Comparison of Two Survival Curves 1 Landmark comparison

Comparison of Two Survival Curves 1. 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-by-Point” Comparisons, p. 329 of FFDRG. Chappell –BMI 542 These slides come after the Kaplan-Meier Mechanics. pdf and before the slides on biasing the Kaplan-Meier. #1

Comparison of Two Survival Curves - Outline 1. Landmark comparison at a pre-specified time t 2. Mantel-Haenszel (Log-rank) Test 3. Peto (old-fashioned version: Gehan) modification of the Wilcoxon Test 4. Restricted mean survival time (= RM life) at time t 5. Cox Proportional hazards model #2

Comparison of Two Survival Curves 2. Mantel-Haenszel (Log-rank) Test the null hypothesis of survival curve equality between two curves H 0: S 1(t) = S 2(t) for all times {t} within the range of the data. It is most powerful in detecting effects which are constant over time (proportional hazards). It fails when the survival curves cross. #3

Comparison of Two Survival Curves 2. Mantel-Haenszel (Log-rank) Test (cont. ) Just like the K-M estimates, it: • Accommodates censoring; • Has no distributional assumptions on failure times; and • Does require independence between censoring and failure times. #4

Comparison of Two Survival Curves 2. Mantel-Haenszel (Log-rank) Test (cont. ) 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. #5

Comparison of Two Survival Curves 3. 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 t = S(t), 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? #6

Comparison of Two Survival Curves 4. 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. : RMOverall. Survival. T(3 years) = RMTto recurrence(3) + RMTfrom. Recurrenceto. Death(3) #7

Comparison of Two Survival Curves • 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. • For short description of general considerations, see Chappell & Zhu, “Describing differences in survival curves. ” JAMA Onc. , published online 4/28/2016. • See Glasziou, Simes, & Gelber, Stat. in Med. 9, 12591276: 1990 for an example in breast cancer. #8

Glasziou, Simes, & Gelber (1990) Ludwig III RCT in metastatic Breast Ca - Cyclophasphamide, methotrexate, 5 -FU, prednisone, tamoxifen vs. Observation - 7 years followup - Outcomes: • Mortality • Progression • Toxicity - How to combine / compare outcomes? #9

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Comparison of Two Survival Curves 5. Cox Proportional Hazard Model Just as a t-test can be generalized to a regression model, the log-rank (MH) test can be generalized to a Cox PH model. It is a log-linear model on hazards, and is used in observational studies rather than clinical trials. Given a covariate x, the hazard is formulated as: h(t) = h(t; x = 0) × e b × x Thus for each unit increase in x, the hazard is multiplied by e b #12