Final
2004.6.22
(Written)
1. (30%)
Suppose we have the following data for the survival times of ovarian
cancer patients:
Subject
Survival Censor
time
indicator
13
1
52
0
6
1
40
1
10
1
7
0
66
1
I
II
III
IV
V
VI
VII
Sex
Age
BUN
1
1
2
1
1
2
1
66
66
53
69
65
57
52
25
13
15
10
20
12
21
(a) Calculate the Kaplan-Meier estimate for the data.
(b) Fit the above data by the Weibull distribution with density function


f t   2t exp  t 2 .
Find the MLE of  and find the estimated survival function.
(c) Suppose the variables Sex, Age, and BUN are the variables of interest.
Using proportional hazards model, derive the partial likelihood and
describe how to obtain the partial likelihood estimate.
2. (30%)
(a) Suppose Y1 ~ P1  and
Y2 ~ P2  and we are interested in
the ratio   1  . Please find the conditional likelihood estimate.
2
(b) Suppose that  1 ,  2 ,,  n are independent and identically distributed
with density
 
f   , depending on the unknown parameter  . Suppose
  
1
also that the observed values y1 , y 2 ,, y n satisfy
1
Y  X  Zr  
,
for fixed known matrices X , Z and unknown parameters  , r . Show that
the distribution of R  I  PY does not depend on  , where

P  X XtX

1
Xt.
3. (20%)
Suppose the independent data Y1 , Y2 ,, Yn have the mean  i and
the variance function. Vi i 
3
(a) If  i   , Vi     , find the quasi-likelihood function and
maximized quasi-likelihood estimate for  .
(b) Suppose  i  xi  , Vi i   i , where  is a single parameter.
Find the quasi-score function and the estimate based on the
quasi-score function.
(Computer)
1. (30%) For the following data,
Patient Time
Cens
281
1
1
604
0
2
457
1
3
384
1
4
341
0
5
842
1
6
1514
1
7
182
0
8
1121
1
9
1411
0
10
814
1
11
Treat
0
0
0
0
0
0
1
1
1
1
1
2
Age
46
57
56
65
73
64
69
62
71
69
77
LBR
3.2
3.1
2.2
3.9
2.8
2.4
2.4
2.4
2.5
2.3
3.8
1071
1
1
58
3.1
12
Cens: censor indicator; Treat: treatment.
(a) Calculate the Kaplan-Meier estimates for two treatment groups and
plot the survival functions in the same Figure.
(b) Test the treatment effect using log-rank test and Wilcoxon test.
(c) Fit the following proportional hazards models
 t   0 t exp  
and comment on the results,

    Treat

  1  Age  2  LBR

  1  Age  2  LBR  12 Age  LBR
2. (40%)The following data concern a type of damage caused by waves
to the forward section of certain cargo-carrying vessels:
Ship
Year of
Period of
Aggregate
Number of
type
construction
operation months service
damage
incidents
1960-64
1960-74
127
0
A
1960-64
1975-79
63
0
A
1965-69
1960-74
1095
3
A
1965-69
1975-79
1095
4
A
1970-74
1960-74
1512
6
A
1970-74
1975-79
3353
18
A
0
1975-79
1960-74
0
A
1975-79
1975-79
2244
11
A
1960-64
1960-74
45
0
B
0 
1960-64
1975-79
0
B
1965-69
1960-74
789
7
B
1965-69
1975-79
437
7
B
1970-74
1960-74
1157
5
B
1970-74
1975-79
2161
12
B
0
1975-79
1960-74
0
B
1975-79
1975-79
542
1
B
1960-64
1960-74
1179
1
C
1960-64
1975-79
552
1
C
3
1965-69
1960-74
781
C
1965-69
1975-79
676
C
1970-74
1960-74
783
C
1970-74
1975-79
1948
C
1975-79
1960-74
0
C
1975-79
1975-79
274
C
*: Necessarily empty cells, **: Accidentally empty cell
0
1
6
2
0
1
(a) Fit the log-linear model for the response the number of damage
incidents, with qualitative factors, Ship type, Year of construction,
Period of operation and quantitative variate Aggregate months
service as offset. What are the conclusions?
(b) Please fit the above data based on quasi-likelihood approach.
4