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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?