Conditional Survival

Updating prognosis for survivors: P(T > t + s | T > s)

Introduction

Conditional survival \(P(T > t + s | T > s)\) gives the probability of surviving an additional time \(t\) given survival up to time \(s\). It is the clinically relevant prognostic metric for patients who have already passed critical milestones.

Prerequisites

Survival function.

Theory

\(P(T > t + s | T > s) = S(t + s) / S(s)\) by the definition of conditional probability.

For a patient who has survived 2 years, 5-year survival conditional on 2-year survival is \(S(5) / S(2)\).

Assumptions

Kaplan-Meier or parametric survival estimate.

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_surv

Output & Results

Conditional survival probability at specified times.

Interpretation

“One-year survival was 0.48; among those who survived 1 year, the additional-year survival was 0.66 – substantially better than the initial estimate.”

Practical Tips

• Report conditional survival for clinically meaningful landmarks (1 year, 5 years).
• condsurv package automates computation.
• Confidence intervals require variance propagation; log or log-log transformation preferred.
• Dynamic predictions are more refined but computationally heavier.
• Useful in survivorship clinics and patient counselling.