Initially, 80 lives are included in an observation of survival times following a specific medical treatment. You are given excerpted information from the study data in the table below.
| [math]j[/math] | [math]t_{(j)}[/math] | Deaths at [math]t_{(j)}[/math] | Exits (Other than deaths) in [math](t_{(j)},t_{(j+1)}][/math] | Entrants in [math](t_{(j)},t_{(j+1)}][/math] |
|---|---|---|---|---|
| 0 | 20 | 4 | ||
| 1 | 0.5 | 1 | 2 | 3 |
| 2 | 1.6 | 1 | 6 | 0 |
| 3 | 1.9 | 1 | 8 | 0 |
| 4 | 2.5 | 1 | 10 | 0 |
Calculate the Kaplan-Meier estimate of [math]S(2)[/math].
Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
Answer: B
[math]r_{1}=80-20+4=64 ; \quad r_{2}=63-2+3=64 ; \quad r_{3}=63-6=57[/math]
[math]\hat{S}(2)=\hat{S}\left(t_{(3)}\right)=\frac{63}{64} \times \frac{63}{64} \times \frac{56}{57}=0.951994[/math]
Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.