3. (Kaplan-Meier and Nelson-Aalen estimators) Consider the following censored lifetime data: 12 23 13 23+ 25 14+ 15+ 16+ 18 25 26 26 ktk 1 2 3 4 Recall that the + sign corresponds to censoring: for e.g., 14+ means exit by censoring at age 14. (a) Find estimates of mortality rate (t), cumulative mortality rate A(t), and the survival probability (Nelson-Aalen and Kaplan-Meier) S (t). Provide your answers in 4 decimal digits. d 18 19 21 22+ 26+ 27+ 29 30 Yk λx Ak SKM SNA where to denotes event times, d, the number of exit by the event at t, y the num- ber of individuals at risk at t,= (t), A = A(t), while SKM and SNA are respectively Kaplan-Meier and Nelson-Aalen estimates of S (tk). Please complete the table in your assignment submission. (b) Use R to plot estimated cumulative hazard rate Ã(t) and survival probability S(t). (c) It is known that the estimated variance Var(SNA (1)) of the estimate SNA(t) of survival probability S(t) for the Nelson-Aalen estimator is given by Var(SNA (1)) = [SNA (1)]² Σ de (yx — de) kast Use the formula to estimate the variance of S (t) at age t = 18. (d) Derive a 95% confidence interval for S(t) aged t = 18 using SNA (1). JL 1