In a clinical trial, Patient A dies at month 6 and Patient B is lost to follow-up at month 6. How does the Kaplan-Meier estimator treat these two observations differently?
ABoth are treated identically — both reduce the survival estimate at month 6
BPatient A causes a step down in the survival curve (an event); Patient B is removed from the risk set at month 6 without causing a step down (censored)
CPatient A is counted; Patient B is excluded from the analysis entirely
DPatient B's survival time is imputed as the median follow-up time
This is the fundamental distinction in survival analysis. Patient A experienced the event (death), so the survival estimate decreases at month 6. Patient B was censored — we know they survived at least 6 months, but not how long after. The Kaplan-Meier estimator uses this information by keeping Patient B in the risk set (the denominator) for all times up to month 6, then removing them. They contribute survival information up to their censoring time without being counted as an event. Excluding them entirely (option C) would waste information; treating them as events (option A) would bias survival downward.
Question 2 True / False
A Kaplan-Meier curve shows that the 1-year survival probability for Treatment A is 70% and for Treatment B is 55%. This means Treatment A is statistically significantly better than Treatment B.
TTrue
FFalse
Answer: False
The Kaplan-Meier curve is a descriptive estimate — it shows the estimated survival function but does not perform a hypothesis test. The apparent difference between 70% and 55% could be due to chance, especially with small samples or wide confidence intervals. A formal statistical test (the log-rank test) is needed to determine whether the difference is statistically significant. The curves might also cross at other time points, complicating interpretation even if the difference at one year appears large.
Question 3 Short Answer
The Kaplan-Meier estimator assumes that censoring is non-informative — that is, censored subjects have the same future survival prospects as those who remain under observation. Why is this assumption critical?
Think about your answer, then reveal below.
Model answer: If censoring is related to prognosis — for example, sicker patients drop out because they seek alternative treatment or are too ill to continue — then censored subjects have different survival prospects than those remaining. The KM estimator treats censored subjects as having the same future risk as continuing subjects, so informative censoring biases the survival estimate upward (if sicker patients leave) or downward (if healthier patients leave). The survival curve would no longer represent the true survival experience of the population.
Non-informative censoring means that the reason for censoring is independent of the event process. Administrative censoring (study ends) and staggered entry (patients enroll at different times) are typically non-informative. Loss to follow-up is potentially informative if sicker or healthier patients are differentially lost. When informative censoring is suspected, sensitivity analyses or joint models of the event and censoring processes are needed.
Question 4 Short Answer
Why does the Kaplan-Meier estimator use a product of conditional probabilities rather than simply dividing the number of survivors by the total number of subjects at each time point?
Think about your answer, then reveal below.
Model answer: Censoring changes the number of subjects at risk over time. A simple proportion (survivors / total enrolled) would either ignore censored subjects (underestimating survival by treating missing data as events) or count them as alive (overestimating survival by assuming they all survived). The product-limit approach computes the conditional probability of surviving each event time given survival up to that point, using only subjects still at risk. Multiplying these conditional probabilities gives the cumulative survival probability that correctly accounts for the changing risk set.
At each event time t_i, the KM estimator computes (n_i - d_i)/n_i, where n_i is the number at risk (alive and uncensored just before t_i) and d_i is the number of events at t_i. The cumulative product S(t) = product of all these conditional survival probabilities up to time t. Censored subjects reduce n_i at the time of censoring but never appear in d_i. This product-limit formula was proposed by Kaplan and Meier in 1958 and remains the standard nonparametric survival estimator.