01/20151 EPI 5344: Survival Analysis in Epidemiology Survival curve comparison (non-regression methods) March 3, 2015 Dr. N. Birkett, School of Epidemiology, Public Health & Preventive Medicine, University of Ottawa

01/20152 Comparing survival (1) A common RCT question: –Did the treatment make a difference to the rate of outcome development? A more general question: –Which treatment, exposure group, etc. has the best outcome lowest mortality, lowest incidence, best recovery

01/20153 Comparing survival (2) Can be addressed through: –Regression methods Cox models (later) –Non-regression methods Log-rank test Mantel-Hanzel Wilcoxon/Gehan

01/20154 dur status treat renal Data for the Myelomatous data set, Allison Does treatment affect survival?

Rank order the data within treatment groups Treatment = 1 dur status treat renal Treatment = 2 dur status treat renal /20155

6 New Rx Old Rx Effect of new treatment

01/20157 No renal disease Renal disease Effect of pre-existing renal disease

Comparing Survival (3) How to tell if one group has better survival? One approach is to compare survival at one point in time –One year survival –Five year survival This is the approach used with Cumulative Incidence Ratios (CIR aka RR). 01/20158

9 Δ Compare the cumulative incidence (1-S(t)) at 5 years using a t-test, etc.

01/ This approach is limited: For both of these situations, the five-year survival is the same for the two groups being compared. BUT, the overall pattern of survival in the study on the left is clearly different between the two groups while for the study on the right, it is not.

01/ Comparing Survival (4) Compare curves at each point Combine across all events Can limit comparison to times when an event happens titi

01/ D D C C  Risk Set – All people under study at time of event – Only include people at risk of having an event Comparing survival (5) D Risk set #1 Risk set #2 Risk set #3

01/ Comparing Survival (6) Nonparametric approaches –Log-rank –Mantel-Hanzel –Wilcoxon/Gehan –Other weighted methods (a wide variety exist) Closely related but not the same ‘Log rank’ is often presented as the Mantel-Hanzel (M- H) method without explanation –They differ slightly in their assumptions (more later) –We will use the M-H approach

01/ Comparing Survival (7) General approach –Tests the null hypothesis that the survival distribution of the 2 groups is the same –Usually assume that the ‘shape’ is the same not specified –But, a ‘location’ parameter is different –Example Both groups follow an exponential survival model Hazard is constant but different in the two groups. Affects the mean survival (location)

01/ Comparing Survival (8) General approach –Under the ‘null’, whenever an event happens, everyone in the risk set has the same probability of being the person having event –Combine all observations into one file –Rank order them on the time-to-event –At each event time, compute a statistic to compare the expected number of events in group 1 (or 2) to the observed number –Combine the results at each time point into a summary statistic –Compare the statistic to an appropriate reference standard.

01/ Comparing Survival (9) Example from Cantor We present the merged and sorted data in the table on the next slide. Group 1Group

01/ 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 ’

GroupDeadAliveTotal / Comparing Survival (10) Consider first event time (t=2). In the risk set at t=2, we have: –5 subjects in group 1 –6 subjects in group 2 We can represent this data as a 2x2 table. GroupDeadAliveTotalO 1,2 O 2,2 E 1,2 E 2,2 V2V Total

01/ Comparing Survival (11) What are the ‘E’ and ‘V’ columns? –E i,t = expected # of events in group ‘i’ at time ‘t’ –V t = Approximate variance of ‘E’ at time ‘t’

01/ Comparing Survival (12) GroupDeadAliveTotalO 1,2 O 2,2 E 1,2 E 2,2 V2V Total

01/ Comparing Survival (13) More generally, suppose we have: –d t1 = # events at time ‘t’ in group 1 –d t2 = # events at time ‘t’ in group 2 –d t+ = # events at time ‘t’ (d t1 +d t2 ) –R t1 = # in risk set at time ‘t’ in group 1 –R t2 = # in risk set at time ‘t’ in group 2 –R t+ = # in risk set at time ‘t’ (R t1 +R t2 ) Then, we have the expected # of events in group 1 is:

01/ Comparing Survival (14) –d t1 = # events at time ‘t’ in group 1 –d t2 = # events at time ‘t’ in group 2 –d t+ = # events at time ‘t’ (d t1 +d t2 ) –R t1 = # in risk set at time ‘t’ in group 1 –R t2 = # in risk set at time ‘t’ in group 2 –R t+ = # in risk set at time ‘t’ (R t1 +R t2 ) And, the ‘V’s are given by this formula:

GroupDeadAliveTotalO 1,2 O 2,2 E 1,2 E 2,2 V2V Total / GroupDeadAliveTotalO 1,3 O 2,3 E 1,3 E 2,3 V3V Total At time ‘2’ At time ‘3’

GroupDeadAliveTotalO 1,5 O 2,5 E 1,5 E 2,5 V5V Total / GroupDeadAliveTotalO 1,t O 2,t E 1,t E 2,t VtVt 1d t1 R t1 2d t2 R t2 Totald t+ (R t+ - d t+ ) R t+ d t1 d t2 At time ‘5’ At time ‘t’

01/ Comparing Survival (15) Compute O 1t – E 1t for each event time ‘t’ Add up the differences across all events to get: This measures how far group ‘1’ differs from what would be expected if survival were the same in the two groups. If you had chosen group ‘2’ instead of group ‘1’, the sum of the differences would have been the same.

01/ Comparing Survival (16) Write this difference as: O + – E + And, let Then, we have: This is the log rank test

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

Comparing Survival (17) The log-rank is nearly the same as the score test from Cox regression. If there are no ties, they will be the same value. –ties: 2 or more subjects with the same event time 01/201528

Comparing Survival (18) The test above essentially applies the Mantel-Hanzel test (covered in Epi 1) to tables created by stratifying the sample into groups based on the event times. The test can be written as: 01/ Log-rank or Mantel-Hanzel test

Comparing Survival (19) The test can be modified by assigning weights to each event time point. –Might be based on size of risk set at ‘t’ Then, the test becomes: 01/201530

Comparing Survival (20) Log-rank: –w t =1 –equally weights events at all points in time Wilcoxon test –w t =R t+ –Weight is the size of the Risk Set at time ‘t’ –Assigns more weight to early events than late events –large risk sets  more precise estimates Other variants exist These tests don’t give the same results. 01/201531

01/ Comparing Survival (21) Some Issues –More than 2 groups Method can be extended –Continuous Predictors Must categorize into groups –Multiple predictors Cross-stratify the predictors Limited # of variables which can be included

01/ Comparing Survival (22) Some Issues –Curves which cross THERE IS NO RIGHT ANSWER!!! Which is ‘better’ depends on the follow-up time –Relates to effect modification –How to weight early/late events Many different approaches –Wilcoxon gives more weight to early events Can give different answers, especially if p-values are close to 0.05

Practical stuff The next slide set looks at implementation in SAS –Strata statement –Test statement Expands the analysis options from the outline given here. 01/201534

Some stuff you may not want to know Each year, questions get raised about things like: –why is it called the ‘log-rank test’? The method doesn’t involve –logs –ranks –What is the difference between the ‘log-rank’ and the ‘Mantel-Haenzel’ tests. So, here’s a summary of that information 01/201535

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) 03/201436

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. 03/201437

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. 03/201438

Theory (2) Theory is based on continuous time –Models the ‘density’ of an event happening at any point in 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 03/201439

Theory (3) 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 03/201440

03/ Theory (4)

03/ Theory (5)

Theory(6) 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! 03/201443

03/ Example from Cantor We presented the log-rank test table earlier in this session. Here are the summary results Group 1Group

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

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