Continuous-Time Modeling of Counterfactual Outcomes Using Neural Controlled Differential Equations

TL;DR

The paper tackles the challenge of estimating time-varying counterfactual outcomes for irregularly sampled clinical data under time-dependent confounding. It introduces TE-CDE, a continuous-time model that learns a latent patient trajectory via Neural Controlled Differential Equations and predicts counterfactuals under hypothetical treatments, with domain adversarial training to balance confounding. The approach is validated in a controllable PK-PD tumor growth simulation, showing superior performance to state-of-the-art methods across irregular sampling patterns, and it demonstrates practical benefits such as uncertainty quantification and data efficiency. The work advances continuous-time causal inference in healthcare, enabling more reliable what-if analyses and informing treatment decisions in realistic, irregular data settings.

Abstract

Estimating counterfactual outcomes over time has the potential to unlock personalized healthcare by assisting decision-makers to answer ''what-iF'' questions. Existing causal inference approaches typically consider regular, discrete-time intervals between observations and treatment decisions and hence are unable to naturally model irregularly sampled data, which is the common setting in practice. To handle arbitrary observation patterns, we interpret the data as samples from an underlying continuous-time process and propose to model its latent trajectory explicitly using the mathematics of controlled differential equations. This leads to a new approach, the Treatment Effect Neural Controlled Differential Equation (TE-CDE), that allows the potential outcomes to be evaluated at any time point. In addition, adversarial training is used to adjust for time-dependent confounding which is critical in longitudinal settings and is an added challenge not encountered in conventional time-series. To assess solutions to this problem, we propose a controllable simulation environment based on a model of tumor growth for a range of scenarios with irregular sampling reflective of a variety of clinical scenarios. TE-CDE consistently outperforms existing approaches in all simulated scenarios with irregular sampling.

Figures (11)

• Figure 1: Illustration of the different paradigms of longitudinal data processing. We contrast the regular sampled setting (left) which RNN-based methods assume vs the irregularly sampled setting (right) which TE-CDE addresses, where data can be observed and evaluations carried out at any time-step.
• Figure 2: An illustration of TE-CDE. We learn a continuous latent path $\mathbf{Z}_{t}$ as the solution to a CDE by encoding historical observations. At future time points, we decode (hypothetical) future treatments to determine the latent state and use this to predict counterfactual outcomes.
• Figure 3: Counterfactual predictions for varying time-varying confounding $\gamma$. Additionally, various values of $\kappa$ are illustrated which controls the intensity of sampling varying over a trajectory based on clinical stage.
• Figure 4: Counterfactual estimation at time $t_{k+n}$ ($n=5$) for varying levels of time-dependent confounding $\gamma$.
• Figure 5: Treatment accuracy under varying degrees of time-dependent confounding $\gamma$.