STAT 510
Homework 10
Due Date: 11:00 A.M., Wednesday, April 13
1. Scientists studied 700 adults who typically smoked at least 15 cigarettes a day but wished to quit
smoking. The subjects were randomly divided into two treatment groups using a balanced and completely randomized design. In treatment group 1, subjects were allowed to smoke as usual for two
weeks and then asked to suddenly quit smoking altogether for as long as possible. In treatment group
2, subjects gradually reduced cigarette smoking for two weeks according to a specific smoking reduction plan and then quit smoking altogether for as long as possible. Four weeks after the quit date, 172
of the 350 people assigned to treatment group 1 were still not smoking, and 137 of the 350 people
assigned to treatment group 2 were still not smoking. All other subjects had resumed smoking within
four weeks of the quite date. Is one treatment more effective than the other at enabling people to quit
smoking for at least four weeks? Use what you learned from Homework 9 Problem 3 to construct one
confidence interval that will help you answer this question.
2. An experiment was designed to compare the effect of three drugs (A, B, and C) on the heart rate of
women. Five women were randomly assigned to each drug. The heart rate (in beats per minute) of
each woman was measured at 0, 5, 10, and 15 minutes after the drug was administered. The data are
provided in the file
http://www.public.iastate.edu/∼dnett/S510/HeartRate.txt
Let yijk denote the heart rate at the kth time point for the jth woman treated with the ith drug. Suppose
yijk = µik + wij + eijk ,
where µik is an unknown constant for each combination of i = 1, 2, 3 and j = 1, 2, 3, 4, 5;
2 ) for all i and j; e
2
wij ∼ N (0, σw
ijk ∼ N (0, σe ) for all i, j, and k; and all random effects are
independent.
(a) Using Kronecker product notation, provide an expression for the variance of
y = [y111 , y112 , y113 , y114 , y121 , y122 , y123 , y124 , . . . , y351 , y352 , y353 , y354 ]0 .
(b) Test the null hypothesis of no drug main effects. Compute a test statistic, state its degrees of
freedom, find a p-value, and provide a brief conclusion.
(c) Determine the distribution of ȳij· based on the given model for y.
(d) Compute ȳij· for all i and j. Using these averages as data, test the null hypothesis of no drug
main effects. Compute a test statistic, state its degrees of freedom, find a p-value, and provide a
brief conclusion.
(e) Compute a 95% confidence interval for the mean heart rate 15 minutes after treatment with drug
B minus the mean heart rate 10 minutes after treatment with drug B.
(f) Compute a 95% confidence interval for the mean heart rate 15 minutes after treatment with drug
B minus the mean heart rate 15 minutes after treatment with drug C.
3. Again consider the data on heart rate from problem 2. Now suppose
yijk = µik + ijk ,
where µik is an unknown constant for each combination of i = 1, 2, 3 and k = 1, 2, 3, 4, 5 and ijk is
a normally distributed error term with mean 0 for all i = 1, 2, 3, j = 1, 2, 3, 4, 5, and k = 1, 2, 3, 4.
For all i = 1, 2, 3 and j = 1, 2, 3, 4, 5, let
ij = (ij1 , ij2 , ij3 , ij4 )0 .
Suppose all the ij vectors are mutually independent, and let W be the variance-covariance matrix of
ij , which is assumed to be the same for all i = 1, 2, 3 and j = 1, 2, 3, 4, 5.
(a) Find the REML estimate of W under the assumption that W is a positive definite, compound
symmetric matrix.
(b) Find AIC and BIC for the case where W is a positive definite, compound symmetric matrix.
(c) Find the REML estimate of W under the assumption that W is a positive definite matrix with
constant variance and an AR(1) correlation structure.
(d) Find AIC and BIC for the case where W is a a positive definite matrix with constant variance
and an AR(1) correlation structure.
(e) Find the REML estimate of W under the assumption that W is a positive definite, symmetric
matrix.
(f) Find AIC and BIC for the case where W is a positive definite, symmetric matrix.
(g) Which of the three structures for W is preferred for this dataset? Explain briefly.
(h) Using the preferred structure for W , compute a 95% confidence interval for the mean heart rate
15 minutes after treatment with drug B minus the mean heart rate 15 minutes after treatment
with drug C.
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