STAT 557
FALL 1998
Instructions:
1.
FINAL EXAM
Name _______________
You may use only the formula sheet provided with this exam. No other
notes or books are allowed. Write your answers in the spaces provided on
this exam. If you need more space use the back of the page or attach
additional sheets of paper, but clearly indicate where this is done. You
should not attempt to complete complex calculations. You will receive
complete credit by showing that you know how to properly solve the
problem.
Use of herbicides to control weeds in grain fields may adversely affect the environment
by inhibiting reproduction in certain organisms. To examine the reproductive toxicity of
a certain herbicide, C. dubia zooplankton, were exposed to five concentrations of the
herbicide: 0, 0.8, 1.6, 2.35, and 3.1 mg/ l The total number of offspring born to the C.
dubia at each concentration are shown below.
Herbicide
Concentration
0
0.8
1.6
2.35
3.1
Number of
Offspring
33
31
28
17
6
Maximum likelihood estimation was used to fit a Poisson regression model to these data.
Using Z to denote the herbicide concentration and mz to denote the corresponding mean
number of offspring, the estimated model is
log ( m̂z ) = 3.5721 (0.1178)
0.1546 Z2
(.0340)
Standard errors are given in parentheses beneath the estimated coefficients. The
estimated covariance matrix for the estimated coefficients is
intercept
Z2
intercept
0.01387
-0.00245
Z2
-0.00245
.0016
(a) Write down a formula for the log-likelihood function.
(b)
Explain how the estimated covariance matrix, shown above, was obtained.
2
(c) One quantity of interest is the reproductive index defined as
RI =
(m 0 - mz )
m0
where m0 is the mean number of offspring when no herbicide is present. Estimate
Z.50, the herbicide concentration at which RI = 0.50. This is the herbicide
concentration corresponding to a 50 percent reduction in the number of offspring.
(d) Show how to construct an approximate 95% confidence interval for Z.50 from part
(c).
2.
Responses from 2500 high school students were obtained from a survey on smoking
habits. These responses were cross classified into a 2x2x3x3 contingency table with
respect to the following factors.
A:
Smoking status of the respondent (i=1 smokers, i=2 nonsmokers)
B:
Sex of respondent (j=1 female, j=2 male)
C:
Socio-economic status of parents (k=1 low, k=2 middle, k=3 high).
D: Smoking status of parents ( l = 1 neither smoke, l =2 one smokes, l =3 both
smoke)
(a)
Consider the following log-linear model
CD
log(m i jk l) =λ +λ iA +λ Bj +λ Ck +λ lD + λ AB
i j +λ k l
What does this model imply about associations among the four factors?
(b) Using the constraints
AB
AB
CD
CD
CD
CD
λA2 = λ2B = λC3 = λD3 = λ 12
= λ AB
21 = λ 22 = λ 13 = λ 23 = λ 31 = λ 32 = 0 ,
3
the maximum likelihood estimates for the interaction terms in the model in part (a)
are shown below:
Parameter
Estimate
AB
λ 11
-.3505
Standard
Error
.0837
CD
λ 11
.0716
.3428
CD
λ 12
-1.9601
.3334
λ CD
21
.0795
.3649
λ CD
22
-1.3838
.3545
Assuming the model is correct, explain how the estimate for λCD
12 should be
interpreted.
(c) Consider the deviance statistic for testing the null hypothesis that the model in part
(a) is correct against the general alternative. How many degrees of freedom are
associated with this test?
(d) List the conditions under which the deviance test in part (c) would approximately
have a central chi-squared distribution.
(e) The following model was also fit to the data:
CD
log(m i jk l) =λ +λ iA +λ Bj +λ Ck +λ lD + λ AB
i j +λ k l + γ1 ui vk + γ2 ui wl
where
(u1, u2) = (-.5, .5)
(v1, v2, v3) = (-1, 0, 1)
(w1, w 2, w 3) = (-1, 0, 1)
and γ1 and γ 2 are unknown parameters. Maximum likelihood estimates are
Parameter
γ1
γ2
Estimate
-.422
.329
Standard Error
.040
.030
assuming that this model is correct, interpret γ̂ 2 as an odds ratio.
(f) What are the degrees of freedom associated with the deviance test of the null
hypothesis that the model in part (a) is correct against the alternative that the model
in part (e) is correct?
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(g) Suppose respondents were obtained by first taking a simple random sample of 100
high schools and then taking a simple random sample of 25 students within each
high school. What effect, if any, would this have on parameter estimates and
standard errors obtained by maximizing a multinomial log-likelihood (as was done
in part (e) of this problem)?
3.
Leukemia patients were treated with chemotherapy and examined at the end of one
year to determine if the disease was in remission. Information was also recorded on the
following variables:
Z1
The percentage of cells undergoing DNA synthesis at the start of the
chemotherapy treatment. This is called the labeling index.
Z2
The highest body temperature of the patient (in ° F ) during the week prior
to the start of chemotherapy.
There were 58 leukemia patients in this study. Let πi denote the conditional probability
that the disease goes into remission for patients with values (Zli, Z2i) for the two
explanatory variables. Maximum likelihood estimation was used to fit the following
logistic regression model to the data:
 πˆ
log i
 1 − πˆ i


=


103.3 + 0.3463 Z1i - 1.0844 Z2i
(42.14)
(.1019)
(.4302)
Standard errors are shown in parentheses beneath the corresponding parameter
estimates. The estimated covariance matrix for the parameter estimates is
intercept
labeling index (Z1)
temperature (Z2)
intercept
1776.05487
1.72931
-18.12329
Labeling
Index
Z1
1.72931
0.01038
-0.01859
Temperature
Z2
-18.12329
-0.01859
0.18505
(a) Clearly explain how 0.3463, the estimated coefficient for Z1, can be interpreted with
respect to a conditional odds ratio for remission.
(b) Estimate the probability that a leukemia patient with Z1 = 10% and Z2 = 99 ° F at
the start of the chemotherapy treatment will experience remission.
(c) Show how to construct a 95% confidence interval for this probability (you need not
complete the calculations).
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d)
Some diagnostic results are shown on page 7. Explain how C and the Df beta
values are computed.
(e) Explain what the diagnostic results from part (d) indicate about cases 23, 30, and
53.
(f) Four plots are shown on page 9. What do these plots indicate about the estimated
model and how it could be improved?
4.
Consider the following study of the effect of incubation temperature on the sex of turtle
eggs. In this study 10 turtle eggs were collected from each of 20 different sites in
Illinois and 10 turtle eggs were also collected from each of 20 different sites in New
Mexico. All eggs at one site were laid by the same female. The ten eggs from one site
in Illinois were placed in a box with the ten eggs from one site in New Mexico. Sites
from Illinois and New Mexico were randomly matched. Four boxes were incubated at
each of five temperatures: 26.5, 27, 27.5, 28, 28.5 °C . Boxes were randomly assigned
to temperatures. The numbers of male and female turtles hatching from Illinois eggs
and the number of male and female turtles hatching from New Mexico eggs were
recorded for each box incubated at each temperature.
Although an incubator can be set at a specific temperature, actual temperatures can vary
across locations inside the incubator. Hence, the temperatures will not be the same in
all boxes incubated at the same temperature setting, and this will affect all of the eggs in
a box. A slightly higher temperature in the box, for example, would increase the
probability of female turtles hatching from the eggs.
Using π to denote the conditional probability that a female turtle hatches from an egg, a
logistic regression model is
 π 
log 
 = β0 + β1 Z1 + β2 Z 2
1 − π 
where
 0 for Illinois eggs
Z1 = 
 1 for New Mexico eggs
and Z2 is the temperature setting on the incubator. Explain how you would estimate the
parameters β0 , β1 , and β2 in this model and obtain standard errors for the estimates.
SCORE ________
COURSE GRADE ________