Statistics 512
Final Exam
Spring 2006
Statistics 512
Final Exam
May 3, 2006
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. You may not use any device, such as a cell-phone or PDA that allows you to
access the internet or other outside information.
5. Failure to comply with items 3 or 4 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
2
3
Total
1
total
points
Statistics 512
Final Exam
Spring 2006
1. In a test of a new drug for blood pressure regulation (ABC), 5 randomly selected rabbits were used
as subjects.
PBG, a drug known to raise blood pressure in rabbits, was used to create a blood pressure increase.
Immediately before the PBG was administered, initial blood pressure was recorded. PBG was then
given at one of the 6 doses: 6.25, 12.50, 25, 50, 100, 200 mg. Fifteen minutes later, either ABC
(ABC=MDL) or a placebo* (ABC=placebo) was administered. After another 15 minutes, the blood
pressure of the rabbit was recorded again. The response is the logarithm of the difference in blood
pressure. The main question of interest was whether or not ABC reduced the change in blood
pressure, and if so, whether the amount of blood pressure reduction depends on the dose of PBG. (In
Midterm 2 was already demonstrated that PBG induces a statistically significant increase in blood
pressure in this population of rabbits.)
Since the effects of PBG and ABC are thought to be very short-term, each rabbit was measured at each
combination of PBG and ABC.
rabbit i:
initial BP PBG dose → ABC dose → final BP →→→→ initial BP PBG dose.....
→
→
time →→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→
*placebo – A treatment that is administered in the same way as the drug under test, but which has no
known biological effect. The placebo is given to control for any effects of administering the drug,
such as stress due to handling the animal or (in humans), psychological effects of treatment.
a) (6) In Midterm 2, we determined that rabbit should be a random effect and that dose of PBG
should be a fixed effect. Write a linear model for this experiment, defining all terms and giving
all constraints and distribution assumptions.
Yijk     i   j   k  ( ) ij  ( ) ik  (  ) jk   ijk , i  1,2,3,4,5 ; j  1,  ,6 ; k  1,2 .
.5 Yijk is the response of measurement of i th rabbit given j th PBC dose and k th ABC dose ,
.5  is the overall mean response,
1  i is the effect of the i th rabbit,  i are iid N (0,  2 )
.5  j is the effect of the j th PBC dose,
6

j 1
.5  k is the effect of the k th ABC dose,
j
 0,
k
0,
2

k 1
2
.5 ( ) ij is the interaction of the i th rabbit and the j th PBC dose, ( ) ij are iid N (0,  
)
.5 ( ) ik is the interaction of the i th rabbit and the k th ABC dose, ( ) ik are iid N (0,  2 )
1 (  ) jk is the interaction of the j th PBC dose and the k th ABC dose,
2
 ( )
k 1
.5  ijk is the random error,  ij are iid N (0,  ) .
2
.5  i , ( ) ij , ( ) ik ,  ijk are independent.
2
2
jk
  (  ) jk 0
j 1
Statistics 512
Final Exam
Spring 2006
No fixed effect interactions 4/6
No constraints or distribution assumptions 3/6
3
Statistics 512
Final Exam
Spring 2006
b) (2) The investigators felt that this experiment should be handled as a repeated measures
design. What does "repeated measures" mean? Are they correct in considering this to be a
repeated measures design?
A study has repeated measures if more than one observation is taken on a sampling unit.
Yes, this is a repeated measures design.
c) (3)The investigators propose to run the experiment as a randomized complete block design.
What are the blocks and experimental units, and how are the treatments assigned?
1 Each rabbit is a block.
1The experiment unit is the rabbit in a given time period.
1There are 6*2=12 combinations of treatment. Assign these treatments to each rabbit in a random
order.
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Statistics 512
Final Exam
Spring 2006
d) (8) Fill in the following table for the randomized complete block design in part c adding any
necessary rows such as interactions and error terms. (Fill as many lines as needed in the table
below – you will not need them all. )
Usual ANOVA
effect
d.f. numerator F-test denominator
rabbit
PBG
ABC
PBG*ABC
Error
Total
4
5
1
5
44
59
3
MSE
MSE
MSE
MSE
3
d.f.
denominator
44
44
44
44
2
ANOVA with block by factor interactions
effect
d.f. numerator F-test denominator
rabbit
PBG
ABC
PBG*ABC
PBG*Rabbit
ABC*Rabbit
Error
Total
4
5
1
5
20
4
20
59
3
MSE
MS(R*P)
MS(R*A)
MSE
MSE
MSE
3
d.f.
denominator
20
20
4
20
20
20
2
5
Statistics 512
Final Exam
Spring 2006
e) (2) The Type 3 tests of fixed effects are given below. Does the effect of ABC depend on the
level of PBG? Briefly justify your answer.
Type 3 Analysis of Variance
F value
18.22
86.88
6.76
Source
ABC
PBG
PBG*ABC
Pr > F
0.0001
0.0001
0.0001
Yes, because the interaction of PBG and ABC is significant.
The treatment means are given below:
type
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
dose*ABC
ABC
level
MDL
Placebo
MDL
Placebo
MDL
Placebo
MDL
Placebo
MDL
Placebo
MDL
Placebo
PBG
level
6.25
6.25
12.5
12.5
25
25
50
50
100
100
200
200
LS mean
0.4206
-0.07044
0.4349
0.7274
0.7167
1.6377
1.2583
2.8198
2.7560
3.2980
3.2395
3.2796
SE of
mean
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
0.2012
tvalue
2.09
-0.35
2.16
3.61
3.56
8.14
6.25
14.01
13.70
16.39
16.10
16.30
p-value
Ho: mean=0
0.0422
0.7279
0.0360
0.0008
0.0009
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
f) (6) Is there a statistically significant effect of ABC (compared to the placebo) at the lowest dose
of PBG? Do a formal test at =0.5 and give a 95% confidence interval for the estimated effect.
(You can assume that t.975 = 2.0 or F.95=4.2)
H 0 :  6.25, MDL   6.25, placebo
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Statistics 512
Final Exam
Spring 2006
H a :  6.25, MDL   6.25, placebo
t* 
1
ˆ
 6.25, MDL  ˆ 6.25, placebo
1

1
0.4206  0.07044
 1.7257  t.975  2.0
1 1
0.2012 * 2
MSE (  )
5 5
1
So, there is no significant effect of ABC at the lowest dose of PBG. 1
Confidence interval: 0.49104  2.0*0.2845 = (-0.078, 1.06) 1
Note that SE= MSE / 5 .
g) (5) Does the effect of ABC at PBG=200 differ significantly from its effect at PBG=100? Do a
formal test at =0.5 and give a 95% confidence interval for the estimated effect. (You can assume
that t.975 = 2.0 or F.95=4.2)
H 0 :  200, MDL   200, placebo  100, MDL  100, placebo
H A :  200, MDL   200, placebo  100, MDL  100, placebo
t* 
1
ˆ 200, MDL  ˆ 200, placebo  ( ˆ 100, MDL  ˆ 100, placebo )
1 1
MSE (4   )
4 5
1
1

(3.2395  3.2796) / 2  (2.756  3.298) / 2
 1.2473
0.2012
|t*|  t.975  2.0
So, there is no significant difference. 1
Confidence interval: -0.251  2.0*0.2012 = (-0.653, 0.151) 1
Note: What is the effect of going on a diet? Surely this is your change in weight, not your final
weight. A treatment effect is generally defined as the deviation from the grand mean due to the
treatment.
7
Statistics 512
Final Exam
Spring 2006
h)(3) The investigators worried that there might be a strong carry-over effect from ABC. What
is a carry-over effect? How could the investigators determine if there is a carry-over effect, and
what should they do if there is a carry-over effect?
Carry-Over effects mean that the effects of a treatment continue into the next time period, even though
the treatment has been withdrawn, and another treatment applied.
To test the presence of carry-over effects, I will design an experiment, giving each experimental unit
different wash-out time, then treat the time as a factor, and test if its effect is significant. If it is, there is
a carry-over effect. If there is carry-over effect, the best thing to do, if possible, is to increase the
wash-out period to avoid it. If that is not possible, it may be possible to use a covariate (e.g. the level
of ABC in the bloodstream when the next dose is administered) or to use a design in which this carryover effect is less important (e.g. no repeated measures, or the split plot design below.)
The investigators decided that they should use different rabbits for the ABC treatment and the placebo,
in case there is a carry-over effect. To ensure that the carry-over effect would also apply in the first
time period, an extra time period was added as follows:
rabbit i:
initial BP saline solution → ABC dose → final BP →→→→ initial BP PBG dose.....
→
→
time →→→→→→→→→→→→→→→→→→→→→→→ | →→→→→→→
rabbit acclimatization time
experiment starts
None of the measurements from the rabbit acclimatization time are used. The remainder of the
experiment proceeds as describes previously. However, now each rabbit in the study is measured only
6 times.
i) (3) The design described above is a split plot design. Explain how to do the randomization for
this design. You may add rabbits if necessary to keep the design balanced.
We need an even number of rabbits – say 2n. First, we randomly divide 2n rabbits into 2 groups with n
in each. MDL and placebo should be randomly assigned to a group. Within each group, we assign PBG
dose to each rabbit in a random order.
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Statistics 512
Final Exam
Spring 2006
j) (6) For the design you have described in part (i) fill out the table below.
effect
PBG
whole
plot/subplot
Subplot
ABC
ABC*PBG
Whole Plot
Subplot
experimental unit F-test
denominator
A rabbit at one
MSE
time period
rabbit
MS(rabbit(ABC))
A rabbit at one
MSE
time period
2
2
2
k) (3) The investigators decided that the wash-out period between treatments should be 1 day.
To minimize day effects, they decided to use a Latin square design for the PBG treatment. (They
are still going to use a split plot design.) Explain how to do the randomization for this design.
You may add rabbits if necessary to keep the design balanced.
We will use 12 rabbits – 6 for each level of ABC.
Within each group of 6 rabbits, we do a Latin square design.
1. Pick a standard 6*6 square at random.
2. Assign 6 PBC dose to letters at random.
3. Rearrange the rows (periods) at random.
4. Rearrange the columns (rabbits) at random.
l) (3) The advisory board for the study suggested that it would be better to give the PBG
treatments in fixed order from the lowest dose on day 1 to the highest dose on day 6 in case there
is a carry-over effect of PBG. The lab technician who handles the animals feels that this is a bad
idea, because the rabbits become more stressed (which raises blood pressure) as the experiment
proceeds. What are the pros and cons of giving the PBG doses in the same order to every
rabbit? What factors might be confounded using the design of the advisory board?
Pros: This design can reduce carry-over effect from the PBG treatments, since any PBG remaining in
the animal simply adds to the higher dose it receives in the next period.
Cons: If there are any period effects, these will be confounded with the PBG effect.
Note: If there are carry-over effects for both drugs, this is probably the best design. It is easier to
avoid period effects in a lab setting, than to account for carry-over effects.
9