# Increasing the efficiency of randomized trial estimates via linear adjustment for a prognostic score

arXiv: https://arxiv.org/abs/2012.09935 · HTML: https://mlsanity.com/paper/2012.09935/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2020.

## Built on by (most cited first, 15 of 15)

- [Efficient Randomized Experiments Using Foundation Models](https://mlsanity.com/paper/2502.04262/) (2025)
- [β-Intact-VAE: Identifying and Estimating Causal Effects under Limited Overlap](https://mlsanity.com/paper/2110.05225/) (2021)
- [Adaptive-TMLE for the Average Treatment Effect based on Randomized Controlled Trial Augmented with Real-World Data](https://mlsanity.com/paper/2405.07186/) (2024)
- [Constructing Confidence Intervals for Average Treatment Effects from Multiple Datasets](https://mlsanity.com/paper/2412.11511/) (2024)
- [Towards Principled Causal Effect Estimation by Deep Identifiable Models](https://mlsanity.com/paper/2109.15062/) (2021)
- [Deconfounding Scores and Representation Learning for Causal Effect Estimation with Weak Overlap](https://mlsanity.com/paper/2604.00811/) (2026)
- [Robust estimation of heterogeneous treatment effects in randomized trials leveraging external data](https://mlsanity.com/paper/2507.03681/) (2025)
- [A Weighted Prognostic Covariate Adjustment Method for Efficient and Powerful Treatment Effect Inferences in Randomized Controlled Trials](https://mlsanity.com/paper/2309.14256/) (2023)
- [Bayesian Prognostic Covariate Adjustment With Additive Mixture Priors](https://mlsanity.com/paper/2310.18027/) (2023)
- [Counterfactual Uncertainty Quantification of Factual Estimand of Efficacy from Before-and-After Treatment Repeated Measures Randomized Controlled Trials](https://mlsanity.com/paper/2411.09635/) (2024)
- [Uncovering Bias Mechanisms in Observational Studies](https://mlsanity.com/paper/2506.01191/) (2025)
- [Machine learning to optimize precision in the analysis of randomized trials: A journey in pre-specified, yet data-adaptive learning](https://mlsanity.com/paper/2512.13610/) (2025)
- [Should I use Synthetic Data for That? An Analysis of the Suitability of Synthetic Data for Data Sharing and Augmentation](https://mlsanity.com/paper/2602.03791/) (2026)
- [Digital Twins as Synthetic Controls in Single-Arm Trials](https://mlsanity.com/paper/2605.12832/) (2026)
- [Improving the Efficiency of Subgroup Analysis in Randomized Controlled Trials with TMLE](https://mlsanity.com/paper/2605.15483/) (2026)

## Abstract (arXiv)

Estimating causal effects from randomized experiments is central to clinical research. Reducing the statistical uncertainty in these analyses is an important objective for statisticians. Registries, prior trials, and health records constitute a growing compendium of historical data on patients under standard-of-care that may be exploitable to this end. However, most methods for historical borrowing achieve reductions in variance by sacrificing strict type-I error rate control. Here, we propose a use of historical data that exploits linear covariate adjustment to improve the efficiency of trial analyses without incurring bias. Specifically, we train a prognostic model on the historical data, then estimate the treatment effect using a linear regression while adjusting for the trial subjects' predicted outcomes (their prognostic scores). We prove that, under certain conditions, this prognostic covariate adjustment procedure attains the minimum variance possible among a large class of estimators. When those conditions are not met, prognostic covariate adjustment is still more efficient than raw covariate adjustment and the gain in efficiency is proportional to a measure of the predictive accuracy of the prognostic model above and beyond the linear relationship with the raw covariates. We demonstrate the approach using simulations and a reanalysis of an Alzheimer's Disease clinical trial and observe meaningful reductions in mean-squared error and the estimated variance. Lastly, we provide a simplified formula for asymptotic variance that enables power calculations that account for these gains. Sample size reductions between 10% and 30% are attainable when using prognostic models that explain a clinically realistic percentage of the outcome variance.