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Homework 6
Stat 158, Fall 2023
Due November 9, 2023 by 11:59 PM PDT on Gradescope
Please remember to keep homework assignments structured and legible, to use a new
page for each problem, and to upload a .pdf with separate pages instead of one long
page. Include all R code written in a code appendix at the end of the homework, and
link pages with relevant code to each question when you upload to Gradescope.
1. (30 points) The model for a split-plot design is given as
yijk = µ + αi + bj(i) + δk + (αδ)ik + ejk(i)
where αi is the effect due to the treatment applied at the block level (e.g.
diabetic or not), bj(i) is the random effect for the block j in the ith group (e.g.
the jth dog in group i), and δk is the effect due to the treatment applied within
each block (e.g. method of infusion/inject). We assume that
2
2
bj(i) ∼ N (0, σblock
), ejk(i) ∼ N (0, σerr
)
and that all of the bj(i) and ejk(i) terms are independent of each other.
The mean sum-of-squares of the blocking factor is given by
I
I
J
J
K
K
1 XXX
1 XXX 2
b̂j(i) =
M Sblock =
(ȳij· − ȳi·· )2 .
IJ − I i=1 j=1 k=1
IJ − I i=1 j=1 k=1
Prove the following result given in the handout:
2
2
+ σerr
.
E(M Sblock ) = Kσblock
2. Analyze the data from the bridge experiment from class (available on bCourses
as file bridge strength.txt).
(a) (5 points) Visualize the data and describe what you see.
(b) (20 points) Perform randomization inference, using F-statistics extracted
from a random effects fit using the nlme package. Include crumpling,
orientation, and their interaction as fixed effects. Report and interpret the
results of your tests.
1
3. We are planning an experiment on the quality of video tape and have purchased
24 tapes, four tapes from each of six types1 . The six types of tape were 1) brand
A high cost, 2) brand A low cost, 3) brand B high cost, 4) brand B low cost, 5)
brand C high cost, 6) brand D high cost. Each tape will be recorded with a series
of standard test patterns, replayed 10 times, and then replayed an eleventh time
into a device that measures the distortion on the tape. The distortion measure
is the response, and the tapes will be recorded and replayed in random order.
Previous similar tests had an error variance of about .25.
(a) (10 points) What is the power when testing at the .01 level if the high cost
tapes have an average one unit different from the low cost tapes?
(b) (15 points) How many tapes per brand would we need to obtain a 95%
confidence interval of no wider than 2 for the mean difference between
tapes of brand A and tapes of brand B?
4. We decide to alter the experiment in the previous question in two ways. First,
in addition to comparing the six types we will measure the effect of storage
temperature on video quality by selecting half the tapes of each type at random
to be stored at a higher temperature than the others. In addition, to speed up
the running of the experiment, we will use two different distortion-measuring
devices concurrently. Since we expect that there may be some variation in measurement across devices, we will assign half the tapes to each device at random,
ensuring that the number of tapes in each brand/temperature combination is
balanced across devices.
(a) (5 points) Identify the design of the experiment, including both the type
and the number of treatment factors at each level. Justify your answer.
(b) (15 points) Repeat the analysis from question 3(a) for the new design.
Comment on any differences between the results and give intuition for
why they do or do not differ.
1
Adapted from Oehlert problem 7.5
2
Run Sheet # Crumpled Orientation Coins at Collaps
1 1 N L 18
2 1 N S 35
3 2 Y L 40
4 2 Y S 42
5 3 Y S 51
6 3 Y L 42
7 4 N S 24
8 4 N L 18
9 5 N S 20
10 5 N L 13
11 6 Y S 33
12 6 Y L 39
13 7 Y L 28
14 7 Y S 66
15 8 N S 33
16 8 N L 13

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