Statistics 512
Final Exam
Spring 2004
Statistics 512
Final Exam
May 3, 2004
10:10-12:00
The following rules apply.
1. You may bring with 3 sheets of paper, double-sided with any
information you need.
2. You may use a calculator.
3. You may not collaborate or copy.
4. Failure to comply with item 3 could lead to reduction in your grade, or
disciplinary action.
I have read the rules above and agree to comply with them.
Signature ________________________________________________
Name (printed) ___________________________________________
problem your score
1
1
total
points
25
2
17
3
8
4
10
Total
60
Statistics 512
Final Exam
Spring 2004
All the problems refer to the following situation:
An investigator is interested in determining the effects of dietary protein P on blood
pressure in mice. 4 levels of protein will be used, 10%, 15%, 20% and 25% protein.
The investigator will use 4 different genotypes G of mouse.
The mice are maintained in the lab on a normal 17.5% protein diet for 3 weeks.
After 3 weeks, blood pressure is measured on each mouse. The mice are then
switched to the controlled protein diet for one week. At the end of the week, the
change in blood pressure is recorded. The response variable is the change in blood
pressure. 4 mice of each genotype are given each diet, so there are 4x4x4=64 mice
in all.
The model is:
Yijk =  + Pi + Gj + (PG)ij+ ijk
i = 1 ... 4,
j=1...4,
k=1...4
Yijk is the change in blood pressure for the kth mouse on protein diet i of genotype j
 is the overall mean change in blood pressure
Pi is the protein level main effect
Gj is the genotype effect
(PG)ijis the protein level by genotype interaction
ijk is random error
a. Write down the constraints and distribution assumptions for this model when both
protein level and genotype are assumed to be fixed effects.
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Statistics 512
Final Exam
Spring 2004
b. Write down the ANOVA table for the experiment and model in part a, noting the F-test
denominators for each of the treatment effects (main effects and interaction). The
numerators and denominators will be in the form "MSXX" where you replace XX by the
appropriate term.
Anova Table
Source
d.f.
F-test Numerator
F-test denominator
protein
genotype
protein*genotype
error
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c. For the model in part a, what are the following in terms of the notation you have used
for your model and ANOVA?
Expected(Y113) =
Var (Y113)
estimated mean for protein level 1, genotype 1
Cov(Y113, Y114)
Cov(Y113, Y124)
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Statistics 512
Final Exam
Spring 2004
d. Write down the constraints and distribution assumptions for this model when protein
level is a fixed effect and genotype is assumed to be a random effect.
e. Write down the ANOVA table for the experiment and model in part d, noting the F-test
denominators for each of the treatment effects (main effects and interaction). The
numerators and denominators will be in the form "MSXX" where you replace XX by the
appropriate term.
Anova Table
Source
d.f.
F-test Numerator
F-test denominator
protein
genotype
protein*genotype
error
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4
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Statistics 512
Final Exam
Spring 2004
f. For the model in part d, what are the following in terms of the notation you have used
for your model and ANOVA?
Expected(Y113) =
Var (Y113)
estimated response for protein level 1, genotype 1
Cov(Y113, Y114)
Cov(Y113, Y124)
g. The laboratory where the investigator works maintains over 500 different genotypes of
mouse. Give an example of a situation in which the investigator should treat G as
a fixed effect
a random effect
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Statistics 512
Final Exam
h. Explain how to randomize the experiment so that it is balanced.
6
Spring 2004
Statistics 512
Final Exam
Spring 2004
2. The description of this experiment is on page 1. The investigator decided to treat
protein and genotype as fixed effects. So the experiment was analyzed as a completely
randomized 2-factor design. However, 3 mice were found to have an infection that may
have affected their blood pressure so these data were not used, causing the analysis to be
unbalanced.
Since there are no random effects in the model, the investigator used a "Type 3" analysis
using SAS PROC GLM. Several relevant outputs are in the "Computing Handout"
a) The investigator started by fitting the full model 2-way ANOVA (Computing Handout
p. 1). Is there a statistically significant genotype by protein interaction? Justify your
answer with a statistical test.
Value of test statistic:
p-value:
conclusion:
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Statistics 512
Final Exam
Spring 2004
b) The investigator expected the effect of protein level to be quadratic. Using the
appropriate outputs, test whether or not there is a statistically significant lack of fit to the
quadratic model at =.10.
SSLOF
d.f. LOF
F-test denominator:
F*
critical value (at =.10)
Conclusion:
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Statistics 512
Final Exam
Spring 2004
c. The investigator hopes to conclude that the response is quadratic in protein level.
Explain how to proceed after doing the lack of fit test above. You may assume that all
testing will be done at =.10.
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Statistics 512
Final Exam
Spring 2004
d. Another investigator decided to use these data as a pilot study for his own
investigation. Because he was interested in other genotypes, he decided to fit the model
with protein level fixed and genotype random. Because the study is unbalanced, he used
PROC MIXED Method=REML for the analysis. He is interested in whether or not there
is evidence of protein by gene interaction and a protein main effect. Some of the output
is on p. 6 of the computing handout.
i Based on these results, is there a statistically significant protein by gene interaction?
Justify your response with a statistical test.
Test statistic:
p-value:
Conclusion
ii) Based on these results, is there a statistically significant protein main effect? Justify
your response with a statistical test.
Test statistic:
p-value:
Conclusion
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Statistics 512
Final Exam
Spring 2004
3. The investigator discovered some problems with the experiment described above, and
decides to redo it. Since blood pressure in mice is associated with weight, the
investigator plans to weigh the mice when they are selected for the experiment (but
before treatment is assigned), and use this information to improve the power of testing
the protein and genotype effects. You may assume that there is no interaction between
initial weight and treatments, and that the different genotypes do not differ in weight
distribution.
a) Write down the model with weight as a covariate and define any additional terms not
defined in problem 1. (You may assume that protein level, genotype and weight are fixed
effects.)
b) How does using weight as a covariate improve the power of testing the effects of
interest? (Explain briefly.)
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Statistics 512
Final Exam
Spring 2004
c) The experimenter wants to block on weight. Briefly explain how to do this, including
the randomization procedure.
d). The experimenter feels that blocks based on initial weight should be a random effect
since the weights were not selected in advance and she wants to make inference to mice
with different initial weights than those in the sample. Should these effects be considered
fixed or random? Briefly justify your answer.
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Statistics 512
Final Exam
Spring 2004
4. The investigator discovers that the mice have very uniform weights, and that weight is
therefore unlikely to be a good covariate or blocking factor. However, she discovers that
there are family similarities in response. She therefore selects 4 litters from each
genotype, selects 4 young mice from each litter and assigns one at random to each protein
level. This is a split plot design experiment.
a) Fill in the following table:
Whole plot
experimental unit
treatment(s)
Subplot
experimental unit
treatment(s)
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Statistics 512
Final Exam
Spring 2004
b. As a pilot experiment, the investigator does the experiment using the least expensive of
the genotypes only (4 litters of 4 mice = 16 mice).
This is a randomized complete block design with litter as the block. Should the blocks be
considered fixed or random? Briefly justify your response.
c. Write down the ANOVA table for the pilot experiment in part b, noting the F-test
denominators for the protein and litter effects. The numerators and denominators will be
in the form "MSXX" where you replace XX by the appropriate term.
Anova Table
Source
d.f.
F-test Numerator
F-test denominator
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protein
litter
protein*litter
error
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