Learning Interactions Between Continuous Treatments and Covariates with a Semiparametric Model

TL;DR

The paper tackles the challenge of estimating effects of continuous treatments on binary outcomes by introducing a semiparametric model that decomposes the treatment effect into a prognostic score and a treatment-interaction score, linked through an unknown nonparametric function g in the form Pr(Y=1|X,τ) = σ(X^Tβ + g(X^Tξ − τ)). Estimation uses Nadaraya-Watson regression to nonparametrically recover residual components, with ξ regularized by an L1 penalty under identifiability constraints and β learned accordingly. The method is validated via four numerical simulation scenarios demonstrating empirical convergence and robustness, and then applied to the IWPC Warfarin pharmacogenomics dataset to derive personalized dosing recommendations by distilling soft labels from an expert model into a student model. The results provide interpretable prognostic and treatment-interaction scores that integrate genetic and clinical covariates, offering practical guidance for safer and more effective anticoagulation therapy. Overall, the work advances interpretable, flexible modeling for continuous treatments in clinical settings and highlights potential for broader use in personalized treatment decision-making.

Abstract

Estimating the impact of continuous treatment variables (e.g., dosage amount) on binary outcomes presents significant challenges in modeling and estimation because many existing approaches make strong assumptions that do not hold for certain continuous treatment variables. For instance, traditional logistic regression makes strong linearity assumptions that do not hold for continuous treatment variables like time of initiation. In this work, we propose a semiparametric regression framework that decomposes effects into two interpretable components: a prognostic score that captures baseline outcome risk based on a combination of clinical, genetic, and sociodemographic features, and a treatment-interaction score that flexibly models the optimal treatment level via a nonparametric link function. By connecting these two parametric scores with Nadaraya-Watson regression, our approach is both interpretable and flexible. The potential of our approach is demonstrated through numerical simulations that show empirical estimation convergence. We conclude by applying our approach to a real-world case study using the International Warfarin Pharmacogenomics Consortium (IWPC) dataset to show our approach's clinical utility by deriving personalized warfarin dosing recommendations that integrate both genetic and clinical data, providing insights towards enhancing patient safety and therapeutic efficacy in anticoagulation therapy.

Figures (12)

• Figure 1: Overview of our proposed pipeline. During training, the model learns $\hat{\beta}$ and $\hat{\xi}$ associated prognosis and treatment effectiveness, respectively. During inference, as the prognostic score $X^T\hat{\beta}$ is fixed conditioning on the patient's attributes, we can directly visualize outcome density using a heatmap with estimated $\hat{g}$. The optimal treatment value is then extrapolated at the maximum point along the y-axis corresponding to a fixed $X^T\hat{\xi}$.
• Figure 2: Directed Acyclic Graph in scenario 3.
• Figure 3: Empirical Convergence of Estimations. The four rows represent scenario 1-4, respectively. The heatmaps show estimated values of the modeled function across two dimensions: the prognostic score ($X^T\beta$, x-axis) and the covariate-treatment term ($X^T\xi - \tau$, y-axis). The colored density of the heatmap represents $\bar{Y} = \log{\frac{Pr(Y=1|X,\tau)}{Pr(Y=0|X,\tau)}}$, the log-odds ratio, where higher values indicate a greater likelihood of the positive outcome. Columns in the heatmap panel represent increasing sample sizes ($n = 10$ to $10^4$), illustrating progressive convergence towards the true underlying function, shown in the rightmost (Ground Truth) column. The far-right plots quantify convergence through Mean Squared Error (MSE), bootstrapped ten times for confidence intervals. Note that the scale of the MSE y-axis differs between scenarios for visual clarity.
• Figure 4: Cohort selection in the IPWC dataset.
• Figure 5: Predictive performance comparison. The arrow indicates the comparison's truth benchmark.