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, rdsurvival, for the R language.
Figures & tables
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 πh and τh . For the population with running variable Zi equal to the threshold c , πh is the vertical distance between to counterfactual survival curves P[Ti(w)>h∣Zi=c] , w∈{0,1}, at time point h , while τh 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 3 (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 π^h using censor-corrections, and an estimate π^∗h using the uncensored complete-data Ti , from the same realization of n=5,000 observations.
Operating Characteristics of DR Approach with Different RD Estimators
Setting
Survival probability πh
RMST τh
Coverage
RMSE
Length
Coverage
RMSE
Length
rdrobust
1
0.94
0.09
0.33
0.94
0.19
0.70
2
0.95
0.08
0.29
0.95
0.26
1.01
3
0.95
0.07
0.26
0.94
0.99
3.58
Table 1: Coverage of 95% confidence intervals for πh and τh , RMSE, and average interval length for the four simulation settings described in Section 3 . The number of Monte Carlo repetitions is 1,000.
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 ).
Figure 6: Event and censoring distributions for prostate cancer-free and overall survival. The gray dashed lines mark the horizons used in our estimation.
Figure 7: Distribution of events and right-censoring for overall survival stratified by subgroups. The density plots show the distribution of events and censoring for overall survival (OS).
Figure 8: Effect of prostate biopsy on survival endpoints. The dashed line indicates the eligibility threshold for prostate biopsy. The survival probabilities are estimated using doubly robust scores ( 3 ). The solid lines represent the bias-corrected linear model ( rdrobust ) fit on each side of the threshold.
Figure 9: Effect of prostate biopsy on overall survival at varying time horizons. Point estimates with 95% confidence intervals for the RD parameter from rdrobust are shown at various time horizons. For each time point, the difference in restricted mean survival time (RMST) when estimated using IPCW and DR scores are presented side-by-side.
Figure 10: Relationship between the horizon h and units’ probabilities of remaining uncensored when analyzing overall survival. At the horizontal dashed line, the probability of remaining uncensored becomes critically low. Units below this line (red color) and near the threshold (vertical dashed line) may cause instabilities in the DR scores and thus RD estimates.
Department of Statistics, TU Dortmund University · Research Center Trustworthy Data Science and Security, University Alliance Ruhr (UA Ruhr) · Institute for Mathematics, University of Augsburg