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EPI 5344:Survival Analysis in

EpidemiologyLog-rank vs. Mantel-Hanzel testing

Dr. N. Birkett,Department of Epidemiology & Community

Medicine,University of Ottawa

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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)

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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.

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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 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 happen03/2014

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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 tests03/2014

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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

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Theory (5)

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Theory (6)

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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!

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• 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

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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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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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