2007 STAT3955 Past Paper 1
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School
The University of Hong Kong *
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Course
3811
Subject
Statistics
Date
Nov 24, 2024
Type
Pages
5
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THE
UNIVERSITY
OF
HONG
KONG
DEPARTMENT
OF
STATISTICS
AND
ACTUARIAL
SCIENCE
STAT3811
SURVIVAL
ANALYSIS
May
22,
2007
Time:
2:30
p.m.
-
5:30
p.m.
Candidates taking examinations that permit the use
of
calculators
may
use any
cal-
culator which fulfils the following criteria: (a)
it
should
be
self-contained, silent,
battery-operated
and
pocket-sized and
{b)
it
should have numeral-display facilities
only and should
be
used only
for
the purposes
of
calculation.
It
is the candidate's responsibility to ensure that the calculator operates satisfactorily
and the candidate
must
record the
name
and type
of
the calculator on the front page
of
the examination scripts. Lists
of
permitted/prohibited calculators will
not
be
made
available to candidates
for
reference, and the onus will
be
on the candidate to ensure
that the calculator used will
not
be
in
violation
of
the criteria listed above.
Answer
ALL
SIX
questions.
Marks
are
shown
in
square
brackets.
1.
The following
data
consist of the times
to
relapse
and
death
of
10
bone marrow
transplant patients.
Table 1
Patient
Relapse Time (months)
Death
time (month)
1
5
11
2
8
12
3
12
15
4
24
33+
5
32
45
6
17
28+
7
16+
16+
8
17+
17+
9
19+
19+
10
30+
30+
In the sample, patients 4 and 6 were still alive
at
the
end of
the
study
and
patients 7-10 were alive, free of relapse
at
the
end of
the
study. Suppose the
time to relapse has
an
exponential distribution with hazard
rate
A
and the
time to
death
has a Weibull distribution with parameters
()
and
a.
S&AS:
STAT3811
Survival
Analysis
2
(a) Construct
the
likelihood function for the relapse
rate
,\ and calculate the
MLE of
A.
(b) Construct
the
likelihood function for the parameters
B
and
a.
(c)
Suppose
we
were only allowed
to
observe a patient's death time if the
patient
relapsed.
Construct the likelihood for
B
and
a
based on this
truncated
sample, and compare
it
to
the
results in (b).
[Total: 15
marks]
2.
Consider
the
data
on
the
time to relapse of
10
bone marrow transplant patients
in Table 1 above.
3.
(i) Construct
the
Kaplan-Meier Product-Limit (PL) estimator and plot the
estimated survival curve.
(ii) Calculate
the
estimated variance of the
PL
estimator and determine the
95% confidence interval for the survival probability
S(t)
at
t
=
20.
(iii) Calculate
the
mean remission survival time
to
30 months.
(iv)
Find
the
median remission survival time.
[Total: 15
marks]
The following
data
consist of the relapse survival time (in days) of twenty
patients after
treatment
for a rare type of cancer
20
48
68
106+
119+
129
132
134+
142
198+
201
208+
215
231+
246
290+
302
304+
(a) Estimate
the
cumulative hazard
rate
using
the
Nelson-Aalen estimator.
(b)
Estimate
the
hazard
rate
at
t
=
80
days, using
the
uniform kernel-with a
bandwidth of
40
days.
(c)
Estimate
the
hazard
rate
at
t
=
250
days, using the uniform kernel with
a bandwidth of 40 days.
Hint: The uniform kernel is given
by
K(x)
=
1/2
for
-1
~
x
~
1,
0 otherwise.
[Total: 15
marks]
4.
Ten female
patients
with breast cancer are randomized to receive either CMF
(cyclic administration of cyclophosphamide, methatrexate, and fluorouracil)
or no
treatment
after a radical mastectomy.
At
the
end of two years, the
fol-
lowing times
to
relapse (or remission times) in months are recorded:
CMF (group
1)
23, 16+, 18+, 20+,
24+
Control (group
2)
15, 18, 19, 19,
20
184+
318
S&AS:
STAT3811
Survival
Analysis
3
(a) Suppose the significance level
a=
0.05. Use
the
Mantel-Haenszel
test
to
test
the
null hypothesis
Ho:
S1(t)
=
S
2
(t)
two treatments are equally effective
against
H
1
:
8
1
(t)
=/:-
S
2
(t)
two treatments are
not
equally effective
(b) Assume
that
the
two distributions are exponential with parameters
.X
1
and
A2,
respectively. Under the same significance level as (a), use the
Cox F-
test
to
test
the null hypothesis
Ho
:
.X
1
=
.X
2
two treatments are equally effective
against
H
1
:
.X
1
=/:-
.X
2
two treatments are
not
equally effective
[Total:
15
marks]
5.
In a
study
of noise level and efficiency, twelve students were given a very
simple
test
under three different noise levels.
It
is known
that
under normal
conditions,
they
should be able
to
finish
the
test
in
10
minutes. The students
were randomly assigned to the three levels. Table 2 gives
the
time required
to finish
the
test
for
the
three levels. Are
the
three noise levels significantly
different
at
the
0.05 level of significance? (See the statistical table on P.5)
Table 2
1
2
3
10.5
10.0
12.0
9.0
12.0
13.0
9.5
12.5
15.5
9.0
11.0
14.0
[Total:
15
marks]
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S&AS:
STAT3811
Survival
Analysis
4
6.
Twelve melanoma patients were studied
to
compare the immunotherapies BCG
(Bacillus Calmette-Guerin) and Corynebacterium parvum for their abilities to
prolong remission. The first group are BCG patients, and the second group
are C. parvum patients. All the patients were resected before treatment began
and thus had no evidence of melanoma
at
the time of first treatment. The
data
(in months) are as follows
BCG group: 33.7+, 3.8, 6.3, 2.3, 6.4, 23.8+, 1.8;
C.
parvum group: 4.3, 26.9+, 21.4+, 18.1+, 5.8.
A proportional hazards model
is
given by
A(tjZ)
=
Ao(t)
exp((3Z),
where
Z
=
1 for BCG group and
Z
=
0 for
C.
parvum group.
(i)
Calculate
the
partial likelihood, the logarithm of the partial likelihood,
the score function and the information matrix for the proportional haz-
ards model.
(ii)
Use
the
score test to test the hypothesis
(3
=
0
at
the 0.05 significance
level.
(iii)
Use
(i)
and
the
Newton-Raphson algorithm with an initial guess
at
(3
=
0
to compute a two step iterative estimator of
(3.
(iv)
Use
the
likelihood ratio test to test the null hypothesis
Ho
:
(3
=
0
at
the
0.05 significance level.
(v)
Use
the
Wald test to test the null hypothesis
Ho
:
(3
=
0
at
the
0.05
significance
leveL
[Total:
25
marks]
**********
END
OF
PAPER**********
S&AS:
STAT3811
Survival
Analysis
5
STATISTICAL TABLES
n
1
•
3,n
2
•5,n
2
=5
n
1
=4,n
2
=.4,n,
s:::4
n
1
=4,n
2
=4,n,
=
4 n
1
=4,n
2
=4,n,
=s
X
P
0
~H;a.x}
X
P
0
{H>x}
X
P
0
{H>x}
X
P
0
{H>x}
6.998
.015
.000
1.000
4.654
.097
.119
.952
1.050
.015
.038
.994
4.769
.094
.132
.937
7.121
.014
.115
.968
4.885
.086
.201
.930
7.209
.014
.154
.941
4.962
.080
.218
.916
7.226
.012
.269
.913
5.115
.074
.228
.903
7.288
.012
.346
.864
5.346
.063
.267
.889
7.306
.012
.462
.840
5.538
.051
.297
.815
7.314
.Oil
.500
.815
6.654
.055
.343
.869
7.437
.011
.615
.770
5.692
.049
.376
.862
7.543
.010
.731
.746
5.808
.044
.382
.849
7.578
.010
.808
.706
6.000
.040
.399
.836
7.622
.009
.962
.667
6.038
.037
.425
.823
7.736
.009
1.038
.648
6.269
.033
.415
.811
7.763
.008
1.077
.630
6.500
.030
.528
.798
7.780
.008
1.192
.592
6.577
.026
.544
.792
1.859
.007
1.385
.551
6.615
.024
.597
.780
7.894
.007
1.423
.540
6.731
.021
.610
.768
7.912
.007
1.500
.510
6.962
.019
.613
.757
8.026
.006
1.654-
.480
7.038
.018
.640
.745
8.079
.006
1.846
.452
7.269
.016
.689
.734
8.106
.006
1.885
.436
1.385
.015
.742
.723
8.237
.005
2.000
.397
7.423
.013
.771
.711
8.264
.005
2.192
.370
7.538
.011
.804
.706
8.316
.005
2.346
.348
7.654
.008
.824
.695
8.334
.005
2.423
.327
7.731
.007
.860
.690
8.545
.004
2.462
.307
8.000
.005
.870
.679
8.571
.oo4
2.577
.296
8.115
.003
.903
.668
8.580
.004
2.808
.277
8.346
.002
.910
.658
8.650
.003
2.885
.260
8.654
.001
.940
.647
8.659
.003
2.923
.252
8.769
.001
1.019
.637
8.791
.002
3.038
.234
9.269
.001
1.058
.627
8.809
.002
3.115
.219
9.846
.000
1.068
.617
8.950
.002
3.231
.212
1.124
.607
9.002
.002
3.500
.197
1.167
.598
9.055
.001
3.577
.173
n
1
=4,n
2
=4,n
3
=5
1.187
.589
9.284
.001
3.731
.162
1.190
.584
9.336
.001
3.846
.151
X
P
0
{H>x}
1.203
.574
9.398
.001
3.962
.145
1.256
.565
9.521
.000
4.154
.136
.000
1.000
1.272
.556
9.635
.000
4.192
.131
.030
.996
1.299
.548
9.916
.000
4.269
.122
.033
.981
1.371
.539
10.057
.000
4.308
.114
.086
.974
1.404
.534
10.550
.000
4.500
.104
.096
.967
1.414
.526
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