Download - 2002 Final
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Final 92.1.16(Written)
1. (30%)Suppose that a experiment is undertaken to compare the effect of a new
process with the current process. The response is the number of
occurrences. For a single stand-alone experiment, the number of
occurrences using the current process and new process may be regarded
as independent Poisson variables ( )ii PY 11 ~ and ( )ii PY 22 ~ ,
respectively, where
( )( ) ii
ii ni
=
=+=
2
1
log
,,2,1,logK
and the parameter of interest, , can characterize the treatment effect.
(a)For every single stand-alone experiment, find the sufficient statisticfor i and the conditional distribution based on the sufficient
statistic.
(b)Derive the conditional likelihood for by combining the conditionallikelihood for every single-alone experiment.(c)Derive the score statistic for no treatment effect based on the
conditional likelihood.
2. (20%)
Suppose the independent data nYYY ,,, 21 K have the mean i and
the variance function. ( )iiV
(a)If =i , ( ) ( ) = 1iV , find the quasi-likelihoodfunction and maximized quasi-likelihood estimate for .
(b)Suppose ii x= , ( ) 2 =iiV , where is a singleparameter. Find the quasi-score function and the estimate based on the
quasi-score function.
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3. (30%)
Suppose we have the following data for the survival times of ovarian
cancer patients:
Subject Survival
time
Censor
indicator
Group Age
I 156 1 T 66
II 1040 0 T 38
III 59 1 T 72
IV 421 0 C 53
V 329 1 T 43
VI 769 0 C 59
T: treatment group; C: control group.
(a)Calculate the Kaplan-Meier estimate for the data.(b)Test the treatment effect using log-rank test and Wilcoxon test.(c)Suppose the variable age is the only variable of interest. Using
proportional hazards model, derive the partial likelihood and describe
how to obtain the partial likelihood estimate.
4. (20%)
The responses nyyy ,,, 21 K are assumed to be the observed values of
independent random variables nYYY ,,, 21 K such that ( )iii mBY ,~
For the linear logistic model, we have ( )
i
i
ii x=
=
1logit ,
where is a single parameter and ix is the covariate corresponding
to iY . Derive the iterated reweighted least square (IRLS) estimate.
(Computer, using SAS or Splus)
1. (30%) For the following data,Patient Time Cens Treat Age RDISEASE PERF
1 156 1 1 66 2 2
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2 1040 0 1 38 2 2
3 59 1 1 72 2 1
4 421 0 2 53 2 1
5 329 1 1 43 2 1
6 769 0 2 59 2 2
7 365 1 2 64 2 1
8 770 0 2 57 2 1
9 1227 0 2 59 1 2
10 268 1 1 74 2 2
11 475 1 2 59 2 2
12 1129 0 2 53 1 1
13 464 1 2 56 2 2
14 1206 0 2 44 2 115 638 1 1 56 1 2
16 563 1 2 55 1 2
17 1106 0 1 44 1 1
18 431 1 1 50 2 1
19 855 0 1 43 1 2
20 803 0 1 39 1 1
21 115 1 1 74 2 1
22 744 0 2 50 1 123 477 0 1 64 2 1
24 448 0 1 56 1 2
25 353 1 2 63 1 2
26 377 0 2 58 1 1
Cens: censor indicator; Treat: treatment.
(a) Calculate the Kaplan-Meier estimate and plot the survival function.
(b) Test the treatment effect using log-rank test and Wilcoxon test.
(c) Fit the following proportional hazards models ( ) ( ) ( ) exp0 tt = ,
z AgeTreatTreatAge ++= 1221 z PERFAge += 21 z DISEASEAge += 21 2. (30%)The following data concern a type of damage caused by waves
to the forward section of certain cargo-carrying vessels:
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Ship
type
Year of
construction
Period of
operation
Aggregate
months service
Number of
damage
incidents
A 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 1975-79 1960-74 0 0
A 1975-79 1975-79 2244 11
B 1960-64 1960-74 45 0
B 1960-64 1975-79 0 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 1975-79 1960-74 0 0
B 1975-79 1975-79 542 1
C 1960-64 1960-74 1179 1
C 1960-64 1975-79 552 1C 1965-69 1960-74 781 0
C 1965-69 1975-79 676 1
C 1970-74 1960-74 783 6
C 1970-74 1975-79 1948 2
C 1975-79 1960-74 0 0
C 1975-79 1975-79 274 1
*: Necessarily empty cells, **: Accidentally empty cell
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?