S-Plus Commands for Survival Estimation > t_c(1,1,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,15,17,22,23) > surv.fit(t,status=rep(1,21)) 95 percent confidence interval time n.risk n.event survival 1 21 2 0.90476190 2 19 2 0.80952381 3 17 1 0.76190476 4 16 2 0.66666667 5 14 2 0.57142857 8 12 4 0.38095238 11 8 2 0.28571429 12 6 2 0.19047619 15 4 1 0.14285714 17 3 1 0.09523810 22 2 1 0.04761905 23 1 1 0.00000000

is of type "log" std.dev lower 95% CI upper 95% CI 0.06405645 0.78753505 1.0000000 0.08568909 0.65785306 0.9961629 0.09294286 0.59988048 0.9676909 0.10286890 0.49268063 0.9020944 0.10798985 0.39454812 0.8276066 0.10597117 0.22084536 0.6571327 0.09858079 0.14529127 0.5618552 0.08568909 0.07887014 0.4600116 0.07636035 0.05010898 0.4072755 0.06405645 0.02548583 0.3558956 0.04647143 0.00703223 0.3224544 NA NA NA

Estimating the Survival Function One-sample nonparametric methods: We will consider three methods for estimating a survivorship function S(t) = P r(T ≥ t) without resorting to parametric methods:

(1) Kaplan-Meier (2) Life-table (Actuarial Estimator) (3) via the Cumulative hazard estimator

1

2

(1) The Kaplan-Meier Estimator

The Kaplan-Meier (or KM) estimator is probably the most popular approach. It can be justified from several perspectives: • product limit estimator • likelihood justification • redistribute to the right estimator We will start with an intuitive motivation based on conditional probabilities, then review some of the other justifications.

3

Motivation: First, consider an example where there is no censoring. The following are times of remission (weeks) for 21 leukemia patients receiving control treatment (Table 1.1 of Cox & Oakes): 1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23 How would we estimate S(10), the probability that an individual survives to time 10 or later? ˜ What about S(8)? Is it

12 21

or

8 21 ?

˜ Let’s construct a table of S(t): ˆ Values of t S(t) t ≤ 1 21/21=1.000 1 < t ≤ 2 19/21=0.905 2 < t ≤ 3 17/21=0.809 3