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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

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Comparing survival (2)

• Can be addressed through:– Regression methods

• Cox models (later)

– Non-regression methods• Log-rank test

• Mantel-Hanzel

• Wilcoxon/Gehan

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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

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New Rx

Old Rx

Effect of new treatment

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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).

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Δ

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

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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.

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Comparing Survival (4)

• Compare curves at each point

• Combine across all events

• Can limit comparison to times when an event happens

ti

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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

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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

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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)

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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.

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Comparing Survival (9)

• Example from Cantor

• We present the merged and sorted data in the table on the next slide.

Group 1 Group 2

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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’

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Group Dead Alive Total

1 0 5 5

2 1 5 6

Total 1 10 11

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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

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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’

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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

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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:

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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:

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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

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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’

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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

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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’

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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.

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Comparing Survival (16)

• Write this difference as: O+ – E+

• And, let

• Then, we have:

This is the log rank test

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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

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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

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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

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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:

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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.

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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

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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.

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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

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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

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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

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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

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Theory (4)

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Theory (5)

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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!

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• Example from Cantor

• We presented the log-rank test table earlier in this session.

• Here are the summary results

Group 1 Group 2

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i t dt1 dt2 dt+ Rt1 Rt2 Rt+ Et1 Et2 Vt

total 4 4 8 3.010 4.990 1.624

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