Unformatted text preview: or Group 1 (in months): 2, 4*, 5, 6, 9, 9, 12,
12*, 15*, 17.
Survival times for Group 2 (in months): 6*, 7, 9*, 10, 13,
15*, 17*, 18
where ∗ denotes the censored.
What can you say about the hazard rates for the 2 groups? 36/42 Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases For k = 10
tj
2
5
6
7
9
10
12
13
17
18
sum 37/42 n1j
10
8
7
6
6
4
4
2
1
0 d1j
1
1
1
0
2
0
1
0
1
0
7 n2j
8
8
8
7
6
5
4
4
2
1 d2j
0
0
0
1
0
1
0
1
0
1
4 nj
18
16
15
13
12
9
8
6
3
1 dj
1
1
1
1
2
1
1
1
1
1
11 Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases We have
χ2 = LR
(Z1 )2
LR
var (Z1 ) = k
j =1 (d1j
n1j
k
j =1 nj 1− − e1j ) n1j
nj 2 nj −dj
nj −1 = 2.42
dj so that
pvalue = Pr (χ2 > 2.42) = 0.12 > 0.05
1
and hence we can not reject the null hypothesis that the hazard
rates for the 2 groups are same. 38/42 Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases Wilcoxon statistic: w (tj ) = nj (Gehan, 1965)
In this particular case
k
W
Z1 = nj (d1j − e1j )
j =1 and
k (nj )2 W
var (Z1 ) =
j =1 n1j
nj 1− n1j
nj nj − dj
nj − 1 Again, under the null hypothesis,
W
(Z1 )2
W
var (Z1 )
39/42 is Chisquare distributed (with 1 degree of freedom). dj . Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases Example
A medical study was performed to investigate the diﬀerence (or
otherwise) in the eﬀectiveness of 2 alternative treatments (A and
B) to a disease. The survival time (in months) are
Survival times for Group 1 (in months):
2, 3∗ , 4, 6, 6, 12∗ , 12
Survival times for Group 2 (in months):
6∗ , 7, 9∗ , 10, 13, 15∗ , 17∗
where ∗ denotes a censored observation.
Suppose the variance of the Wilcoxon statistic was calculated as
160.40. Calculate the Wilcoxon statistic and perform the Wilcoxon
test.
40/42 Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases Solution
j
1
2
3
4
5
6
7 tj
2
4
6
7
510
12
13 d1j
1
1
2
0
0
1
0 n1j
7
5
4
2
2
2
0 d2j
0
0
0
1
1
0
1 n2j
7
7
7
6
4
3
3 dj
1
1
2
1
1
1
1 nj
14
12
11
8
6
5
3 n e1j = n1jj dj
0.5
0.416666667
0.727272727
0.25
0.333333333
0.4
0 nj (d1j − e1j )
7
7
14
2
2
3
0 and hence the Wilcoxon statistic is 27.
The null hypothesis is that there is no diﬀerence in all points of the
survival function vs the alternative of at least 1 point of diﬀerence.
272
The Wilcoxon chisquared statistic is 160.4 = 4.545 which is
signiﬁcant.
Hence on the basis of the Wilcoxin test result we reject the null
and conclude that there is evidence of a diﬀerent survival function.
41/42 Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases Discussion
Some other weight functions may be appropriate. The choice
of weight function depends on the investigator’s desire to give
diﬀerent weights to diﬀerent types of error.
For instance, when comparing 1 vs nj (logrank vs Wilcoxon),
the latter gives more weight to early times (because
n1 > n2 > n3 > · · · )
A practical note: logrank statistic is more powerful for
detecting diﬀerences in the hazard rates when the hazard
rates are proportional (h1 (t ) = rh2 (t )) i.e.,
S1 (t ) = [S2 (t )]r
for some constant r .
The above tests can be generalised to involve
more than 2 groups.
42/42...
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