Conditional Survival
Introduction
Conditional survival is the probability of surviving an additional time \(t\) given that a patient has already survived to time \(s\). It answers the clinically relevant question that absolute survival cannot: a five-year survival probability of 30 % at diagnosis says little to a patient who has already lived three years. The conditional five-year-from-now estimate may be 70 % or higher, dramatically updating the prognosis. Survivorship clinics, oncology long-term follow-up, and patient counselling all rely on conditional rather than unconditional survival.
Prerequisites
A working understanding of the survival function \(S(t)\), conditional probability, and the Kaplan-Meier estimator.
Theory
By the definition of conditional probability,
\[\mathrm P(T > t + s \mid T > s) = \frac{\mathrm P(T > t + s)}{\mathrm P(T > s)} = \frac{S(t + s)}{S(s)}.\]
A patient who has survived \(s\) years is now in the risk set at \(s\); their next-\(t\)-year survival is the ratio of survival probabilities. For Kaplan-Meier estimates, this is computed from \(\hat S(s)\) and \(\hat S(t + s)\) directly. Confidence intervals require variance propagation through the ratio; the log or log-log transformation gives intervals that respect the \([0, 1]\) bounds.
Assumptions
A reliable Kaplan-Meier or parametric survival estimate at both \(s\) and \(t + s\); in practice this means sufficient follow-up so that \(\hat S(t + s)\) is not based on a tiny remaining cohort.
R Implementation
library(survival)
data(lung)
fit <- survfit(Surv(time, status) ~ 1, data = lung)
# Unconditional 2-year survival
summary(fit, times = c(500, 1000))$surv # approximate 2- and 3-year
# Conditional 1-year survival given 1-year survivorship
times <- summary(fit, times = c(365, 730))
cond_surv <- times$surv[2] / times$surv[1]
cond_survOutput & Results
The ratio of two KM survival estimates at chosen landmark times. For confidence intervals, transform onto the log-log scale, compute SEs by the delta method, and back-transform. The condsurv package automates the computation including bootstrap intervals.
Interpretation
A reporting sentence: “Unconditional one-year survival was 48 %; conditional on having survived one year, the probability of surviving an additional year was 66 % (95 % CI 56 % to 75 %); conditional on having survived two years, the next-year survival rose to 78 %.” Always report the landmark time \(s\) explicitly; conditional survival is meaningless without it.
Practical Tips
- Report conditional survival at clinically meaningful landmarks: one-year, five-year, ten-year survivor checkpoints in oncology, post-transplant 90-day survival in transplantation.
condsurvautomates the computation and produces clean tables and plots; it handles confidence intervals via the log-log transformation by default.- Confidence intervals must respect the \([0, 1]\) bounds; the log-log transformation is the standard choice and is preferable to the naive Wald interval.
- For dynamic prediction (conditional survival as a function of \(s\)), landmark analyses or joint longitudinal-survival models give more refined estimates that incorporate post-baseline information.
- Conditional survival is particularly useful in patient counselling because it answers the question patients actually ask: “Given that I’m still alive, how much longer can I expect to live?” rather than the population-level survival reported at diagnosis.
- Pair conditional-survival reports with the underlying KM curve so readers can see how the conditional estimate evolves across the cohort’s experience.
R Packages Used
survival for the Kaplan-Meier baseline; condsurv for tabular conditional survival with CIs; survminer::ggsurvplot() for KM curves to display alongside conditional summaries; dynpred and JM for dynamic prediction beyond the simple ratio of survival probabilities.