Description
1. Consider the data set lathe1 from package alr4.
(a) [2 pts] Fit a full second-order polynomial model of the response Life versus the
variables Feed and Speed. That is, your model should include both quadratic terms
and the interaction term. Produce an R summary of your fitted model. Does the
interaction term appear to be statistically significant?
(b) [2 pts] Produce the usual R diagnostic plots for your model from the previous part.
(c) [2 pts] Briefly assess the diagnostic plots.
(d) [2 pts] Using the boxcox function from the package MASS, produce a plot of the
(profile) Box-Cox log-likelihood, versus λ.
(e) [2 pts] Approximately what λ value appears to be selected by the Box-Cox procedure?
(Hint: boxcox(…, plotit=FALSE) shows you the λ (x) values and log-likelihood (y)
values used in the plot.)
(f) [2 pts] Find the most “simple” λ value that is still within the confidence limits shown
on your plot from part (d). To what kind of a “simple” transformation (i.e. what
function) does this correspond?
(g) [2 pts] Using your “simple” transformation from the previous part, refit the model.
Produce an R summary of your fitted model. Does the interaction term appear to be
statistically significant?
(h) [2 pts] Produce the usual R diagnostic plots for your model from the previous part.
Have they improved?
2. The data set trees1
contains the Girth (diameter, inches), Height (feet), and Volume
(cubic feet) of timber in 31 felled black cherry trees. Natural physical considerations would
suggest that
Volume ∝ (Girth)
2
· Height (1)
Allowing for variation among trees, and a more general relationship, we might consider the
model
Volume = γ · (Girth)
β1
· (Height)
β2
· e (2)
where e is multiplicative error.
(a) [2 pts] Use transformations to linearize the model (2), i.e. transform it to have the
form of a linear regression model in some transformed variables.
(b) [2 pts] Fit the linearized model, and produce a summary of your results.
1Data set trees is actually in the datasets package, which is automatically available, so you do not need any
other package to load it.
1
(c) [2 pts] Briefly assess the fit of your model using diagnostic plots.
(d) [2 pts] Form individual 95% confidence intervals for β1 and β2. Do they contain the
theoretical values suggested by the relationship (1)?
(e) [2 pts] Consider a new tree (of the same kind) that has a girth of 10.9 inches and a
height of 75 feet. Using the fitted model, form a 95% prediction interval for its
log-volume.
(f) [2 pts] Transform your prediction interval (from the previous part) back to the original
volume scale (in cubic feet).
3. Use the ais data set (in package alr4) with Bfat as the response and only the variables
Sex, Ht, Wt, LBM, BMI, and SSF as possible predictors. Implement the following variable
selection methods to determine a model. In each case, (i) show appropriate R output, and
(ii) list the independent variables in the final model.
(a) [2 pts] forward selection (use Fin = 3)
(b) [2 pts] backward elimination (use Fout = 3)
(c) [2 pts] selection with the R function leaps, according to minimum Cp
(Hint: When you use model.matrix, use the formula Bfat ~ Sex + Ht + Wt + LBM +
BMI + SSF – 1.)
(d) [2 pts] stepwise selection with the R function step, using the arguments object=
lm(Bfat ~ 1, data=ais), scope= ~ Sex + Ht + Wt + LBM + BMI + SSF, and
direction= “both”. (Consult help(step) for more information.)
(Note: This will perform stepwise selection using AIC as the selection criterion.)
4. [ GRADUATE SECTION ONLY ] The so-called arcsine transformation, often used to
transform binomial proportions, is given by
h(y) = sin−1
(
√
y), 0 ≤ y ≤ 1
(a) [2 pts] Compute the first derivative of h(y) (for 0 < y < 1).
(b) [2 pts] Using the method demonstrated during lecture, verify that the arcsine
transformation is (approximately) variance-stabilizing for the situation
Var(Y ) ∝ E(Y )
1 − E(Y )
(c) [4 pts] Y = W/n is a binomial proportion if W ∼ binomial(n, p). Derive the mean and
variance of Y (using the mean and variance of W, which you know).
(d) [2 pts] Briefly explain why binomial proportions (with the same n) satisfy the
condition of part (b).
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.
