[Paper Review] Inference for treatment-specific survival curves using machine learning

Ted Westling, Alex Luedtke+2 more|arXiv (Cornell University)|Jun 11, 2021

Advanced Causal Inference Techniques4 citations

TL;DR

This paper proposes a cross-fitted doubly robust estimator for treatment-specific survival curves using machine learning to estimate conditional survival functions, ensuring consistency and asymptotic normality under weak regularity conditions. The method enables valid inference in continuous or discrete time and includes a novel ensemble learner for combining multiple survival estimators, improving robustness and efficiency in observational studies with confounding.

ABSTRACT

In the absence of data from a randomized trial, researchers often aim to use observational data to draw causal inference about the effect of a treatment on a time-to-event outcome. In this context, interest often focuses on the treatment-specific survival curves; that is, the survival curves were the entire population under study to be assigned to receive the treatment or not. Under certain causal conditions, including that all confounders of the treatment-outcome relationship are observed, the treatment-specific survival can be identified with a covariate-adjusted survival function. Several estimators of this function have been proposed, including estimators based on outcome regression, inverse probability weighting, and doubly robust estimators. In this article, we propose a new cross-fitted doubly-robust estimator that incorporates data-adaptive (e.g. machine learning) estimators of the conditional survival functions. We establish conditions on the nuisance estimators under which our estimator is consistent and asymptotically linear, both pointwise and uniformly in time. We also propose a novel ensemble learner for combining multiple candidate estimators of the conditional survival estimators. Notably, our methods and results accommodate events occurring in discrete or continuous time (or both). We investigate the practical performance of our methods using numerical studies and an application to the effect of a surgical treatment to prevent metastases of parotid carcinoma on mortality.

Motivation & Objective

• To develop a doubly robust estimator for treatment-specific survival curves in observational studies where confounding is present.
• To incorporate data-adaptive machine learning methods for estimating conditional survival functions, improving robustness to model misspecification.
• To establish uniform asymptotic linearity and weak convergence to a Gaussian process for inference over time.
• To handle both discrete and continuous time-to-event outcomes within a unified framework.
• To propose an ensemble learning strategy that combines multiple candidate estimators of conditional survival functions for improved performance.

Proposed method

• Uses cross-fitting to reduce overfitting in machine learning-based nuisance estimators, ensuring asymptotic normality.
• Employs a doubly robust estimating equation that combines outcome regression and inverse probability weighting, with the estimator remaining consistent if either the conditional survival or censoring mechanism is consistently estimated.
• Derives a loss function for estimating the conditional survival function via minimization of an estimating equation involving observed event indicators and inverse probability weights.
• Applies Fubini’s theorem to justify interchanging integrals over time and conditional expectations, ensuring validity of the estimating equation.
• Introduces an ensemble learner that combines multiple candidate estimators of the conditional survival function, enhancing robustness and efficiency.
• Establishes conditions under which the final estimator is asymptotically linear and uniformly convergent over time, enabling valid confidence bands and hypothesis tests.

Experimental results

Research questions

• RQ1 Can machine learning-based estimators of conditional survival functions be used to construct a doubly robust estimator of treatment-specific survival curves with valid inference?
• RQ2 Under what regularity conditions is the proposed estimator uniformly asymptotically linear over time?
• RQ3 How can ensemble learning be leveraged to combine multiple survival estimators while preserving double robustness and asymptotic normality?
• RQ4 Does the method maintain consistency and valid inference when events occur in discrete or continuous time?
• RQ5 Can the proposed estimator achieve parametric convergence rates even when using flexible machine learning methods for nuisance estimation?

Key findings

• The proposed cross-fitted doubly robust estimator is uniformly asymptotically linear under mild regularity conditions, enabling valid inference over the entire time interval.
• The estimator achieves weak convergence to a tight mean-zero Gaussian process, supporting the construction of confidence bands for survival curves.
• The method remains consistent if either the conditional survival function or the censoring mechanism is consistently estimated, ensuring robustness to model misspecification.
• The ensemble learner effectively combines multiple candidate estimators, improving finite-sample performance without sacrificing theoretical guarantees.
• Theoretical results hold for both discrete and continuous time-to-event outcomes, broadening applicability to real-world data.
• Numerical studies and a real-world application to parotid carcinoma surgery demonstrate the method’s practical utility and improved performance over existing approaches.