Comparison of Two Survival Curves Landmark comparison at a pre-specified time t: divide S^T(t) – S^C(t) by the standard error of the difference computed using Greenwood’s estimator. See “Point-byPoint” Comparisons, p. 329 of FFDRG.

Comparison of Two Survival Curves 2. Restricted mean survival time (= RM life) at time t: Mean survival up to time t; = Mean[min(t, survival time)] = Area under S^(.) between 0 and t. Interpreted as “Mean number of years lived out of t” Not “Mean number of years lived given death before t”: what is the RMST(5 years) for US newborns? Can also be added & subtracted, as with unrestricted means, e.g.: RMOverallSurvivalT(3 years) = RMTto recurrence(3) + RMTfromRecurrencetoDeath(3)

Comparison of Two Survival Curves 2. Restricted mean survival time (cont.): Not in FFDRG, but see Uno, et al., “Alternatives to hazard ratios for comparing efficacy or safety of therapies in noninferiority studies.” Ann Intern Med. 163, pp. 127-134: 2015 for references and examples in diabetes and colorectal cancer. See Glasziou, Simes, & Gelber, Stat. in Med. 9, 1259-1276: 1990 for an example in breast cancer. For short description of general considerations, see Chappell & Zhu, “Describing differences in survival curves.” JAMA Onc., published online 4/28/2016.

Comparison of Two Survival Curves 3. Mantel-Haenszel (Log-rank) Test Ref: Mantel & Haenszel (1959) J Natl Cancer Inst Mantel (1966) Cancer Chemotherapy Reports - Mantel and Haenszel (1959) showed that a series of 2 x 2 tables could be combined into a summary statistic, based on the work of Cochran and Cox. - Mantel (1966) applied this procedure to the comparison of two survival curves. - Basic idea is to form a 2 x 2 table at each distinct death time, determining the number in each group who were at risk and number who died.

Suppose we have K distinct times for a death occurring at tj j = 1,2, .., K. For each death time tj, we have a table such as on p. 330 of FFDRG. They use standard “contingency table notation”. To translate into “survival analysis notation”: DjI = # of deaths at time tj in Intervention group = aj DjC = # of deaths at time tj in Control group = cj RjI = # at risk at time tj in Intervention group = aj + bj RjC = # at risk at time tj in Control group = cj + dj . Consider aj, the observed number of deaths in the TRT group, under H0

Mantel-Haenszel Statistic E(aj) = (aj + bj)(aj + cj)/Ni Mantel-Haenszel Statistic

Table 15.3: Comparison of Survival Data for a Control Group and an Intervention Group Using the Mantel-Haenszel Procedure Rank Event Intervention Control Total Times j tj aj + bj aj lj cj + dj cj lj aj + cj bj + dj 1 0.5 20 0 0 20 1 1 1 39 2 1.0 20 1 0 18 0 0 1 37 3 1.5 19 0 2 18 2 1 2 35 4 3.0 17 0 1 15 1 2 1 31 5 4.5 16 1 0 12 0 0 1 27 6 4.8 15 0 1 12 1 0 1 26 7 6.2 14 0 1 11 1 2 1 24 8 10.5 13 0 1 8 1 1 20 aj + bj = number of subjects at risk in the intervention group prior to the death at time tj cj + cj = number of subjects at risk in the control group prior to the death at time tj aj = number of subjects in the intervention group who died at time tj cj = number of subjects in the control group who died at time tj lj = number of subjects who were lost or censored between time tj and time tj+1 aj + cj = number of subjects in both groups who died at time tj bj + dj = number of subjects in both groups who are at risk minus the number who died at time tj

Mantel-Haenszel Test Operationally 1. Rank event times for both groups combined 2. For each failure, form the 2 x 2 table a. Number at risk (ai + bi, ci + di) b. Number of deaths (ai, ci) c. Losses (lTi, lCi) Example (See table 15-3 FFDRG) - Use previous data set Trt: 1.0, 1.6+, 2.4+, 4.2+, 4.5, 5.8+, 7.0+, 11.0+, 12.0+'s Control: 0.5, 0.6+, 1.5, 1.5, 2.0+, 3.0, 3.5+, 4.0+, 4.8, 6.2, 8.5+, 9.0+, 10.5, 12.0+'s

2. At t1 = 0.5 (k = 1) 1. Ranked Failure Times - Both groups combined 0.5, 1.0, 1.5, 3.0, 4.5, 4.8, 6.2, 10.5 C T C C T C C C 8 distinct times for death (k = 8) 2. At t1 = 0.5 (k = 1) T: a1 + b1 = 20 a1 = 0 lT1 = 0 c1 + d1 = 20 c1 = 1 lC1 = 1 D A R T 0 20 20 C 1 19 20 1 39 40 E(a1)= 1•20/40 = 0.5 V(a1) = 1•39 • 20 • 20 402 •39

3. At t2 = 1.0 (k = 2) T: a2 + b2 = (a1 + b1) - a1 - lT1 a2 = 1.0 = 20 - 0 - 0 = 20 lT2 = 3 C. c2 + d2 = (c1 + d1) - c1 - lC1 c2 = 0 = 20 - 1 - 1 = 18 lC2 = 0 so D A R T 1 19 20 C 0 18 18 1 37 38 E(a2)= 1•20 38 V(a2) = 1•37 • 20 • 18 382 •37

Eight 2x2 Tables Corresponding to the Event Times Used in the Mantel-Haenszel Statistic in Survival Comparison of Treatment (T) and Control (C) Groups 1. (0.5 mo.)* D† A‡ R§ 5. (4.5 mo.)* D A R T 0 20 20 T 1 15 16 C 1 19 20 C 0 12 12 1 39 40 1 27 28 2. (1.0 mo) D A R 6. (4.8 mo.) D A R T 1 19 20 T 0 15 15 C 0 18 18 C 1 11 12 1 37 38 1 26 27 3. (1.5 mo.) D A R 7. (6.2 mo.) D A R T 0 19 19 T 0 14 14 C 2 16 18 C 1 10 11 2 35 37 1 24 25 4. (3.0 mo.) D A R 8. (10.5 mo.) D A R T 0 17 17 T 0 13 13 C 1 14 15 C 1 7 8 1 31 32 1 20 21 * Number in parentheses indicates time, tj, of a death in either group † Number of subjects who died at time tj ‡ Number of subjects who are alive between time tj and time tj+1 § Number of subjects who were at risk before the death at time tj R=D+A)

Compute MH Statistics Recall K = 1 K = 2 K = 3 t1 = 0.5 t2 = 1.0 t3 = 1.5 D A 0 20 20 1 19 20 1 39 40 D A 1 19 20 0 18 18 1 37 38 D A 0 19 19 2 16 18 2 35 37 a. ai = 2 (only two treatment deaths) b. E(ai ) = 20(1)/40 + 20(1)/38 + 19(2)/37 + . . . = 4.89 c. V(ai) = = 2.22 d. MH = (2 - 4.89)2/2.22 = 3.76 or ZMH =

Comparison of Two Survival Curves 4. Peto (old-fashioned version: Gehan) modification of the Wilcoxon Test to account for censored data. Derived as a modification of the Wilcoxon two-sample rank test. Equivalent to a weighted MH (logrank) test, with the weight at time tj = S^(tj), the Kaplan-Meier curve for the combined sample. When would we want to use decreasing weights? When (if ever) would we want to use increasing weights?