Description
1. The file BFHS.dat contains data from the British Family Heart Study.1
In each of thirteen
different towns, two general medical practices were selected. In each pair of practices from
the same town, one was chosen at random to receive an Intervention: a subset of males
aged 40–59, randomly chosen from the practice’s patient registry, received an initial
screening and advice and support for leading a healthier lifestyle. The other practice
(ExternalComparison) did not receive any such intervention. After one year, the males in
the intervention groups and all males aged 40–59 in the comparison groups were screened,
and their serum cholesterol levels (mmol/l) were recorded. The data set consists of, for each
town, the average cholesterol level of the subset of males in the intervention practice, and
the average cholesterol level of the males in the comparison practice.
(a) [2 pts] Create an appropriate R data set for this data. Display a summary of it.
(b) [2 pts] Perform an appropriate t-test for whether there is any mean difference in the
cholesterol levels of the intervention and comparison groups.2 Display a summary of
the results. What do you conclude?
(c) [2 pts] Perform an approximate randomization test (based on simulating the t-statistic
under re-randomization) for whether there is any mean difference. What is your
approximate p-value? (Show the R code you used to produce it.) Does your conclusion
change?
(d) [2 pts] Create a histogram of the randomization distribution of the t-statistic, based on
your simulation of the previous part. (Show the R code you used to produce it.)
2. The file Barley1928.csv contains data from an experiment to examine the effect of five
different soil treatments (1: no nitrogen, 2: cyanamide, single dressing, 3: sulphate of
ammonia, single dressing, 4: cyanamide, double dressing, 5: sulphate of ammonia, double
dressing) on the yield of straw (in quarter pounds). The experiment was conducted in a
randomized complete block design (where blocks are designated by the variable Block).
(Note: The R function read.csv might be useful for reading this data into R.)
(a) [2 pts] Examine the data. How many blocks are there? How many experimental units
in total?
(b) [2 pts] Fit the linear model appropriate for analysis of this experiment. Display a
summary of the results. (Remember: Make sure the treatment is a factor variable!)
(c) [2 pts] Produce an ANOVA table for your model. Are treatment effects statistically
significant? (Remember to state the p-value.)
1From Thompson, Simon G., Pyke, Stephen D., and Hardy, Rebecca J. (1997) “The design and analysis of paired
cluster randomized trials: An application of meta-analysis techniques,” Statistics in Medicine, 16, 2063–2079.
2Even though a one-sided test might be more appropriate, you should do a two-sided test.
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(d) [2 pts] Produce Tukey simultaneous 95% confidence intervals for all mean differences
between pairs of treatments.
(e) [2 pts] According to your Tukey intervals, which pairs of treatments have significantly
different means (after adjusting for multiple comparisons)?
3. The file Spelling1941.csv contains data from an experiment to compare four different
ways of testing the spelling abilities of children: Multiple Choice (MC), Second Dictation
(SD), Wrongly Spelled (WS), and Skeleton Word (SW). In the experiment, there was an
original dictation test of the words to be spelled, followed by a test using one of the four
ways (MC, SD, WS, or SW). The response (Number) was “the number of words wrong in
the original dictation but correct in the later test.” The experiment was conducted in a
Latin square design, with different lists of words as one blocking factor and different groups
of children as the other blocking factor.
(a) [2 pts] Examine the data. What are the names of the two blocking factors?
(b) [2 pts] Display the particular Latin square that was used in this experiment. Identify
which factor corresponds to the rows and which to the columns.
(c) [2 pts] Fit the linear model appropriate for analysis of this experiment. Display a
summary of the results. (Remember to convert variables to factor as needed.)
(d) [2 pts] Produce an ANOVA table for your model. Are treatment effects statistically
significant? (Remember to state the p-value.)
(e) [2 pts] Produce Tukey simultaneous 95% confidence intervals for all mean differences
between pairs of treatments.
(f) [2 pts] According to your Tukey intervals, which pairs of treatments have significantly
different means (after adjusting for multiple comparisons)?
4. [ GRADUATE SECTION ONLY ] Consider the following two possible assignments of
treatments A, B, C, and D to a 4 × 4 square grid of experimental units:
Assignment I
A B C D
C A D B
B D A C
A D C B
Assignment II
A B C D
D A B C
C D A B
B C D A
For each of the assignments (I and II) separately, determine the probability of that
particular assignment being chosen at random if the design is:
(a) [2 pts] completely randomized (with 4 each of A, B, C, and D).
(b) [2 pts] a randomized complete block design with rows as blocks.
(c) [2 pts] a randomized complete block design with columns as blocks.
(d) [2 pts] a Latin square design (with rows and columns as blocks), with the square
chosen randomly from among the 576 total distinct 4 × 4 Latin squares.
Some reminders:
• Unless otherwise stated, all data sets are either automatically available or can be found in
either the alr4 package or the faraway package in R.
• Unless otherwise stated, use a 5% level (α = 0.05) in all tests.
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