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7.3 Comparison of Two Groups of Survival Data

Suppose )()2()1( mt t t

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( ) ( ) ===

==

=

m

j j

j j j

m

j j j

c L n

d nd ed

lU

1

11

111

0,

where is the conditional log-likelihood function and( )cl

j

j j j n

d ne 11 = is the conditional mean of . Further, jd

( ) ( )=

==m

j j j

j j j j j L L nn

d nd nnU Var V

12

21

1 .

As the number of death time is not too small, then

( )1,0~ N V

U Z

L

L=.

Note:

We can also use the continuity-correct value

L

L

V

U Z 2

1=

.

We can use Z or Z to test 0:0 = H . The test using

L

L

V U

Z 2

2 = is referred to as Log-rank test or Mantel-Haenzel

procedure. The other test is based on

( )=

=m

j j j jw ed nU

111 ,

which is referred to as the Wilcoxon test . The difference between

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LU and is that the Wilcoxon test, each difference

is weighted by . The effect of this is to give less

weight to difference between and at those times when

the total number of individuals who are still alive is small, that is, at

the longest survival time. The variance of is

wU

j j ed 11 jn

jd 1 je 1

wU

( ) ( )( )=

==m

j j j

j j j j j jww nn

d nd nnnU Var V

12

212

1 .

Thus, the Wilcoxon statistic to test is0:0 = H

w

ww V

U W

2

= . Note that under .21~ wW 0:0 = H

Example (continue):

In the motivating example, we have dataGroup T C T C T T C C

Survival time

6 6 7 7 7 5.9 10 11

Then, we have the following tables:

6)1( =t Group # of death # alive

beyond# alive

T 111 =d 3 411 =n C 021 =d 4 421 =n

11 =d 7 81 =n

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7)2( =t Group # of death # alive

beyond# alive

T 112 =d 2 312 =n C 122 =d 2 322 =n

22 =d 4 62 =n

11)3( =t Group # of death # alive

beyond# alive

T 013 =d 0 013 =n C 123 =d 0 123 =n

13 =d 0 13 =n Thus,

( ) ( ) ( )

5.0 1

100

623

18

141

131312121111

=

+

+

=

++= ed ed ed U L

and

( ) ( ) ( )

4

110

016

2316

814

18

131331212211111

=

+

+

=

++= ed ned ned nU w

Similarly, and can be obtained. LV wV

4