1 Analysis of Survival Data with Demographic Applications (Spring term 2006) Lecture 3: Non-Parametric Comparison of two or more Survival Curves

2 Non-parametric comparisons of two or more survival curves. Part of this lecture may be based on hand- written material.

3 Non-Parametric Methods for Comparing Two or More Survival Curves Consider two groups of patients (say, taking two different treatments) Their Survival functions are S 1 (t) and S 2 (t) The aim is to test H 0 : S 1 (t) = S 2 (t) against any alternative H 1 : S 1 (t) > S 2 (t) H 1 : S 1 (t) < S 2 (t) H 1 : S 1 (t)  S 2 (t) If there is no censoring, we can use standard non-parametric methods. Since H 0 : S 1 (t) = S 2 (t) implies F 1 (t) = F 2 (t), we can use non-parametric tests for the equality of two distribution functions.

4 Non-Parametric Methods for Comparing Two or More Survival Curves (contd…) In the presence of censoring we need special non-parametric tests: Log-Rank test Generalized Wilcoxon Test (Breslow & Gehan Test) Tarone-Ware test Cox’s F-test Cox-Mantel test Apparently different names (in different sources) for the same test

5 Log-Rank Test Consider two groups Let R tj be the risk set (individuals exposed to risk) at time t in the j th population (j=1,2). Let E tj be the observed events at time t in the j th population. Define E t = E t1 + E t2 and R t = R t1 + R t2 Then, the expected events in the first population are computed as

6 Log-Rank Test (contd…) At each event time t, the expected event will have variance The Log-Rank test statistic is, then, given by or, equivalently

7 Log-Rank Test (Example) Group 1: 7, 14, 24, 27, 27, 50, 51, 56, 58, 60, 62, 69+, 71, 74, 74, 76, 80+, 80+, 80+, 88+, 93, 98, 104+ Group 2: 1, 1, 2, 8, 9, 9, 14, 17, 20, 27, 34, 43, 45, 47, 55, 56, 57+, 62, 64, 78+, 82+, 86+, 92+

8 Kaplan-Meier Estimates of S(t) & h(t) by group

9 Log-Rank Test (Example, contd…)

10 Log-Rank Test (Example, contd…) The Log-Rank test statistic is, then, given by

11 Results from SPSS

12 Generalized Wilcoxon Test (Breslow & Gehan Test) Log-Rank test is based on the differences The Generalized Wilcoxon Test uses, instead, and replaces the variance V t in the Log-Rank test by The test statistic is then given by Exercise: Apply the above formula on the example on two groups and compare with the value of Breslow in the SPSS output