A proposed way of training deep neural networks could make image-based models less vulnerable to omitted-variable bias, the distortion that can arise when an important observed factor is left out. In simulations, the approach consistently recovered the true image-related effects, while competing methods either needed more data or failed to recover them. In a separate UK Biobank MRI test, synthetic age confounding reduced an uncontrolled model’s balanced-test AUC from about 0.70 to about 0.62; controlling for age recovered most of the gap when the network and final refit used disjoint data.
The document is an arXiv preprint dated 26 Aug 2026. It investigates whether observed covariates can be modeled explicitly in deep neural networks to address omitted-variable bias and estimate the image-related component.
The proposed fix
The proposed estimator begins with a pretrained network and refits its final layer. It adds covariate effects to that final layer in an additive form, uses ridge penalization to regularize the fit, and applies cross-fitting, in which different folds of the data are rotated through the fitting procedure. The goal is to separate the image-related component from modeled covariate contributions.
The same framework can create covariate-controlled predictions by marginalizing model predictions over the covariate distribution. In ordinary language, it averages predictions across the modeled covariate distribution so the reported output reflects a prediction after that covariate contribution has been controlled. The result remains tied to the additive covariate model used by the estimator.
Evidence from simulations
For each simulation setting, performance was estimated on a test set of 800 observations and across 50 independently drawn training datasets. Each training dataset was divided into equal training and validation parts. The tests examined whether explicit covariate modeling could recover the intended image-related effect.
Across the simulation comparisons, the proposed procedure showed better convergence than competing approaches. It consistently estimated true effects in simulated images, whereas existing methods were more data-hungry or did not recover the target effect. Exact numerical effect sizes, convergence rates and confidence intervals were not reported in the supplied analysis.
The theoretical analysis also warns that prediction-covariate decorrelation is not enough. Model outputs can remain biased even when predictions are uncorrelated with an omitted covariate. A simple correlation check, therefore, is not by itself a guarantee that the omitted-variable problem has been removed.
The MRI test
The neuroimaging application predicted high alcohol consumption from UK Biobank T1-weighted MRI after synthetic age confounding was introduced. It used 14,617 original training observations and 4,505 original test observations.
On an age-balanced held-out test set, the uncontrolled deep-neural-network model’s area under the curve, or AUC, fell from about 0.70 before age confounding to about 0.62 afterward. AUC measures how well a classifier separates two outcome groups. Age control recovered most of the gap only when the network backbone and final refit used disjoint data. The analysis did not report confidence intervals or exact per-fold AUC values.
The application also found lower fold-to-fold variance with three-fold cross-fitting than with two-fold cross-fitting. That points to a difference in result stability depending on how the data are rotated, although exact variance estimates were not reported.
The method’s boundaries
The correction is limited to confounders that have actually been observed and included. A factor that was never measured remains outside the method’s scope. The noisy-control tests likewise found that imperfect controls removed only part of omitted-variable bias, and additional data did not eliminate the remaining bias.
The reported separation is also built around additive covariate effects. The authors present the final-layer procedure as a way to separate image and covariate components and reduce omitted-variable bias, but that interpretation depends on the covariate effects being modeled correctly and on the additive framework being suitable.
The MRI demonstration used synthetic age confounding, so its reported recovery belongs to that constructed application. The broader evidence in the paper comes from simulated image benchmarks and this neuroimaging test, rather than from a wider set of applications.
Disclosure and access
The work was funded by the Deutsche Forschungsgemeinschaft, the German Research Foundation, under project number 459422098. The UK Biobank resource was used under application number 33073, and the paper reports that source code is available at its cited GitHub repository.
Paper data and sources
Original title: Controlling for Omitted Variable Bias in Deep Neural Networks
Authors: Manuel Pfeuffer, Roshan Prakash Rane, Kerstin Ritter, Sonja Greven
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text