In Kaplan-Meier estimation, the survival probability up to time T is given by which product?

• A. Product over t_j <= T of (1 - d_j / r_j)
• B. Product over t_j <= T of (d_j / r_j)
• C. Sum over t_j <= T of (1 - d_j / r_j)
• D. Sum over t_j <= T of (d_j / r_j)

Answer: (A)

Kaplan-Meier builds the survival function by multiplying the successively conditional survival probabilities across all observed event times. At each time when an event occurs, among the r_j individuals at risk just before that time, d_j experience the event. The chance of surviving past that time given survival up to that moment is 1 − d_j/r_j. Multiplying these conditional survival probabilities for every event time up to T yields the overall survival probability at T:

S(T) = ∏_{t_j ≤ T} (1 − d_j/r_j)

Here r_j is the number at risk just before time t_j, with censoring reducing the at-risk set accordingly. The other forms—summing the terms or using d_j/r_j directly—do not provide the correct survival probability, since they do not reflect the product of consecutive conditional survivals.