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Propensity score weighted Cox regression for survival outcomes in observational studies with multiple or factorial treatments

Zixian Zhao, Chengxin Yang, Fan Li

TL;DR

This work develops a propensity-score weighted Cox framework for causal inference with survival outcomes when there are multiple treatments, defining the marginal hazard ratio (MHR) via $\\lambda_j(t)=\\lambda_0(t)\\exp(\\tau_j)$. It introduces inverse-probability weighting (IPW) and overlap weighting (OW) to estimate $\\tau^w$ in a marginal Cox model, and proves consistency with a robust sandwich variance estimator, accounting for weight estimation via a multinomial propensity score $e_j(x;\\gamma)$. A special case for two-way factorial designs treats the combined treatment group as a separate entity to avoid confounding between main effects and interactions, rather than using an interacted Cox model. Through simulations, OW demonstrates stable performance across overlap scenarios, while IPW is sensitive to extreme propensities; an application to GLP-1 obesity medications on heart failure shows GLP-1 having favorable outcomes relative to NB and PT, with IPW unstable in the presence of extreme weights. An accompanying R package PSsurvival provides practical implementation for these methods, including visualization of weighted survival curves and estimation of counterfactual hazards.

Abstract

In observational studies with survival or time-to-event outcomes, a propensity score weighted marginal Cox proportional hazard model with the treatment variable as the only predictor is commonly used to estimate the causal marginal hazard ratio between two treatments. Observational studies often have more than two treatments, but corresponding analysis methods are limited. In this paper, we combine the propensity score weighting method for multiple treatments and a marginal Cox model with indicators for each treatment to estimate the causal hazard ratios between multiple treatments and a common reference treatment. We illustrate two weighting schemes: inverse probability of treatment weighting and overlap weighting. We prove the consistency of the maximum weighted partial likelihood estimator of the causal marginal hazard ratio and derive a robust sandwich variance estimator. As an important special case of multiple treatments, we elaborate the Cox model for two-way factorial treatments. We apply the method to evaluate the real-world comparative effectiveness of three types of anti-obesity medications on heart failure. We develop an associated R package 'PSsurvival'.

Propensity score weighted Cox regression for survival outcomes in observational studies with multiple or factorial treatments

TL;DR

This work develops a propensity-score weighted Cox framework for causal inference with survival outcomes when there are multiple treatments, defining the marginal hazard ratio (MHR) via . It introduces inverse-probability weighting (IPW) and overlap weighting (OW) to estimate in a marginal Cox model, and proves consistency with a robust sandwich variance estimator, accounting for weight estimation via a multinomial propensity score . A special case for two-way factorial designs treats the combined treatment group as a separate entity to avoid confounding between main effects and interactions, rather than using an interacted Cox model. Through simulations, OW demonstrates stable performance across overlap scenarios, while IPW is sensitive to extreme propensities; an application to GLP-1 obesity medications on heart failure shows GLP-1 having favorable outcomes relative to NB and PT, with IPW unstable in the presence of extreme weights. An accompanying R package PSsurvival provides practical implementation for these methods, including visualization of weighted survival curves and estimation of counterfactual hazards.

Abstract

In observational studies with survival or time-to-event outcomes, a propensity score weighted marginal Cox proportional hazard model with the treatment variable as the only predictor is commonly used to estimate the causal marginal hazard ratio between two treatments. Observational studies often have more than two treatments, but corresponding analysis methods are limited. In this paper, we combine the propensity score weighting method for multiple treatments and a marginal Cox model with indicators for each treatment to estimate the causal hazard ratios between multiple treatments and a common reference treatment. We illustrate two weighting schemes: inverse probability of treatment weighting and overlap weighting. We prove the consistency of the maximum weighted partial likelihood estimator of the causal marginal hazard ratio and derive a robust sandwich variance estimator. As an important special case of multiple treatments, we elaborate the Cox model for two-way factorial treatments. We apply the method to evaluate the real-world comparative effectiveness of three types of anti-obesity medications on heart failure. We develop an associated R package 'PSsurvival'.
Paper Structure (20 sections, 4 theorems, 69 equations, 9 figures, 4 tables)

This paper contains 20 sections, 4 theorems, 69 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Propensity score weighted hazard ratio estimator for multiple treatments
  3. Weighted marginal Cox model and estimator
  4. A special case: two-way factorial treatments
  5. Consistency and robust estimator of variance
  6. Simulation
  7. Setting 1: Multiple Treatments
  8. Setting 2: Two-way factorial treatments
  9. Application: Real-world effectiveness of GLP-1 on heart failure
  10. Discussion
  11. Proof of Theorem 1
  12. Setup and notations
  13. Key identities for weighted population moments
  14. Unbiasedness of the population score at $\tau^w$
  15. Asymptotic linear representation
  16. ...and 5 more sections

Key Result

Theorem 1

Under Assumptions A1-A4 and standard regularity conditions, the weighted estimator $\widehat{\tau}^w$ with a balancing weight $w$ is consistent to the target log marginal hazard ratio $\tau^w$. The asymptotic covariance matrix of $(\widehat{\tau}^{w'},\hat{\gamma}^{'})^{'}$ is consistently estimated where and and

Figures (9)

  • Figure 1: Distributions of the estimated generalized propensity scores in the HFpEF application.
  • Figure 2: Kaplan-Meier Curves by treatment of the HFpEF study: unadjusted (left) and overlap weighted (right).
  • Figure 3: Overlap-Weighted Kaplan-Meier Curves of the COMBINE Study
  • Figure 4: Multiple Treatment and Strong Overlap $\psi=1$
  • Figure 5: Multiple Treatment and Moderate Overlap $\psi=2$
  • ...and 4 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1: Integral representation of $R_j(t\mid X)$
  • proof
  • Lemma 2: Weighted population moments
  • proof
  • Lemma 3: Population score vanishes at $\tau^w$
  • proof