STAT 511
Spring 1999
Instructions:
1.
FINAL EXAM
NAME ____________
This is a closed book exam. No books or notes are allowed. We have not left
enough space to write your answers on this exam. Please write your answers on
separate sheets of paper. Be sure to write your name on each sheet of paper that
you submit.
A pharmacologist modeled the responsiveness of patients to a drug using the following
model
β0
Yj = β0 + εj
β2
Xj 

1 + 
β1 
where
Xj
is the dosage level of the drug
Yj
is the observed responsiveness expressed as a percent of the
maximum possible responsiveness
εj
denotes independent and identically distributed random errors with
E(εj) = 0 and Var(εj) = σ 2 for all j = 1, 2, ...., n.
and β0 > 0, β1 > 0 and β2 > 0. Data were obtained from 19 patients at the dosage levels
shown in the following table.
Patient
(j)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Dosage Level
(Xj)
1.0
2.0
3.0
3.5
3.5
4.0
4.0
4.5
4.5
5.0
5.0
5.5
5.5
6.0
6.0
6.5
7.0
8.0
9.0
Observed
Responsiveness
(Yj)
0.5
2.3
3.4
11.5
10.9
24.0
25.3
39.6
37.9
54.7
56.8
70.8
68.4
82.1
80.6
87.2
92.8
94.2
94.4
2
3
4
Least squares estimates of the parameters, β0, β1, and β2 were obtained from the
nls( ) function in S-PLUS. S-PLUS code and some of the results are shown on
pages 2 and 3.
2.
(a)
Give an interpretation of the parameters β0 and β1. What do these parameters
represent?
(b)
The least squares estimates of β0, β1 and β2 are shown in the results listed on
page 2 along with values for standard errors. Describe how the standard errors
are computed. Define any notation you introduce in your answer.
(c)
How is the deviance value listed on the top of page 3 computed? Give a formula.
(d)
Suppose the model is correct. Use the least squares estimates of the parameters to
estimate the dosage level of the drug at which the mean responsiveness is 80% of
the maximum. Show how to compute a standard error for your estimate, but a
numerical value for the standard error is not needed.
(e)
A plot of the least squares estimate of the curve defined by our model is included
in the results from the S-PLUS code. This plot suggests that the model may not
be entirely correct. Describe how you would perform a test of the fit of this
model.
Let tn denote the 20% trimmed mean computed from a sample of n values, Y1, Y2, ...,
Yn. A 20% trimmed mean is evaluated by ordering the data from smallest to largest,
deleting the largest 20% of the observed values and the smallest 20% of the observed
values, and computing the average of the remaining values. Explain how you would use
the bootstrap to obtain a standard error of a 20% trimmed mean for a random sample of
n = 100 observations.
5
3.
Females of a certain species of fresh water turtle lay their eggs in the sand and
immediately return to the water. The eggs are kept warm through the effect of the sun
warming the sand. The sex of the young that hatch from the eggs is affected by the
temperature at which the eggs are incubated. To study this phenomenon, a sample of 120
eggs was obtained from eggs deposited in the sand by turtles in New Mexico. The eggs
were randomly divided into 6 groups, with 20 eggs in each group. Each group of eggs
was incubated in a laboratory at a different temperature until the eggs hatched. Then, the
numbers of males and females were counted. The results are show below.
Incubation
Temperature (ºC)
27.0
27.5
28.0
28.5
29.0
29.5
Number
Females
3
5
4
12
17
18
Number
of Males
17
15
16
8
3
2
(a)
What are the basic features of a generalized linear model?
(b)
Let Yj denote the number of males that emerge from the nj = 20 eggs incubated at
the j-th temperature. Assume that Yj ~ Bin(nj, πj), where
log(1 - log(1 - πj)) = β0 + β1Xj,
j = 1, ..., 6
and Xj denotes the j-th temperature level. Maximum likelihood estimates were
computed with the glm( ) function in S-PLUS. The estimates and their
standard errors are as follows:
Estimate
βˆ0 = − 49.74
βˆ1 =
1.76
Standard Error
8.95
0.32
6
The formula for the residual deviance is
6
[
(
2 ∑ Yj log( Yj / n jπˆ j ) + ( n j − Yj ) log (n j − Yj ) /(n j − n j πˆ j )
j =1
)]
where π̂ j is the maximum likelihood estimator of π j for the model in part (a).
This statistic can be used to test a hypothesis. Clearly state the null and
alternative hypotheses, and state what you know about the distribution of this test
statistic.
(c)
4.
An independent sample of 120 eggs was obtained from turtles of the same
species living in Illinois. The researchers want to know if the relationship
between sex of the hatchlings and incubation temperature is the same for turtles in
New Mexico and Illinois. Describe how this hypothesis could be tested?
Three rabbits were used in an experiment to examine the effectiveness of the drug MDL
in controlling blood pressure. First, each rabbit was exposed to a stimulant (PBG) and
the increase in blood pressure was recorded, as a percentage of blood pressure measured
before PBG was administered. After waiting for two weeks, to allow any effect of the
first exposure to PBG to wear off, each rabbit was treated with MDL and once again
exposed to the same level of PBG. The resulting increase in blood pressure was
recorded. The goal is to determine if treatment with MDL is effective in suppressing the
effect of the stimulant (PBG) on blood pressure. The observations are as follows:
Rabbit
1
2
3
Control
(Not Treated
with MDL)
Y11
Y21
Y31
Treated
with
MDL
Y12
Y22
Y32
Here, the rabbits are considered to be a sample from a larger population of rabbits used in
the experiment, and the following model was proposed.
Yij = µj + ηi + εij
where
ηi ~ NID(0, σ η2 )
i = 1, 2, 3
εij ~ NID(0, σ ε2 )
i = 1, 2, 3 and j = 1, 2
and any ηi is independent of any εij.
7
(a)
Show how to write the model in the form
Y = Xβ + Zu + ε
~
~
~
~
where β is a vector of non-random parameters, u is a vector of random
~
~
effects, and ε is a vector of random errors. Report formulas for covariance
~
matrices for u and ε .
~
(b)
~
Suppose the values of the variance components, σ η2 and σ ε2 , are known. Give a
formula for the best linear unbiased estimator for β . Report what you know
~
about the distributional properties of this estimator.
(c)
Show how to construct a set of "error contrasts" to use in REML estimation of the
variance components. What is the motivation for using REML estimates of
variance components?
(d)
Suppose the REML estimates for σ η2 and σ ε2 are inserted into your formula for
the estimator for β from part (b). Report what you know about the
~
distributional properties of the resulting estimator.
Exam Score ________
Course Grade ______