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I had a couple of exercises about building a table for survival analysis under two different hypothesis: actuarial method and Kaplan-Meier. The only differences I know between this two approaches are the length of the intervals (given in actuarial method and "endogenous" in Kaplan-Meier) and the hypothesis on the censored cases (at risk for half interval in actuarial method and at risk for the full interval in Kaplan-Meier, given that there is at least one event).

In the solution of the exercises I also noticed that there is a difference in how the survival function is computed: in actuarial method, in the first interval the survival function is equal to 1, whereas in Kaplan-Meier it is equal to $1-q_i$, where $q_i$ is the probability of experiencing the event.

From these tables it seems that in actuarial method, we compute the survival at the beginning of the interval, whereas in the Kaplan-Meier case at the end of the interval. Is this true? Or is it just a usual way to report tables?

P.S. I reported here the solutions to my exercises. In the first table, the notation is the following: $E_i$ is the number of events, $Z_i$ the number of censored cases, $N_i$ the sample size, $R_i$ the risk set, $q_i=\frac{E_i}{R_i}$ the probability of experiencing the event and $G_i$ the survivor function.

Actuarial method

Actuarial Method

Kaplan-Meier

Kaplan-Meier estimation

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