1 Introduction
The regression discontinuity (RD) design has transformed empirical economics by exploiting sharp eligibility thresholds to generate quasi-experimental variation in treatment assignment [Imbens and Lemieux, 2008, Calonico et al., 2014]. The bulk of the RD literature, however, focuses on outcomes that are fully observed and can be summarised by their conditional mean — wages, test scores, conviction rates, earnings.
Many important policy questions concern time-to-event outcomes: how long does a patient survive after a cancer diagnosis? How long before a previously incarcerated individual reoffends? How long until a firm exits after bankruptcy? These are survival outcomes characterised by (a) a non-negative, right-skewed distribution, (b) right censoring — the event may not yet have occurred for some individuals at the time of data collection, and (c) hazard functions that summarise the instantaneous probability of the event conditional on survival to time t.
Applying standard RD methods to survival outcomes in the presence of censoring is problematic. The conditional mean 𝔼[Tᵢ | Xᵢ] is not identified from censored data without additional assumptions, invalidating naive mean-regression approaches. This article surveys the problem of RD identification with survival outcomes and describes nonparametric estimation strategies that accommodate censoring.
2 Why Standard RD Methods Fail with Censored Survival Data
In a standard RD, we estimate the conditional expectation functions 𝔼[Yᵢ | Xᵢ = x] on either side of the cutoff c and take the difference at x = c. With a fully observed outcome Yᵢ, this is straightforward using local polynomial regression [Calonico et al., 2014].
When Yᵢ = min(Tᵢ, Cᵢ) is the observed time (the minimum of the true event time Tᵢ and a censoring time Cᵢ), and Δᵢ = 1(Tᵢ ≤ Cᵢ) is the event indicator, the conditional mean 𝔼[Yᵢ | Xᵢ = x] is the conditional mean of the observed time — a mixture of event times and censoring times — not the conditional mean of the event time Tᵢ. Naively regressing Yᵢ on Xᵢ recovers the effect of crossing the threshold on observed follow-up time, which mixes the causal effect on survival with the effect on censoring probability.
Furthermore, 𝔼[Tᵢ | Xᵢ = x] cannot be recovered from censored data without assuming that censoring is independent of the event time (non-informative censoring). Standard non-informative censoring assumptions require Tᵢ ⊥ Cᵢ | Xᵢ, which may or may not be satisfied depending on the application.
3 Potential Outcomes Framework for Survival Outcomes
In the RD potential outcomes framework, let Tᵢ(1) and Tᵢ(0) denote the potential survival times under treatment and control. The sharp RD treatment effect at the cutoff is:
Under continuity of 𝔼[Tᵢ(1) | Xᵢ = x] and 𝔼[Tᵢ(0) | Xᵢ = x] at x = c, this is identified by the jump in 𝔼[Tᵢ | Xᵢ = x] at the cutoff. However, with censored data, we cannot directly estimate 𝔼[Tᵢ | Xᵢ = x].
3.1 Survival Function Approach
Rather than the conditional mean, consider the conditional survival function:
The causal effect can be expressed through differences in survival functions:
where c⁺ and c⁻ denote limits from the right and left, respectively. This gives a curve of treatment effects — the effect on the probability of surviving past time t — rather than a single scalar. Under non-informative censoring, the conditional survival function S(t | x) is identifiable from censored data using Kaplan-Meier-type estimators, and the RD can be conducted at each time horizon t separately.
3.2 Hazard Rate Approach
An alternative summary is the hazard function:
the instantaneous risk of the event at time t, conditional on survival to t and on the running variable x. Proportional hazard models [Cox, 1972] postulate λ(t | x, D) = λ₀(t) exp(βD + f(x)), where D = 1(x ≥ c) is the treatment indicator. However, Cox models are parametric in the covariate effects and impose the proportional hazards assumption, which may be violated.
4 Nonparametric Estimation Strategies
4.1 Local Kaplan-Meier Estimation
A natural nonparametric approach uses kernel-weighted Kaplan-Meier estimators for S(t | x) on each side of the cutoff:
where dⱼ± is the number of events at time tⱼ and nⱼ± is the number at risk, computed using kernel weights that give higher weight to observations with Xᵢ closer to c. The RD effect on survival at horizon t is then τ̂(t) = Ŝ⁺(t) − Ŝ⁻(t).
Bandwidth selection for local Kaplan-Meier poses new challenges compared to local polynomial regression, because the optimal bandwidth depends on the time horizon t and on the smoothness of the conditional survival function. Cross-validation procedures adapted for censored data, or optimal bandwidth formulas derived from the asymptotic bias-variance trade-off of the local Kaplan-Meier, are needed.
4.2 Local Accelerated Failure Time Models
An alternative is the accelerated failure time (AFT) model, which specifies:
where μ(Xᵢ) is a flexible function of the running variable and εᵢ has a specified or nonparametrically estimated distribution. Near the cutoff, μ(x) is approximated by separate local polynomials on each side, and the censored observations are handled via the partial likelihood or accelerated failure time semiparametric estimators.
The AFT-RD estimator recovers the treatment effect on log Tᵢ at the cutoff:
which can be interpreted as a proportional effect on median survival time if εᵢ is log-normal or as an additive effect on log-survival time more generally.
5 Application: Criminal Justice and Recidivism
Recidivism studies are an important application. Incarceration policies often operate via eligibility thresholds (criminal history scores, conviction counts) that create RD designs, and recidivism is naturally a survival outcome — the time until re-offence, subject to censoring at the end of the study period.
A researcher studying the causal effect of incarceration on the time-to-reoffence faces two sources of censoring: (a) administrative censoring, when study follow-up ends before an individual reoffends, and (b) "competing risks" censoring, when an individual dies or is incapacitated before reoffending.
The competing risks issue is particularly important for survival RD. Standard methods treating death as non-informative censoring produce sub-distribution hazards that mix the effect on the cause-specific hazard (reoffence) with the effect on the competing event (death). Cause-specific hazard RDs — focusing solely on the reoffence hazard, treating death as censoring — and Fine-Gray sub-distribution hazard RDs provide complementary perspectives [Fine and Gray, 1999].
6 Inference and Bandwidth Selection
Uniform confidence bands for the survival function difference τ̂(t) = Ŝ⁺(t) − Ŝ⁻(t) over a range of horizons t ∈ [0, T_max] are more challenging than pointwise bands and require either bootstrap procedures or derivations of the limiting process of τ̂(·) under the local Kaplan-Meier asymptotics.
For bandwidth selection, Calonico et al. [2014] propose mean-squared-error optimal bandwidths for continuous outcomes; adapting these to survival outcomes requires estimating the bias and variance of the local Kaplan-Meier estimator at the boundary, which depends on the censoring distribution. Cross-validation approaches based on integrated Brier scores for censored data provide a practical alternative.
7 Available Software
- The
survivalpackage in R provides Cox proportional hazard and Kaplan-Meier estimation, which can be combined with the local polynomial approach manually. - The
rdrobustpackage in R supports nonparametric local polynomial regression; adaptation for survival outcomes requires pre-processing the data (e.g., via restricted mean survival time as a scalar summary). - Restricted Mean Survival Time (RMST) — the integral ∫₀^T_max S(t | x)dt — is increasingly used as a scalar summary of the survival distribution that is estimable from censored data. An RD on RMST using
rdrobustprovides a convenient and interpretable alternative to the full survival curve approach.
8 Conclusion
Survival outcomes are common in economics, medicine, criminology, and finance, yet their analysis in regression discontinuity designs requires care. The key challenge is right censoring, which prevents direct application of standard conditional-mean RD estimators. Nonparametric approaches — local Kaplan-Meier survival curves, AFT-RD models, and RMST-based scalar summaries — provide valid alternatives under standard continuity and non-informative censoring assumptions. As RD applications continue to expand into health, criminal justice, and insurance settings where time-to-event outcomes are natural, methodological tools for handling survival outcomes in the RD framework will become increasingly important.
References
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