HSRP 734: Advanced Statistical Methods July 10, 2008

1. HSRP 734: Advanced Statistical MethodsJuly 10, 2008

2. Objectives • Describe the Kaplan-Meier estimated survival curve • Describe the log-rank test • Use SAS to implement

3. Kaplan-Meier Estimate of Survival Function S(t) • The Kaplan-Meier estimate of the survival function is a simple, useful and popular estimate for the survival function. • This estimate incorporates both censored and noncensored observations • Breaks the estimation problem down into small pieces

4. Kaplan-Meier Estimate of the Survival Function S(t) • For grouped survival data, • Let interval lengths Ljbecome very small – all of length L=Dt and let t1, t2, … be times of events (survival times)

5. Kaplan-Meier Estimate of the Survival Function S(t) • 2 cases to consider in the previous equation • Case 1. No event in a bin (interval) • does not change — which means that we can ignore bins with no events

6. Kaplan-Meier Estimate of the Survival Function S(t) • Case 2. yj events occur in a bin (interval) Also: nj persons enter the bin assume any censored times that occur in the bin occur at the end of the bin

7. Kaplan-Meier Estimate of the Survival Function S(t) • So, as Dt → 0, we get the Kaplan- Meier estimate of the survival function S(t) • Also called the “product-limit estimate” of the survival function S(t) • Note: each conditional probability estimate is obtained from the observed number at risk for an event and the observed number of events (nj-yj) / nj

8. Kaplan-Meier Estimate of Survival Function S(t) • We begin by • Rank ordering the survival times (including the censored survival times) • Define each interval as starting at an observed time and ending just before the next ordered time • Identify the number at risk within each interval • Identify the number of events within each interval • Calculate the probability of surviving within that interval • Calculate the survival function for that interval as the probability of surviving that interval times the probability of surviving to the start of that interval

9. Example - AML + indicates a censored time to relapse; e.g., 13+ = more than 13 weeks to relapse

10. Example – AML • Calculation of Kaplan-Meier estimates: In the “not maintained on chemotherapy” group:

11. Example – AML (cont’d) In the “maintained on chemotherapy” group:

12. Example – AML (cont’d) • The “Kaplan-Meier curve” plots the estimated survival function vs. time — separate curves for each group

13. Example – AML (cont’d) • Notes — Can count the total number of events by counting the number of steps (times) — If feasible, picture the censoring times on the graph as shown above.

14. Kaplan-Meier Estimate Using SAS

15. Comments on the Kaplan-Meier Estimate • If the event and censoring times are tied, we assume that the censoring time is slightly larger than the death time. • If the largest observation is an event, the Kaplan-Meier estimate is 0. • If the largest observation is censored, the Kaplan-Meier estimate remains constant forever.

16. Comments on the Kaplan-Meier Estimate • If we plot the empirical survival estimates, we observe a step function. If there are no ties and no censoring, the step function drops by 1/n. • With every censored observation the size of the steps increase. • When does the number of intervals equal the number of deaths in the sample? • When does the number of intervals equal n?

17. Comments on the Kaplan-Meier Estimate • The Kaplan-Meier is a consistent estimate of the true S(t). That means that as the sample size gets large, KM estimate converges to the true value. • The Kaplan-Meier estimate can be used to empirically estimate any cumulative distribution function

18. Comments on the Kaplan-Meier Estimate • The step function in K-M curve really looks like this: • If you have a failure at t1 then you want to say survivorship at t1 should be less than 1. • For small data sets it matters, but for large data sets it does not matter.

19. Confidence Interval for S(t) – Greenwood’s Formula • Greenwood’s formula for the variance of : • Using Greenwood’s formula, an approximate 95% CI for S(t) is • There is a “problem”: the 95% CI is not constrained to lie within the interval (0,1)

20. Confidence Interval for S(t) – Alternative Formula • Based on log(-log(S(t)) which ranges from -∞ to ∞ • Find the standard error of above, find the CI of above, then transform CI to one for S(t) • This CI will lie within the interval [0,1] • This is the default in SAS

21. Log-rank test for comparing survivor curves • Are two survivor curves the same? • Use the times of events: t1, t2, ... (do not include censoring times) • Treat each event and its “set of persons still at risk” (i.e., risk set) at each time tjas an independent table • Make a 2×2 table at each tj

22. Log-rank test for comparing survivor curves • At each event time t j, under assumption of equal survival (i.e., SA(t) = SB(t) ), the expected number of events in Group A out of the total events (dj=aj +cj) is in proportion to the numbers at risk in group A to the total at risk at time tj: Eaj= dj x njA / nj • Differences between ajand Eajrepresent evidence against the null hypothesis of equal survival in the two groups

23. Log-rank test for comparing survivor curves • Use the Cochran Mantel-Haenszel idea of pooling over events j to get the log-rank chi-squared statistic with one degree of freedom

24. Log-rank test for comparing survivor curves • Idea summary: • Create a 2x2 table at each uncensored failure time • The construct of each 2x2 table is based on the corresponding risk set • Combine information from all the tables • The null hypothesis is SA(t) = SB(t) for all time t.

25. Comparisons across Groups • Extensions of the log-rank test to several groups require knowledge of matrix algebra. In general, these tests are well approximated by a chi-squared distribution with G-1 degrees of freedom. • Alternative tests: • Wilcoxon family of tests (including Peto test) • Likelihood ratio test (SAS)

26. Comparison between Log-Rank and Wilcoxon Tests • The log-rank test weights each failure time equally. No parametric model is assumed for failure times within a stratum. • The Wilcoxon test weights each failure time by a function of the number at risk. Thus, more weight tends to be given to early failure times. As in the log-rank test, no parametric model is assumed for failure times within a stratum. • Between these two tests (Wilcoxon and log-rank tests), the Wilcoxon test will tend to be better at picking up early departures from the null hypothesis and the log-rank test will tend to be more sensitive to departures in the tail.

27. Comparison with Likelihood Ratio Test in SAS • The likelihood ratio test employed in SAS assumes the data within the various strata are exponentially distributed and censoring in non-informative. Thus, this is a parametric method that smoothes across the entire curve.