STAT 557
FALL 2000
FINAL EXAM
NAME ________________
Instructions: You may use a calculator and the formula sheets you brought to this exam. No
other notes or books are allowed. Write your answers in the spaces provided
below. 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 need not complete
numerical computations, you will receive complete credit by showing that you
know how to solve the problem. Be sure to define any notation you use that is not
defined in the statement of a problem.
1. To determine if the incidence rates of a particular bad side effect are different for two
anesthetics, patients undergoing a certain surgical procedure were randomly assigned to
one of the two anesthetics, labeled A or B. Each patient was classified according to
whether or not they experienced the particular side effect. One of the 10 patients who
received anesthetic A experienced the side effect, and three of the 8 patients who received
anesthetic B experienced the side effect. Describe an appropriate test procedure and show
how to compute the p-value.
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2. A company that sells its products by mailing offers to potential customers currently has
about a 4% response rate. It has been purchasing address lists for potential costumers from
supplier A. Supplier B claims that they can provide lists of addresses for potential
customers that will generate a higher response rate. To investigate this claim, the company
will mail offers to a random sample of n addresses provided by supplier A, and they will
mail the same offers to a random sample of n addresses provided by supplier B. How big
should n be? The company would like to have a good chance (at least 80%) of showing a
difference at the .05 level of significance if addresses from Supplier B can increase their
response rate by at least one percentage point. Display the formula you would use to
determine n.
3. To study the potential benefit of a new drug for treating a breathing condition, 400 subjects
who suffered from the breathing condition were randomly divided into two groups of equal
size. Subjects in one group received the drug in pill form, and subjects in the other group
received a placebo (a pill that did not contain the drug). Each subject took two pills each
day for six weeks. Each of the subjects used an inhaler to help relieve the effects of the
breathing condition. At the beginning of the study each subject was examined and
classified as either a heavy or a moderate inhaler user. At the end of the six week study
period each subject was re-examined and classified as either a heavy or moderate inhaler
user. The placebo group was included in this study because changes in weather and other
environmental conditions during the study period could have an impact on changes in
inhaler use. Show how you would determine if the drug was effective in reducing inhaler
use.
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4. In a study of a chemotherapy treatment for leukemia, the result for each of 170 treated
patients was coded as
1 for remission
0 for no remission
The covariates are
X1 = percentage of cells undergoing DNA synthesis in the presence of chemotherapy
X 2 = highest recorded patient temperature (°F) prior to the start of the chemotherapy
treatment
Data for 12 of the patients are given in the following table.
Patient
1
2
3
4
5
6
7
8
9
10
11
12
Result
X1
X2
0
1
0
0
0
1
0
0
1
0
1
1
0.11
0.19
0.05
0.10
0.06
0.11
0.04
0.06
0.10
0.16
0.17
0.09
99.0
101.4
102.0
100.4
99.0
98.6
101.0
102.0
100.2
98.8
98.6
98.6
The researchers fit the following model to these data:
 π 
log i  = β 0 + β1 X1i + β 2 X 2i
 1 − πi 
where π i is the conditional probability of remission given the values of (X1i, X2i). They
assumed that each leukemia patient responded independently of any other patient.
(a) Write out a formula for the likelihood function they maximized to obtain maximum
likelihood estimates for the parameters in this model.
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(b) The values of maximum likelihood estimates for the parameters and the estimated
covariance matrix are as follows:
βˆ 0  45.4
 
β = βˆ1  = 33.0
~
 ˆ  - 0.5

β 2  
∧
 2193.92
V ∧ =  294.89
β
~
 - 22.43
∧
294.89
186.29
- 3.18
- 22.43 
- 3.18 
0.23 
Use these results to obtain the maximum likelihood estimate of the value of X1
needed to achieve a 0.90 probability of remission when X2 = 100.0°F.
(c) Show how to compute a standard error for the estimate in part (b).
(d) Describe the steps you would take to assess the fit of the proposed model, and, if
necessary, find a better model.
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5. In a study of the relationships between car size and severity of accident injuries, a simple
random sample of n= 1200 accident reports was selected, without replacement, from a file
of over 2 million automobile accident reports for maintained by the State of California. The
sampled records were classified into a 4×3×2×2 contingency table with respect to the levels
of the following four factors:
Accident type (T):
(i=1)
(i=2)
(i=3)
(i=4)
Collision with another vehicle, no rollover
Collision with an object, no rollover
Rollover with no collision
Rollover involving a collision
Accident Severity
for the driver (S):
(j=1) Not severe
(j=2) Moderately severe
(j=3) Severe
Car Size (C):
(k=1) Compact (smaller cars)
(k=2) Standard (larger cars)
Ejection of driver
from the vehicle (E):
( l =1) No
( l =2) Yes
In answering the following questions, let π ijkl denote the probability that a randomly
selected accident record is classified as the i-th accident type, the j-th level of severity for the
driver, the k-th car size, and the l -th ejection category. Let mijkl and Yijkl denote the
corresponding expected and observed counts, respectively.
A. Write out the formula for the largest (least parsimonious) log-linear model that satisfies
the following null hypothesis:
H 0 : Given the type of accident (T), the accident severity for the
driver is conditionally independent of both car size (C)
and whether or not the driver is ejected (E).
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B.
Consider the log-linear model
TC
log( m ijkl ) = λ + λTi + λSj + λCk + λEl + λTS
ij + λ ik
SC
SE
TSC
+ λTE
il + λ jk + λ jl + λ ijk
E
where λT1 = λS1 = λC
1 = λ1 = 0 and any interaction parameter is constrained to be
zero when any factor involved in the interaction is at its lowest level. Maximum
likelihood estimates of the parameters and their standard errors are shown in the table
on the next page. Use this information to answer the following questions. If you do
not have enough information to complete an answer, describe the formula or method
you would use and the additional information that you would need to complete the
answer.
(i)
The value of the Pearson chi-square test statistic for testing the fit of this model
against the general alternative is 20.65. All of the estimated expected counts are
larger then 5.8. What are the degrees of freedom for the chi-square
approximation to the null distribution of this test statistic?
(ii)
Compute the maximum likelihood estimate for the odds ratio corresponding to
the odds that a driver who is ejected from the vehicle is severely injured divided by
the odds that a driver who is not ejected from the vehicle is severely injured.
(iii)
Show how to construct a 95% confidence interval for the odds ratio in part (ii).
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Estimate
-------3.666
Standard
Error
--------0.029
Estimate /
(Std. Error)
---------126.86
-0.324
0.040
-8.09
λT3
-0.186
0.024
-7.65
λT4
-0.033
0.016
-2.00
λS2
0.078
0.032
2.4
-0.218
0.023
-9.79
0.742
0.026
28.61
-0.673
0.023
-29.12
0.073
0.030
2.45
0.230
0.022
10.49
0.177
0.017
10.21
-0.014
0.029
-0.49
0.003
0.020
0.16
0.075
0.012
6.32
0.007
0.033
0.22
-0.166
0.023
-7.35
-0.015
0.015
-1.00
0.062
0.031
2.01
0.253
0.021
12.52
0.094
0.013
7.29
0.057
0.026
2.18
0.051
0.021
2.42
λSE
22
0.179
0.027
6.60
λSE
32
0.206
0.017
12.29
λTSC
222
-0.076
0.030
-2.58
λTSC
322
0.041
0.022
1.93
λTSC
422
-0.024
0.017
-1.40
0.013
0.028
0.47
0.016
0.019
0.86
0.012
0.012
1.07
λ
λT2
λS3
λC2
λE2
λTS
22
TS
λ 32
λTS
42
TS
λ 23
λTS
33
TS
λ 43
λTC
22
TC
λ 32
λTC
42
TE
λ 22
λTE
32
TE
λ 42
λSC
22
SC
λ 32
λTSC
232
TSC
λ 332
λTSC
432
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(iv)
What do the parameter estimates for this model imply about associations
between severity of driver injuries and car size and accident type?
C. Show how you could express the model in part B as a logistic regression model.
EXAM SCORE_____________
COURSE GRADE __________