Proximal Causal Inference for Conditional Separable Effects

TL;DR

This work extends conditional separable effects (CSE) to settings with unmeasured confounding by embedding proximal causal inference into the CSE framework, introducing treatment- and outcome-confounding proxies and confounding bridge functions to achieve nonparametric identification. It develops a semiparametric inference approach based on an influence function under a flexible model with unrestricted nuisance functions, and constructs a practical estimator via cross-fitting and a PMMR-based nuisance estimation procedure. Theoretical results establish the existence and identification of CSE, derive the efficient influence function, and prove asymptotic normality with a consistent variance estimator under mild regularity and source conditions. Simulations and an application to the Southwest Oncology Group trial demonstrate accurate finite-sample performance and meaningful treatment effects on quality of life under post-treatment conditioning, even when unmeasured confounding may bias standard analyses.

Abstract

Scientists regularly pose questions about treatment effects on outcomes conditional on a post-treatment event. However, causal inference in such settings requires care, even in perfectly executed randomized experiments. Recently, the conditional separable effect (CSE) was proposed as an interventionist estimand that corresponds to scientifically meaningful questions in these settings. However, existing results for the CSE require no unmeasured confounding between the outcome and post-treatment event, an assumption frequently violated in practice. In this work, we address this concern by developing new identification and estimation results for the CSE that allow for unmeasured confounding. We establish nonparametric identification of the CSE in observational and experimental settings with time-varying confounders, provided that certain proxy variables for hidden common causes of the post-treatment event and outcome are available. For inference, we characterize an influence function for the CSE under a semiparametric model where nuisance functions are a priori unrestricted. Using modern machine learning methods, we construct nonparametric nuisance function estimators and establish convergence rates that improve upon existing results. Moreover, we develop a consistent, asymptotically linear, and locally semiparametric efficient estimator of the CSE. We illustrate our framework with simulation studies and a real-world cancer therapy trial.

Figures (5)

• Figure 1: A Single World Intervention Graph compatible with \ref{['(A4)']}-\ref{['(A6)']} when $T=1$.
• Figure 2: A Graphical Summary of the Simulation Results. The left and right panels show results for observational and experimental settings, respectively. In the top panel, each column gives boxplots of biases of $\widehat{\tau}_{\text{CSE}}$ and $\widetilde{\tau}_{\text{CSE}}$ for $N\in \{500,1000,1500,2000\}$, respectively. The bottom panel provides numerical summaries informing the performance of $\widehat{\tau}_{\text{CSE}}$. Each row represents the empirical bias, empirical standard error (ESE), asymptotic standard error based on the proposed variance estimator (ASE), bootstrap standard error (BSE), empirical coverage rates of 95% confidence intervals based on the ASE and BSE. The bias and standard errors are scaled by factors of 10.
• Figure 3: Single World Intervention Graph in the Experimental Setting
• Figure 4: A Graphical Summary of the Simulation Results. The left and right panels show results for observational and experimental settings, respectively. In the top panel, each column gives boxplots of biases of $\widehat{\tau}_{\text{CSE}}$ and $\widetilde{\tau}_{\text{CSE}}$ for $N\in \{500,1000,1500,2000\}$, respectively. The bottom panel provides numerical summaries informing the performance of $\widehat{\tau}_{\text{CSE}}$. Each row represents the empirical bias, empirical standard error (ESE), asymptotic standard error based on the proposed variance estimator (ASE), bootstrap standard error (BSE), empirical coverage rates of 95% confidence intervals based on the ASE and BSE. The bias and standard errors are scaled by factors of 10.
• Figure 5: A Graphical Summary of Estimation Error. Each plot shows the relationship between $N$ and the RMSE of the estimated nuisance functions. The red line represents the best linear fit between $\log(N)$ and $\log(\text{RMSE})$, and the number shown in each figure indicates the slope of this line.

Theorems & Definitions (14)

• Theorem 4.1
• Theorem 5.1
• Theorem 5.2
• Theorem 5.3
• Theorem A.1
• Theorem A.2
• Theorem A.3
• Theorem A.4
• Theorem A.5
• Theorem A.6