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Final 2004.6.22

(Written)

1. (30%)

Suppose we have the following data for the survival times of ovarian

cancer patients:

Subject Survival

time

Censor

indicator

Sex Age BUN

I 13 1 1 66 25

II 52 0 1 66 13

III 6 1 2 53 15IV 40 1 1 69 10

V 10 1 1 65 20

VI 7 0 2 57 12

VII 66 1 1 52 21

(a)Calculate the Kaplan-Meier estimate for the data.(b)Fit the above data by the Weibull distribution with density function

( ) ( )2exp2 tttf = .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 ( )11 ~ PY and ( )22 ~ PY and we are interested in

the ratio2

1

= . Please find the conditional likelihood estimate.

(b) Suppose that n ,,, 21 K are independent and identically distributed

with density

f

1, depending on the unknown parameter . Suppose

also that the observed values nyyy ,,, 21K

satisfy

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2

++= ZrXY ,

for fixed known matrices ZX, and unknown parameters r, . Show that

the distribution of ( )YPIR = does not depend on , where

( ) tt XXXXP 1= .

3. (20%)

Suppose the independent data nYYY ,,, 21 K have the mean i and

the variance function. ( )iiV

(a)If =i , ( ) 3 =iV , find the quasi-likelihood function andmaximized quasi-likelihood estimate for .

(b)Suppose ii x= , ( ) iiiV = , where is a single parameter.Find the quasi-score function and the estimate based on thequasi-score function.

(Computer)

1. (30%) For the following data,

Patient Time Cens Treat Age LBR

1 281 1 0 46 3.2

2 604 0 0 57 3.13 457 1 0 56 2.2

4 384 1 0 65 3.9

5 341 0 0 73 2.8

6 842 1 0 64 2.4

7 1514 1 1 69 2.4

8 182 0 1 62 2.4

9 1121 1 1 71 2.5

10 1411 0 1 69 2.3

11 814 1 1 77 3.8

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12 1071 1 1 58 3.1

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 ( ) ( ) ( ) exp0 tt =

and comment on the results,

z Treat= z LBRAge += 21 z LBRAgeLBRAge ++= 1221 2. (40%)The following data concern a type of damage caused by waves

to the forward section of certain cargo-carrying vessels:

Ship

type

Year of

construction

Period of

operation

Aggregate

months service

Number of

damage

incidents

A 1960-64 1960-74 127 0A 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 0B 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 1

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C 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

(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 asoffset. What are the conclusions?

(b) Please fit the above data based on quasi-likelihood approach.