported findings include approximately nominal coverage around $m(X)\approx \mathbb E[Y_0\mid X],$6, greater stability than historical prognostic adjustment under covariate shift, and highest feasible power for trials with $m(X)\approx \mathbb E[Y_0\mid X],$7 in the heterogeneous-effect setting among the feasible methods (Højbjerre-Frandsen et al., 31 Jul 2025).

More generally, automated within-trial prognostic adjustment has been developed using data-adaptive outcome prediction and influence-function-based estimation (Lancker et al., 2024). This work allows stepwise selection, lasso, random forests, gradient boosting, neural networks, and Super Learner-type methods for outcome prediction, while preserving valid marginal treatment-effect inference. It presents non-split canonical-GLM estimators, sample-splitting or cross-fitting estimators, and TMLE or CV-TMLE variants (Lancker et al., 2024). A key claim is that with the true randomization probability plugged in, sample splitting can yield exact finite-sample unbiasedness even if the predictions are biased (Lancker et al., 2024). The paper emphasizes that the algorithm itself can be prespecified rather than the exact selected covariates and functional form.

Adaptive Prespecification within TMLE extends this selection logic by choosing among a prespecified menu of adjustment strategies using cross-validated estimated influence-curve-squared loss (Balzer et al., 2022). The candidate set includes the unadjusted estimator, working GLMs adjusting for one covariate, main-terms GLMs, stepwise regression, stepwise regression with pairwise interactions, LASSO, and MARS (Balzer et al., 2022). In simulations with $m(X)\approx \mathbb E[Y_0\mid X],$8 trials and $m(X)\approx \mathbb E[Y_0\mid X],$9, large-trial APS yielded relative efficiency around $M_i = m(X_i)$0–$M_i = m(X_i)$1 for binary outcomes and $M_i = m(X_i)$2–$M_i = m(X_i)$3 for continuous outcomes, corresponding to approximately $M_i = m(X_i)$4–$M_i = m(X_i)$5 and $M_i = m(X_i)$6–$M_i = m(X_i)$7 sample-size savings, respectively; the abstract summarizes this as $M_i = m(X_i)$8–$M_i = m(X_i)$9 reductions in sample size for the same power (Balzer et al., 2022). The requirement that the unadjusted estimator always be included underscores a common practical principle: adjustment should be allowed to default to no adjustment if the selected prognostic structure is weak.

A related efficient-estimation extension uses historical prognostic scores within nonparametric efficient estimators such as TMLE (Liao et al., 2023). Its stated result is that asymptotic efficiency cannot improve merely by adding a fixed function of baseline covariates, because $\delta=g(\mu_1)-g(\mu_0)$0 adds no information beyond $\delta=g(\mu_1)-g(\mu_0)$1, but finite-sample point estimation and standard-error estimation can improve because the historical score helps learn nuisance regressions more accurately (Liao et al., 2023). Simulation findings include about an $\delta=g(\mu_1)-g(\mu_0)$2 increase in power at $\delta=g(\mu_1)-g(\mu_0)$3 and about an $\delta=g(\mu_1)-g(\mu_0)$4 power gain at $\delta=g(\mu_1)-g(\mu_0)$5 when prognostic adjustment was used with efficient estimators in small trials (Liao et al., 2023). This suggests a useful distinction between asymptotic efficiency bounds and finite-sample operating characteristics.

5. Outcome-type-specific implementations

The methods differ substantially by outcome scale, particularly in whether adjustment preserves a marginal estimand automatically or requires standardization.

Continuous outcomes

For continuous outcomes, linear ANCOVA or augmented estimators are prominent. The Schuler-type prognostic covariate adjustment uses linear regression with robust sandwich variance (Schuler et al., 2020). A separate theory paper quantifies sample-size reduction directly in the normal linear setting by the factor $\delta=g(\mu_1)-g(\mu_0)$6 (Siegfried et al., 2021). Weighted PROCOVA extends the linear model to heteroskedastic settings using not only a prognostic score but also a personalized precision derived from a digital twin generator: $\delta=g(\mu_1)-g(\mu_0)$7 The variance model is

$\delta=g(\mu_1)-g(\mu_0)$8

and the weighted least-squares estimator is

$\delta=g(\mu_1)-g(\mu_0)$9

with $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$0 and $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$1 in the specific implementation (Vanderbeek et al., 2023). The paper reports power increases from $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$2 under PROCOVA to roughly $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$3–$$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$4 under Weighted PROCOVA when DTG-based variances explain $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$5–$$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$6 of variation in outcomes (Vanderbeek et al., 2023).

Longitudinal continuous outcomes motivate PROCOVA-MMRM, which uses time-matched prognostic scores in a repeated-measures mixed model: $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$7 The paper recommends REML with an unstructured covariance matrix and robust standard errors, and targets $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$8 at the final visit (Ross et al., 2024). In an Alzheimer’s disease reanalysis, unadjusted MMRM variance for ADAS-Cog11 was $$\psi()=g'(\mu_1)\frac{A(Y-\mu_1)}{\pi} - g'(\mu_0)\frac{(1-A)(Y-\mu_0)}{1-\pi}.$$9, PROCOVA-MMRM reduced it to $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$0, and PROCOVA-MMRM plus baseline ADAS-Cog11 reduced it further to $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$1; for CDR-SB the corresponding variances were $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$2, $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$3, and $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$4 (Ross et al., 2024).

Binary outcomes

For binary outcomes with risk-difference estimands, logistic-regression-based standardization or G-computation is used: $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$5 Predicted treatment and control risks are averaged over the full trial sample to estimate the marginal risk difference (Liu et al., 2023). That paper argues that the usual conditional delta-method variance does not capture variability from the covariate distribution and proposes the unconditional variance estimator

$\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$6

Its simulations show that methods using prognostic covariates have substantially smaller standard errors than unadjusted methods, that HC2 performs best overall in large samples, and that HC3 is preferred in smaller samples (Liu et al., 2023).

For binary outcomes with odds-ratio modeling, PROCOVA-LR addresses non-collapsibility. It states that the asymptotic relative efficiency of the unadjusted coefficient estimator relative to the adjusted one at the null is

$\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$7

with

$\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$8

From this it defines an efficiency factor $\widehat\delta_{\text{aug}(b)} = \widehat\delta_{\text{emp} - \frac1n\sum_{i=1}^n (A_i-\pi)b(X_i),$9 and prospective sample-size reduction

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$00

The paper emphasizes that covariate adjustment can increase Wald-test power for the conditional odds ratio even if the adjusted coefficient’s variance appears larger than the unadjusted one, because the estimands differ under non-collapsibility (Li et al., 2024). It then uses g-computation to estimate marginal risk difference, relative risk, and odds ratio from the fitted adjusted model (Li et al., 2024).

Binary outcomes also motivate stratification-based approaches. PROCOVA-CMH uses a prognostic score learned from historical data, discretizes it into strata, and analyzes the binary endpoint with a Cochran–Mantel–Haenszel estimator for the marginal risk ratio

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$01

It provides two closed-form prospective asymptotic variance estimators and reports variance reduction around $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$02 when $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$03 in the baseline simulation case (Vanderbeek et al., 2022).

Win odds and pairwise composite outcomes

Covariate adjustment has also been adapted to the win-odds estimand by exploiting its relation to the marginal probabilistic index

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$04

The win odds is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$05

and the conditional probabilistic index is modeled with a logit PIM,

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$06

If the PIM is correctly specified, the adjusted estimator is stated to achieve the semiparametric efficiency bound; even under misspecification it remains consistent and asymptotically normal (Scheidegger et al., 18 Nov 2025). The same paper notes a slight inflation of type I error for small sample sizes.

Marginal-preserving adjustment for nonlinear outcomes

A broader challenge is that standard adjustment in nonlinear models may change the estimand. One recent solution embeds the marginal treatment effect directly in a joint nonparanormal model for outcome and covariates (Wuethrich et al., 22 May 2026). It begins with a marginal transformation model

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$07

then constructs a joint Gaussian-copula model

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$08

with

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$09

This preserves the marginal treatment effect $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$10 while allowing explicit prognostic and predictive effects to be ranked on a common latent scale (Wuethrich et al., 22 May 2026).

6. Randomization design, stratification, and common misconceptions

A major theoretical clarification concerns the relation between covariate adjustment in analysis and stratified randomization in design. One paper shows that stratified randomization improves a given approximation to the optimal augmentation function by projecting away the component of approximation error explainable by the stratification factor $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$11 (Zhang, 2024). Its asymptotic variance under stratified randomization is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$12

so stratification cannot increase variance (Zhang, 2024). A key identity is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$13

interpreted as stratified randomization being asymptotically equivalent to adding the best possible augmentation term depending only on $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$14 (Zhang, 2024).

The practical conclusion is not that all important covariates must be stratified on. Rather, the paper states that in designing a trial with stratified randomization, it is not essential to include all important covariates in the stratification because their prognostic information can be incorporated through covariate adjustment (Zhang, 2024). It further states that under stratified randomization, adjusting only for the stratification factor in the analysis is not expected to improve efficiency; the real gains come from baseline covariates not used for stratification, or richer functions of covariates underlying $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$15 (Zhang, 2024). This addresses a common misconception that post hoc adjustment for the stratification factor alone should always improve precision.

Group sequential designs add another nuance. In that setting, within-trial prognostic adjustment can use baseline variables $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$16 and short-term outcomes $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$17. The asymptotic relative efficiency of an efficient adjusted estimator versus unadjusted is

$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$18

where $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$19 quantifies prognostic value of baseline variables, $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$20 is additional prognostic value of short-term outcomes beyond $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$21, $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$22 is the limiting fraction with $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$23 observed, $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$24 the fraction with $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$25 observed, and $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$26 treatment-effect heterogeneity (Qian et al., 2019). That paper states that adjusting for a prognostic baseline variable leads to at least as much asymptotic precision gain as adjusting for an equally prognostic short-term outcome (Qian et al., 2019). This suggests that baseline prognostic adjustment remains the primary efficiency device even when interim information is available.

Another subtle issue is first-stage score-estimation uncertainty in two-stage PROCOVA. A 2026 paper studies PROCOVA as a two-sample, two-stage estimator and shows that for the ATE coefficient in the randomized-trial ANCOVA, the asymptotic variance is the same whether the prognostic score is treated as known or estimated (Seya et al., 2 Apr 2026). Thus the simpler fixed-score variance estimator is asymptotically justified for ATE inference, though the paper recommends the estimator that explicitly accounts for score estimation if conservative inference is preferred when historical data are small (Seya et al., 2 Apr 2026).

7. Empirical findings, software, and practical orientation

Simulation studies and case studies in the cited literature generally support the same qualitative pattern: when prognostic covariates or prognostic scores are informative, adjusted estimators are more precise than unadjusted ones, and when the adjustment object is weak, gains attenuate without introducing bias.

The geometric covariate-adjustment paper reports negligible bias throughout its simulations, covariate-adjustment gains when covariates were prognostic, lack of benefit from adjusting only for the stratification factor under stratified randomization, and frequent superiority of Super Learner augmentation under model misspecification (Zhang, 2024). In Scenario A with $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$27 and a continuous outcome, empirical SD under simple randomization was $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$28 for the empirical estimator, $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$29 for regression on $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$30, and $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$31 for regression on all $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$32 and augmentation (Zhang, 2024). In the NINDS rt-PA stroke trial illustration, correcting standard errors for stratified randomization made only minimal difference, whereas adjusting for other covariates reduced standard errors more materially; for the Barthel Index, empirical SE was $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$33, regression on covariates $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$34, augmented $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$35, and calibrated augmented $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$36 (Zhang, 2024).

The theoretical quantification paper found that empirical variance reduction in nonlinear simulations closely followed the factor $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$37, and that the design factor based on $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$38 tended to underestimate the actual variance reduction in its setup (Siegfried et al., 2021). The weighted heteroskedastic PROCOVA paper reports Type I error rates around $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$39–$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$40 and coverage around $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$41–$$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$42 across 10,000 simulated datasets per setting, with variance reductions up to about $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$43 in favorable regimes (Vanderbeek et al., 2023). PROCOVA-MMRM simulations report, for example, under the alternative in the linear scenario, average variance $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$44 for MMRM versus $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$45 for PROCOVA-MMRM and power increasing from $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$46 to $$\Psi_a = \mathbb{E}[Y(a)], \qquad \Psi = r(\Psi_1,\Psi_0),$$47 (Ross et al., 2024).

Software support now exists. The R package postcard implements the GLM plug-in method with or without prognostic score adjustment, including rctglm() and rctglm_with_prognosticscore(), influence-function-based variance estimation, optional cross-validated variance estimation via cv_variance = TRUE, power approximation through power_marginaleffect(), and integration of a Discrete Super Learner via fit_best_learner() (Jeppesen et al., 15 Oct 2025). The package article describes it as implementing the GLM plug-in procedure for estimating marginal effects and variance with or without prognostic adjustment and approximating statistical power (Jeppesen et al., 15 Oct 2025). This reflects a broader movement in the literature toward pre-specifiable, machine-learning-compatible adjustment workflows.

Across papers, the practical recommendations converge. Adjustment should use pre-treatment variables only. Prognostic scores are most useful when they are genuinely predictive and sufficiently transferable from historical controls to the trial population. Stratification should be used for a manageable set of strong predictors, with additional prognostic information recovered analytically in the outcome model (Zhang, 2024). In nonlinear models, analysts should distinguish carefully between conditional and marginal estimands and use standardization, plug-in marginalization, or joint marginal-preserving models where appropriate (Liu et al., 2023, Jeppesen et al., 15 Oct 2025, Wuethrich et al., 22 May 2026). When within-trial scores are estimated from the trial itself, the appropriate conceptual framework is TMLE or closely related targeted or influence-function-based estimation rather than a naive two-stage regression view (Højbjerre-Frandsen et al., 31 Jul 2025).

Taken together, the literature presents within-trial prognostic adjustment as a unified efficiency program rather than a single estimator class. Its core forms—covariate adjustment, prognostic-score adjustment, stratified design, influence-function augmentation, and targeted updating—are best understood as different mechanisms for exploiting baseline prognostic information while preserving the randomized clinical trial estimand and inferential validity.