03/20141 EPI 5344: Survival Analysis in Epidemiology Log-rank vs. Mantel-Hanzel testing Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

Peto, Pike, et al, 1977 The name " logrank " derives from obscure mathematical considerations (Peto and Pike, 1973) which are not worth understanding; it's just a name. The test is also sometimes called, usually by American workers who cite Mantel (1966) as the reference for it, the " Mantel- Haenszel test for survivorship data [Peto, Pike, et al, 1977) 03/20142

Peto et al, 1973 In the absence of ties and censoring, we would be able to rank the M subjects from M (the first to fail) down to 1 (the last to fail). To the accuracy with which, as r varies between 2 and M + 1, the quantities are linearly related to the quantities, statistical tests based on the x i can be shown to be equivalent to tests based on group sums of the logarithms of the ranks of the subjects in those groups, and the x i are therefore called "logrank scores" even when, because of censoring, actual ranks are undefined. 03/20143

Theory (1) We looked at survival curves when we developed the log-rank test Actually, the test is examining an hypothesis related to the distribution of survival times: –Assume that the two groups have the same ‘shape’ or distribution of survival –BUT, they differ by the ‘location’ parameter or ‘mean’ Test can either assume proportional hazards or accelerated failure time model Can also be derived using counting process theory. 03/20144

Theory (2) Theory is based on continuous time –Models the ‘density’ of an event happening at any point no time, not an actual event. –Initial development ignored censoring Need to convert this theoretical model to the ‘real’ world. –Censored events –Events happen at discrete point in time –Ties happen 03/20145

Theory (3) All methods come up with essentially the same test, the one we covered in class. It does depend on assumptions –Distributions are the same –PH or AFT is true Test is derived assuming –no censoring –no tied event times Methods for handling ties and censoring leads to slight variants in the tests 03/20146

Theory (4) Machin’s book presents 2 versions of this test, calling one the ‘log-rank’ and the other the ‘Mantel-Hanzel’ test This is incorrect. His ‘log rank’ is just an easier way to do the correct log-rank –Approximation which underestimates the true test score 03/20147

8 Theory (5)

03/20149 Theory (6)

Theory(7) Tests are generally similar. They can differ if there are lots of tied events. There is more but you don’t really want to know it! 03/201410

03/ Example from Cantor We present the merged and sorted data in the table on the next slide. Group 1Group

03/ itR1R1 R2R2 R+R+ d1d1 d2d2 d+d itR1R1 R2R2 R+R+ d1d1 d2d2 d+d itR1R1 R2R2 R+R+ d1d1 d2d2 d+d itR1R1 R2R2 R+R+ d1d1 d2d2 d+d d i = # events in group ‘I’; R i = # members of risk set at ‘t i ’

03/ itd t1 d t2 d t+ R t1 R t2 R t+ E t1 E t2 VtVt total

03/201414