Description
1. Consider the ais data set (in package alr4), which contains data on athletes from the
Australian Institute of Sport. We will be using only the variables Wt (weight, kg), Ht
(height, cm), and Sex (0 = male, 1 = female).
(a) [2 pts] Consider the simple linear regression of Wt on Ht (formula Wt ~ Ht). Using the
boxcox function from the package MASS, what (simple) transformation of Wt is
suggested by the Box-Cox procedure? (Show the plot and also state what value of λ
you would choose.)
(b) [2 pts] Compute the RSS value (sum of the squared residuals) for the following two
models: (i) the simple linear regression of log(Wt) on Ht (formula log(Wt) ~ Ht) and
(ii) the simple linear regression of log(Wt) on log(Ht) (formula log(Wt) ~ log(Ht)).
Which has the smaller RSS?
(Remark: When transforming an independent variable, the transformation
minimizing RSS is generally the one that should be chosen. See Weisberg 4th,
Section 8.1.2.)
(c) [2 pts] Fit the regression of log(Wt) on log(Ht), Sex, and the interaction between
log(Ht) and Sex. Give a summary of the results. Is the interaction significant?
(d) [2 pts] Plot log(Wt) versus log(Ht) with different symbols for the males and the
females. Then plot the two (not parallel) regression lines representing the relationship
between log(Wt) and log(Ht) for the male and female athletes, according to your
model from the previous part that includes an interaction. (The two lines should be on
the same plot as the points.)
(e) [2 pts] Fit the regression of log(Wt) on log(Ht) and Sex, without interaction. Give a
summary of the results. Is Sex significant?
(f) [2 pts] Plot log(Wt) versus log(Ht) as before, then plot the two (parallel) regression
lines representing the relationship between log(Wt) and log(Ht) for the male and
female athletes, according to your model from the previous part. Which line is higher:
the male or the female?
2. The data set turk0 (in package alr4) contains the results of an experiment with a
completely randomized design: 35 pens of turkeys were randomly allocated into 6 groups,
and each group was given a different level of supplementation with methionine.1 You are to
analyze how the average weight Gain (grams) of the turkeys in a pen depends on the
amount A of methionine supplement that was assigned to the pen.
(a) [2 pts] Is the design balanced? How do you know? (Hint: Examine the structure of the
data. How many experimental units are in each treatment group?)
1See Weisberg 4th, Sec. 11.3 for a more complete description.
1
(b) [2 pts] Fit an appropriate ANOVA model. (Note: You will need to convert A into a
factor variable.) Display a summary of your results.
(c) [2 pts] Produce the usual diagnostic plots for your model, and draw conclusions.
(d) [2 pts] Produce an ANOVA table.
(e) [2 pts] Test whether there are any differences among the mean weight gains of the
groups (based on an F-test at α = 0.05).
(f) [2 pts] Produce Tukey simultaneous 95% confidence intervals for all mean differences
between pairs of groups.
(g) [2 pts] According to your Tukey intervals, which pairs of methionine levels have
significantly different means (after adjusting for multiple comparisons)? (List the
pairs.)
3. The data in the file pine.dat2 are from an experiment to investigate how production of
pine oleoresin, obtained by tapping pine trees, is affected by shape of the hole (1=circular,
2=diagonal slash, 3=check, 4=rectangular) and whether or not acid treatment (trt) was
used (1=no, 2=yes). The experiment was performed with 24 pine trees as experimental
units, in a completely randomized design. The response y is the amount (g) of resin
collected from an individual tree.
(a) [2 pts] Examine the data. How many treatment groups are there? How many
experimental units are in each treatment group?
(b) [2 pts] Fit the linear model appropriate for analysis of this experiment, with y as the
response. Display a summary of the results. (Note: You will need to convert shape
and trt into factor variables.)
(c) [2 pts] Produce and interpret the usual diagnostic plots for your model. Do you notice
any problems?
(d) [2 pts] Perform a Box-Cox analysis on your model. (Show the graph produced by the
boxcox function.) What simple transformation seems most appropriate?
(e) [2 pts] Fit the linear model appropriate for analysis of this experiment, with sqrt(y)
as the response. Display a summary of the results.
(f) [2 pts] Create an interaction plot (with the square root of y as the response). Use
shape as the x-axis factor.
(g) [2 pts] Produce an ANOVA table for your model of part (e).
(h) [2 pts] Using your ANOVA table, test for interaction effects. (What is your
conclusion?)
(i) [2 pts] If it is appropriate, test for the main effects. If that is NOT appropriate, briefly
explain why not.
4. [ GRADUATE SECTION ONLY ] Consider the means model for a one-way ANOVA:
Yij = µi + eij where the errors are independent and follow the distribution eij ∼ N
0, σ2
.
Index i indicates the group, ranging from 1 to T, and index j the observation within the
group, ranging from 1 to ni
. The total number of observations is N =
PT
i=1 ni
.
2From Oehlert (2000) A First Course in Design and Analysis of Experiments, New York: W. H. Freeman.
