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Homework 5
STAT 8320
Due May 4, 2006
Problem 1: An animal study is designed to address the effects of a set of different drugs
on the amount of immuno-response to an allergen. It is believed that the age of the animal
may effect this response, and to address this, eight animals from each of three ages (young,
adolescent, adult) are used in the study. Each animal will receive the four drugs of interest,
and measures of their immuno-response will be taken. Hypothetical data from a study of
this sort are included in the attached file. Address the following questions:
a. Describe an appropriate model for this data. First, define the model for Y
ijk
and describe
which terms in the model are fixed and which are random. Also indicate how we could write
this model in the mixed model form.
There are multiple answers for this part. One possible correct answer is
Y
ijk = μ + α i
ijk
where αi is the fixed effect of age i, βj is the fixed effect of drug j, (αβ)ij is the fixed interaction
between age i and drug j, and ≤ ijk is the random error, with ≤ ∼ N(0, σ
2 Σ). Here, observations on
the same animal (those which have the same value k) are correlated, and Σ is block diagonal.
We could write this model in mixed model form according to
Y = Xβ + ≤,
where
Y =
Y
1 , 1 , 1
Y
1 , 2 , 1
Y
3 , 4 , 24
1 , 1 , 1
1 , 2 , 1
3 , 4 , 24
, β =
μ
α 1
α 3
β 1
β 4
(αβ) 11
(αβ) 34
, and X =
b. What assumptions do you think are reasonable about the structure of the covariance
within an animal? To test your intuition, run your model in SAS for CS, AR(1), and un-
structured covariance matrices. Which model(s) do the fit statistics suggest are best?
It seems reasonable to think that the correlations between the observations on a given animal do
not depend upon the order of the drugs. Thus, either a compound symmetric or an unstructured
covariance seem appropriate, depending upon whether or not the different drugs have the same
variance.
See the attached output for the exact numbers. Based upon both AIC and BIC, it appears
that the compound symmetric structure provides a good fit with a sufficiently small number of
parameters.
c. For your favorite model, interpret the output in SAS. Be sure to mention which of your
fixed effects are significant, and describe which drug(s) are best (smallest response).
Based upon the compound symmetric model, there is a significant interaction between the
age and the drug. Due to this interaction, it is inappropriate to test for main effects. Also, this
interaction means that we need to look at the effects of drug for each age group separately.
where α i is the fixed effect of battery type i, β j is the fixed effect of time j, (αβ) ij is the fixed
interaction between type i and time j, and ≤ ijk is the random error, with ≤ ∼ N(0, σ
2 Σ). Here,
observations on the same battery (those which have the same value k) are correlated, and Σ is
block diagonal.
We could write this model in mixed model form according to
Y = Xβ + ≤,
where
Y =
Y 1 , 1 , 1
Y
1 , 2 , 1
Y
3 , 4 , 30
1 , 2 , 1
3 , 4 , 30
, β =
μ
α 1
α 3
β 1
β 4
(αβ) 11
(αβ) 34
, and X =
b. What assumptions do you think are reasonable about the structure of the covariance
within a battery? To test your intuition, run your model in SAS for CS, AR(1), and un-
structured covariance matrices. Which model(s) do the fit statistics suggest are best?
It seems reasonable to think that the correlations between the observations on a particular bat-
tery should depend upon the time between the measurements. Thus, some form of an autoregressive
model would seem appropriate, although if the variance is changing over time, and unstructured
matrix might be needed.
See the attached output for the exact numbers. Based upon the AIC, it appears that the
unstructured covariance matrix is best, although none of the numbers are drastically different.
Based upon the BIC, it appears that the autoregressive model provides a good fit with a sufficiently
small number of parameters.
Based upon this information and my intuition, I am going to select the autoregressive model
for further study (you could have chosen the unstructured).
c. For your favorite model, interpret the output in SAS. Be sure to mention which of your
fixed effects are significant, and describe which battery type you feel would be the best pur-
chase.
Based upon the autoregressive model, there is a significant interaction between the type of
battery and the time. Due to this interaction, it is inappropriate to test for main effects. Also, this
interaction means that we need to look at the effects of time on each type of battery separately.
For “A” batteries, the predicted responses for the four times are 150.77, 125.41, 100.42, and
For “B” batteries, the predicted responses for the four times are 131.49, 118.13, 116.35, and
For “C” batteries, the predicted responses for the four times are 143.13, 128.65, 117.48, and
Depending upon the photographer’s goals, your answer may be different. However, assuming
that the photographer is wanting to use these batteries for a long time, I would suggest battery
type “B”. It starts out with about 20 less than battery A, but has close to 40 more at the end of
the time considered. Similarly, it has about 10 less that battery C at the start, but has 18 more
at the end of the time considered. I feel that the photographer is out to get as much use from a
battery as possible, and type “B” seems to hold up the best.
