---
title: Within-Trial Prognostic Adjustment
url: https://www.emergentmind.com/topics/within-trial-prognostic-adjustment
type: topic
---

# Within-Trial Prognostic Adjustment

Searching arXiv for recent 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 [2401.11352], [2308.15688], [2404.11150], [2507.23446], [2605.23691].

## 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 [2111.03391]. In randomized settings, the benefit is efficiency: unexplained residual variation is reduced, which can improve precision and power or lower the required sample size [2111.03391], [2404.11150].

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 [2308.15688]. 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 [2510.13347], [2503.22284]. For continuous outcomes in linear models, the target is often the average treatment effect \(\tau = \mu_1-\mu_0\) [2012.09935], 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 [2605.23691].

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 [2510.13347]. Another paper writes the historical prediction target as
\[
m(X)\approx \mathbb E[Y_0\mid X],
\]
with trial-level score \(M_i = m(X_i)\) [2012.09935]. 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 [2401.11352]. 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 \(b(X)\) is any square-integrable function of baseline covariates [2401.11352].

Under simple randomization, the asymptotic variance is
\[
\sigma^2(b)=\operatorname{Var}\{\psi()-(A-\pi)b()\},
\]
and the optimal augmentation function is
\[
b_{\text{opt}(X) = \frac{g'(\mu_1)\{m_1(X)-\mu_1\}{\pi} + \frac{g'(\mu_0)\{m_0(X)-\mu_0\}{1-\pi},
\]
with
\[
m_a(X)=E\{Y\mid X,A=a\}=E\{Y(a)\mid X\},\qquad a=0,1.
\]
The paper’s interpretation is that efficient covariate adjustment is equivalent to approximating \(b_{\text{opt}}\) [2401.11352].

The same work gives a geometric identity,
\[
\sigma^2(b)=\sigma^2(b_{\text{opt})+\pi(1-\pi)\|b-b_{\text{opt}\|_2^2,
\]
which implies that efficiency loss is proportional to the squared \(L_2\) 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 [2401.11352]. 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
\[
\frac{Z_i}{\pi}(Y_i-h_1(X_i)) + h_1(X_i), \qquad
\frac{1-Z_i}{1-\pi}(Y_i-h_0(X_i)) + h_0(X_i),
\]
and attributes robustness to misspecification to this orthogonalized form [2404.11150]. In GLM plug-in analyses, the influence-function-based arm-specific term is
\[
\hat\phi_{a}(A_i, W_i, Y_i) =
\frac{I(A_i = a)}{\pi_a}(Y_i-\hat\mu(a, W_i))+(\hat\mu(a, W_i)-\hat\Psi_a),
\]
and marginal effects are obtained by delta-method combination [2510.13347], [2503.22284]. 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 \(1{:}1\) normal-outcome setting, one paper parameterizes the utility of prognostic adjustment by two quantities: \(R^2\), the explained variance of the true prognostic component on historical data, and \(\rho\), the correlation between the estimated and true prognostic scores [2111.03391]. The data-generating model is
\[
Y = \alpha + \beta z + \{\pi s(X) + \sqrt{\sigma^2-\pi^2}\,\epsilon\}, \qquad \epsilon \sim N(0,1),
\]
with
\[
R^2 = \frac{\pi^2}{\sigma^2},
\]
and adjusted residual variance
\[
\sigma^2 - \pi^2 \rho^2.
\]
The resulting residual-variance ratio is
\[
1 - R^2\rho^2,
\]
so the planned total sample size \(n\) for an unadjusted analysis reduces to
\[
(1 - R^2\rho^2)\, n
\]
under prognostic-score adjustment [2111.03391]. 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 \(R^2 > 0.3\) and \(\rho\) is around \(0.8\) or more [2111.03391].

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 \(\mathcal M\) on historical data, generating \(M_i=m(X_i)\), 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 \(X\) and \(M\) [2012.09935]. Under constant treatment effects and exact control regression \(m(X)=\mathbb E[Y_0\mid X]\), oracle prognostic covariate adjustment is stated to be semiparametrically efficient; under \(L_2\) convergence of the learned score and comparable growth of historical and trial samples, the feasible estimator is asymptotically equivalent to the oracle one [2012.09935].

That paper also gives a simplified bound for asymptotic variance,
\[
\mathrm{AVar}(\hat\tau) \;\le\; \frac{\sigma_0^2}{\pi_0} + \frac{\sigma_1^2}{\pi_1}
- \pi_0 \pi_1 \left( \frac{\rho_1\sigma_1}{\pi_1} + \frac{\rho_0 \sigma_0}{\pi_0} \right)^2,
\]
and in the special case of \(1{:}1\) randomization and common variance/correlation assumptions,
\[
\frac{n^\dagger}{n}=1-\rho^2.
\]
It reports that sample size reductions between \(10\%\) and \(30\%\) are attainable when prognostic models explain a clinically realistic percentage of outcome variance, and in a DHA Alzheimer’s disease example under \(3{:}2\) randomization found \(n=402\) versus \(n=321\) for 80% power under prognostic adjustment, described as about a \(20\%\) reduction in enrollment [2012.09935].

Subsequent work extends the external-score idea to binary and non-Gaussian settings. PROCOVA-LR uses a prognostic score \(m_i\) from historical control data in logistic regression for binary endpoints,
\[
\mathrm{Pr} \left (y_i = 1 \mid w_i, m_i \right )
= \frac{\mathrm{exp} \left ( \beta_0 + \beta_1 w_i + \beta_2 m_i \right )}
{1 + \mathrm{exp} \left ( \beta_0 + \beta_1 w_i + \beta_2 m_i \right )},
\]
and derives power and sample-size calculations for the Wald test for the conditional odds ratio [2402.18900]. 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 [2503.22284], [2510.13347].

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
\[
Y_i \mid M_1,\dots,M_n,W_1,\dots,W_n
\sim \beta_0 + \beta_1 W_i + \beta_2(M_i-\bar M)+\bar M + N(0,\sigma^2),
\]
with priors on \(\beta_0/\sigma\) controlling trust in prognostic-model bias [2012.13112]. Another Bayesian PROCOVA paper uses a digital twin generator, prognostic score
\[
m_i = \int_{-\infty}^{\infty} y\, dG_i(y),
\]
and an additive mixture prior combining informative and weakly informative components [2310.18027]. 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 [2507.23446]. The setup is a randomized trial with i.i.d. observations
\[
O_i = (W_i, A_i, Y_i), \qquad i=1,\dots,n,
\]
targeting
\[
\Psi^* = \mathbb{E}[Y(1) - Y(0)].
\]
The standard plug-in uses \(\hat\mu(a,W)\) to estimate
\[
\hat\Psi = \frac{1}{n}\sum_{i=1}^n \hat\mu(1,W_i)-\hat\mu(0,W_i),
\]
while the within-trial prognostic score is taken to be
\[
\widehat{\rho}_1(A,W)=\hat\mu(A,W).
\]
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 [2507.23446]. The reported findings include approximately nominal coverage around \(95\%\), greater stability than historical prognostic adjustment under covariate shift, and highest feasible power for trials with \(n>100\) in the heterogeneous-effect setting among the feasible methods [2507.23446].

More generally, automated within-trial prognostic adjustment has been developed using data-adaptive outcome prediction and influence-function-based estimation [2404.11150]. 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 [2404.11150]. 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 [2404.11150]. 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 [2210.17453]. 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 [2210.17453]. In simulations with \(5000\) trials and \(N=500\), large-trial APS yielded relative efficiency around \(0.57\)–\(0.71\) for binary outcomes and \(0.58\)–\(0.80\) for continuous outcomes, corresponding to approximately \(29\)–\(43\%\) and \(20\)–\(42\%\) sample-size savings, respectively; the abstract summarizes this as \(20\)–\(43\%\) reductions in sample size for the same power [2210.17453]. 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 [2305.19180]. Its stated result is that asymptotic efficiency cannot improve merely by adding a fixed function of baseline covariates, because \(R=g(W)\) adds no information beyond \(W\), but finite-sample point estimation and standard-error estimation can improve because the historical score helps learn nuisance regressions more accurately [2305.19180]. Simulation findings include about an \(11\%\) increase in power at \(n=250\) and about an \(80\%\) power gain at \(n=100\) when prognostic adjustment was used with efficient estimators in small trials [2305.19180]. 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 [2012.09935]. A separate theory paper quantifies sample-size reduction directly in the normal linear setting by the factor \((1-R^2\rho^2)\) [2111.03391]. 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:
\[
m_i = \int r\, dF_{i,0}(r), \qquad
s_i^2 = \int (r-m_i)^2\, dF_{i,0}(r).
\]
The variance model is
\[
\log(\sigma_i^2) = \gamma_0 + \gamma_1 \log(s_i^2) + \xi_i,
\]
and the weighted least-squares estimator is
\[
\hat{\beta} = \left(\mathbf V^\top \widehat{\Omega}^{-1}\mathbf V\right)^{-1}
\mathbf V^\top \widehat{\Omega}^{-1} y,
\]
with \(v_i = (1,w_i,m_i)^\top\) and \(u_i = (1,\log(s_i^2))^\top\) in the specific implementation [2309.14256]. The paper reports power increases from \(80\%\) under PROCOVA to roughly \(85\%\)–\(90\%\) under Weighted PROCOVA when DTG-based variances explain \(5\%\)–\(10\%\) of variation in outcomes [2309.14256].

Longitudinal continuous outcomes motivate PROCOVA-MMRM, which uses time-matched prognostic scores in a repeated-measures mixed model:
\[
\left [ \mathbf{y}_i \mid w_i, \mathbf{x}_i \right ]
\sim \mathrm{Normal}_{T_i}
\left (
\begin{pmatrix} \beta_{0,1} \\ \vdots \\ \beta_{0,T_i} \end{pmatrix}
+ w_i \begin{pmatrix} \beta_{w,1} \\ \vdots \\ \beta_{w,T_i} \end{pmatrix}
+ \mathbf{B}_{x}\mathbf{x}_i, \mathbf{R}_i \right ).
\]
The paper recommends REML with an unstructured covariance matrix and robust standard errors, and targets \(\beta_{w,T}\) at the final visit [2404.17576]. In an Alzheimer’s disease reanalysis, unadjusted MMRM variance for ADAS-Cog11 was \(1.024\), PROCOVA-MMRM reduced it to \(0.907\), and PROCOVA-MMRM plus baseline ADAS-Cog11 reduced it further to \(0.857\); for CDR-SB the corresponding variances were \(0.113\), \(0.099\), and \(0.093\) [2404.17576].

### Binary outcomes

For binary outcomes with risk-difference estimands, logistic-regression-based standardization or G-computation is used:
\[
\mathrm{logit}\left\{\Pr(Y=1\mid Z,\mathbf W)\right\}
= \beta_0+\beta_1 Z+\boldsymbol\beta_2^\top \mathbf W.
\]
Predicted treatment and control risks are averaged over the full trial sample to estimate the marginal risk difference [2308.15688]. That paper argues that the usual conditional delta-method variance does not capture variability from the covariate distribution and proposes the unconditional variance estimator
\[
\widehat{\mathrm{Var}(\widehat{\mathrm{RD}) =
\left(\mathbf d_{(1)}-\mathbf d_{(0)}\right)^\top
V_{\text{sandwich}}
\left(\mathbf d_{(1)}-\mathbf d_{(0)}\right)
+ \frac{\widehat{\sigma}^2_{\mathrm{RD}}}{n}.
\]
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 [2308.15688].

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
\[
1-\frac{\mathrm{Var}(\mu_{0,i})}
{E(\mu_{0,i}) \left\{ 1 - E(\mu_{0,i}) \right\} },
\]
with
\[
\mu_{0,i} = \frac{\mathrm{exp} \left ( \beta_0 + \beta_2 x_i \right )}
{1 + \mathrm{exp} \left ( \beta_0 + \beta_2 x_i \right )}.
\]
From this it defines an efficiency factor \(f_{\mathrm{EFF}}\) and prospective sample-size reduction
\[
N_{\text{P-LR} = f_{\mathrm{EFF}^2 N_{\mathrm{UN}.
\]
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 [2402.18900]. It then uses g-computation to estimate marginal risk difference, relative risk, and odds ratio from the fitted adjusted model [2402.18900].

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
\[
\hat\psi = \frac {\sum_k Z_{1k} N_{0k} / N_k}
{\sum_k Z_{0k} N_{1k} / N_k}.
\]
It provides two closed-form prospective asymptotic variance estimators and reports variance reduction around \(20\%\) when \(r^2_{XY}\approx 0.2\) in the baseline simulation case [2212.09903].

### 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
\[
\nu = P(Y_i \preceq Y_j \mid A_i=0, A_j=1).
\]
The win odds is
\[
\theta = \frac{\nu}{1-\nu},
\]
and the conditional probabilistic index is modeled with a logit PIM,
\[
P(Y_i \preceq Y_j \mid A_i, A_j, X_i, X_j)
= \operatorname{expit}\!\left(\tau_A(A_j-A_i)+\tau_X^\top(X_j-X_i)\right).
\]
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 [2511.14292]. 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 [2605.23691]. It begins with a marginal transformation model
\[
F_w(y) = G\!\big(h(y) - \tau w\big),
\]
then constructs a joint Gaussian-copula model
\[
\Pr(\mathbf X \le \mathbf x, Y \le y \mid W=w)
= \Phi_{\Sigma(w)}\!\big(\,h_1(x_1),\dots,h_{J-1}(x_{J-1}), h_J(y\mid w)\big),
\]
with
\[
h_J(y\mid w) = \Phi^{-1}\!\Big(G\!\big(h(y)-\tau w\big)\Big).
\]
This preserves the marginal treatment effect \(\tau\) while allowing explicit prognostic and predictive effects to be ranked on a common latent scale [2605.23691].

## 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 \(S=s(X)\) [2401.11352]. Its asymptotic variance under stratified randomization is
\[
\sigma_{\text{st}^2(b) = \sigma^2(b)
- \frac{1}{\pi(1-\pi)}
E\!\left[ \left(
E[(A-\pi)\{\psi()-(A-\pi)b()\}\mid S]
\right)^2 \right],
\]
so stratification cannot increase variance [2401.11352]. A key identity is
\[
\min_{c\in\mathcal C}\sigma^2(b+c)=\sigma_{\text{st}^2(b),
\]
interpreted as stratified randomization being asymptotically equivalent to adding the best possible augmentation term depending only on \(S\) [2401.11352].

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 [2401.11352]. 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 \(S\) [2401.11352]. 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 \(W\) and short-term outcomes \(L\). The asymptotic relative efficiency of an efficient adjusted estimator versus unadjusted is
\[
ARE = \frac{1}
{1 + (p_y/2)\gamma - R^2_W - (1-p_y/p_l)R^2_{L\mid W},
\]
where \(R^2_W\) quantifies prognostic value of baseline variables, \(R^2_{L\mid W}\) is additional prognostic value of short-term outcomes beyond \(W\), \(p_l\) is the limiting fraction with \(L\) observed, \(p_y\) the fraction with \(Y\) observed, and \(\gamma\) treatment-effect heterogeneity [1910.05800]. 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 [1910.05800]. 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 [2604.01911]. 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 [2604.01911].

## 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 [2401.11352]. In Scenario A with \(n=200\) and a continuous outcome, empirical SD under simple randomization was \(0.25\) for the empirical estimator, \(0.22\) for regression on \(S\), and \(0.16\) for regression on all \(X\) and augmentation [2401.11352]. 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 \(0.033\), regression on covariates \(0.028\), augmented \(0.028\), and calibrated augmented \(0.027\) [2401.11352].

The theoretical quantification paper found that empirical variance reduction in nonlinear simulations closely followed the factor \(1-R^2\hat\rho^2\), and that the design factor based on \(\widehat R^2_{\text{OOS}}\) tended to underestimate the actual variance reduction in its setup [2111.03391]. The weighted heteroskedastic PROCOVA paper reports Type I error rates around \(0.050\)–\(0.056\) and coverage around \(0.944\)–\(0.950\) across 10,000 simulated datasets per setting, with variance reductions up to about \(50\%\) in favorable regimes [2309.14256]. PROCOVA-MMRM simulations report, for example, under the alternative in the linear scenario, average variance \(0.165\) for MMRM versus \(0.122\) for PROCOVA-MMRM and power increasing from \(0.827\) to \(0.920\) [2404.17576].

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()` [2510.13347]. 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 [2510.13347]. 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 [2401.11352]. 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 [2308.15688], [2510.13347], [2605.23691]. 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 [2507.23446].

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.

Source: https://www.emergentmind.com/topics/within-trial-prognostic-adjustment