Nonparametric Regression Discontinuity Designs with Survival Outcomes

Abstract

Quasi-experimental evaluations are central for generating real-world causal evidence and complementing insights from randomized trials. The regression discontinuity design (RDD) is a quasi-experimental design that can be used to estimate the causal effect of treatments that are assigned based on a running variable crossing a threshold. Such threshold-based rules are ubiquitous in healthcare, where predictive and prognostic biomarkers frequently guide treatment decisions. However, standard RD estimators rely on complete outcome data, an assumption often violated in time-to-event analyses where censoring arises from loss to follow-up. To address this issue, we propose a nonparametric approach that leverages doubly robust censoring corrections and can be paired with existing RD estimators. Our approach can handle multiple survival endpoints, long follow-up times, and covariate-dependent variation in survival and censoring. We discuss the relevance of our approach across multiple areas of applications and demonstrate its usefulness through simulations and the prostate component of the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial where our new approach offers several advantages, including higher efficiency and robustness to misspecification. We have also developed an open-source software package, $\texttt{rdsurvival}$, for the $\texttt{R}$ language.

Figures (10)

• Figure 1: Illustration of the canonical RD design. Survival is declining as the prognostic running variable increases. At the threshold (dashed line) an intervention is triggered, resulting in a substantial improvement in the prognosis.
• Figure 2: Illustration of the estimands $\pi^h$ and $\tau^h$. For the population with running variable $Z_i$ equal to the threshold $c$, $\pi^h$ is the vertical distance between to counterfactual survival curves $\mathbb{P}_{}\left[T_i(w) > h \mid Z_i = c\right]$, $w \in \{0,1\},$ at time point $h$, while $\tau^h$ is the area between the counterfactual survival curves up to time point $h$.
• Figure 3: The figure shows two scenarios with a total of around 70 months of follow-up. In the first scenario (panel a), censoring poses no challenge up to 28 months (dashed gray line). In the second scenario (panel b), estimating the effect on $h = 60$ months (dashed red line) involves remapping censoring and event times observed past $h$.
• Figure 4: Example of bias using an IPCW approach compared with a more flexible doubly robust (DR) approach that uses nonparametric survival estimators, using simulation settings described in Section \ref{['sec:simulation']} (uniform censoring is "setting 1" and complicated censoring is "setting 2") and a sample size of $n=5,000$. The boxplots show censoring bias defined as the difference between an estimate $\hat{\pi}^{h}$ using censor-corrections, and an estimate $\hat{\pi}_*^{h}$ using the uncensored complete-data $T_i$, from the same realization of $n=5,000$ observations.
• Figure 5: Uptake of prostate biopsy as a function of the running variable PSA. The density and bar plots (top) shows the distribution and median (with IQR) of PSA. The red dashed line indicates the eligibility threshold for prostate biopsy and the solid lines the linear fits on each side of the threshold (rdrobust).

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Remark 2