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101/2015
EPI 5344:Survival Analysis in
EpidemiologySurvival curve comparison(non-regression methods)
March 3, 2015
Dr. N. Birkett,School of Epidemiology, Public Health &
Preventive Medicine,University of Ottawa
201/2015
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
301/2015
Comparing survival (2)
• Can be addressed through:– Regression methods
• Cox models (later)
– Non-regression methods• Log-rank test
• Mantel-Hanzel
• Wilcoxon/Gehan
01/2015 4
dur status treat renal 8 1 1 1 180 1 2 0 632 1 2 0 852 0 1 0 52 1 1 12240 0 2 0 220 1 1 0 63 1 1 1 195 1 2 0 76 1 2 0 70 1 2 0 8 1 1 0 13 1 2 11990 0 2 01976 0 1 0 18 1 2 1 700 1 2 01296 0 1 01460 0 1 0 210 1 2 0 63 1 1 11328 0 1 01296 1 2 0 365 0 1 0 23 1 2 1
Data for the Myelomatous data set, Allison
Does treatment affect survival?
5
Rank order the data within treatment groups
Treatment = 1
dur status treat renal 8 1 1 1 8 1 1 0 52 1 1 1 63 1 1 1 63 1 1 1 220 1 1 0 852 0 1 0 365 0 1 01296 0 1 01328 0 1 01460 0 1 01976 0 1 0
Treatment = 2
dur status treat renal 13 1 2 1 23 1 2 1 18 1 2 1 70 1 2 0 76 1 2 0 180 1 2 0 195 1 2 0 210 1 2 0 632 1 2 0 700 1 2 01296 1 2 01990 0 2 02240 0 2 0
01/2015
601/2015
New Rx
Old Rx
Effect of new treatment
701/2015
No renal disease
Renal disease
Effect of pre-existing renal disease
8
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).
01/2015
901/2015
Δ
Compare the cumulative incidence (1-S(t)) at 5 years using a t-test, etc.
1001/2015
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.
1101/2015
Comparing Survival (4)
• Compare curves at each point
• Combine across all events
• Can limit comparison to times when an event happens
ti
1201/2015
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
1301/2015
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
1401/2015
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)
1501/2015
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.
1601/2015
Comparing Survival (9)
• Example from Cantor
• We present the merged and sorted data in the table on the next slide.
Group 1 Group 2
358+1015
2511+13+1416
1701/2015
i t R1 R2 R+ d1 d2 d+
1 2
2 3
3 5
4 8
5 10
6 11
7 13
8 14
9 15
10 16
i t R1 R2 R+ d1 d2 d+
1 2 5 6 11 0 1 1
2 3
3 5
4 8
5 10
6 11
7 13
8 14
9 15
10 16
i t R1 R2 R+ d1 d2 d+
1 2 5 6 11 0 1 1
2 3 5 5 10 1 0 1
3 5
4 8
5 10
6 11
7 13
8 14
9 15
10 16
i t R1 R2 R+ d1 d2 d+
1 2 5 6 11 0 1 1
2 3 5 5 10 1 0 1
3 5 4 5 9 1 1 2
4 8 3 4 7 0 0 0
5 10 2 4 6 1 0 1
6 11 1 4 5 0 0 0
7 13 1 3 4 0 0 0
8 14 1 2 3 0 1 1
9 15 1 1 2 1 0 1
10 16 0 1 1 0 1 1
di= # events in group ‘I’; Ri= # members of risk set at ‘ti’
18
Group Dead Alive Total
1 0 5 5
2 1 5 6
Total 1 10 11
01/2015
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.
Group Dead Alive Total O1,2 O2,2 E1,2 E2,2 V2
1 0 5 5
2 1 5 6
Total 1 10 11 0 1
1901/2015
Comparing Survival (11)
• What are the ‘E’ and ‘V’ columns?– Ei,t = expected # of events in group ‘i’ at time ‘t’– Vt = Approximate variance of ‘E’ at time ‘t’
2001/2015
Comparing Survival (12)
Group Dead Alive Total O1,2 O2,2 E1,2 E2,2 V2
1 0 5 5
2 1 5 6
Total 1 10 11 0 1 0.455 0.545 0.248
2101/2015
Comparing Survival (13)
• More generally, suppose we have:
– dt1 = # events at time ‘t’ in group 1
– dt2 = # events at time ‘t’ in group 2
– dt+ = # events at time ‘t’ (dt1+dt2)
– Rt1 = # in risk set at time ‘t’ in group 1
– Rt2 = # in risk set at time ‘t’ in group 2
– Rt+ = # in risk set at time ‘t’ (Rt1+Rt2)
• Then, we have the expected # of events in group 1 is:
2201/2015
Comparing Survival (14)
– dt1 = # events at time ‘t’ in group 1
– dt2 = # events at time ‘t’ in group 2
– dt+ = # events at time ‘t’ (dt1+dt2)
– Rt1 = # in risk set at time ‘t’ in group 1
– Rt2 = # in risk set at time ‘t’ in group 2
– Rt+ = # in risk set at time ‘t’ (Rt1+Rt2)
• And, the ‘V’s are given by this formula:
23
Group Dead Alive Total O1,2 O2,2 E1,2 E2,2 V2
1 0 5 5
2 1 5 6
Total 1 10 11 0 1 0.455 0.545 0.248
01/2015
Group Dead Alive Total O1,3 O2,3 E1,3 E2,3 V3
1 1 4 5
2 0 5 5
Total 1 9 10 1 0 0.5 0.5 0.25
At time ‘2’
At time ‘3’
24
Group Dead Alive Total O1,5 O2,5 E1,5 E2,5 V5
1 1 3 4
2 1 4 5
Total 2 7 9 1 1 0.889 1.111 0.432
01/2015
Group Dead Alive Total O1,t O2,t E1,t E2,t Vt
1 dt1 Rt1
2 dt2 Rt2
Total dt+ (Rt+-dt+) Rt+ dt1 dt2
At time ‘5’
At time ‘t’
2501/2015
Comparing Survival (15)
• Compute O1t – E1t 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.
2601/2015
Comparing Survival (16)
• Write this difference as: O+ – E+
• And, let
• Then, we have:
This is the log rank test
2701/2015
i t dt1 dt2 dt+ Rt1 Rt2 Rt+ Et1 Vt
1 2 0 1 1 5 6 11 0.455 0.248
2 3 1 0 1 5 5 10 0.500 0.250
3 5 1 1 2 4 5 9 0.889 0.432
4 8 0 0 0 3 4 7 0 0
5 10 1 0 1 2 4 6 0.333 0.222
6 11 0 0 0 1 4 5 0 0
7 13 0 0 0 1 3 4 0 0
8 14 0 1 1 1 2 3 0.333 0.222
9 15 1 0 1 1 1 2 0.500 0.250
10 16 0 1 1 0 1 1 0 0
total 4 4 8 3.010 1.624
28
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
01/2015
29
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:
01/2015
Log-rank or Mantel-Hanzel test
30
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:
01/2015
31
Comparing Survival (20)
• Log-rank:
– wt=1
– equally weights events at all points in time
• Wilcoxon test
– wt=Rt+
– 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.
01/2015
3201/2015
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
3301/2015
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
34
Practical stuff
• The next slide set looks at implementation in SAS– Strata statement– Test statement
• Expands the analysis options from the outline given here.
01/2015
35
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
01/2015
36
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)
03/2014
37
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 xi 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 xi are therefore
called "logrank scores" even when, because of censoring, actual
ranks are undefined.
03/2014
38
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.03/2014
39
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 happen03/2014
40
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
03/2014
4103/2014
Theory (4)
4203/2014
Theory (5)
43
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!
03/2014
4403/2014
• Example from Cantor
• We presented the log-rank test table earlier in this session.
• Here are the summary results
Group 1 Group 2
358+1015
2511+13+1416
4503/2014
i t dt1 dt2 dt+ Rt1 Rt2 Rt+ Et1 Et2 Vt
total 4 4 8 3.010 4.990 1.624
4601/2015