Stat/For/Hort 572 Larget April 21, 2008
Assignment #8 — Due Friday, April 25, 2008, by 4:00 P.M.
Turn in homework in lecture, discussion, or your TA’s mailbox (just inside the main entrance to MSC). Please
circle the discussion section you expect to attend to pick up this assignment.
311: Tues. 1:00–2:15 312: Wed. 2:30–3:45 313: Tue. 4:00–5:15
The first several questions revisits the data from the file larch.txt, which contains measurements from 26
twenty-four-year-old larch trees, an evergreen tree related to pines, native to central Europe, that drop their needles
every year. The observation number is in column 1 id. The explanatory variables are in columns 2–5 and are
the percent content of nitrogen nitro, the percent content of phosphorus phos, the percent content of potassium
potas, and the percent content of residual ash ash. The percent content of the minerals is determined from dried
needles from the tree. The response variable, tree height (height) is in column 6 and is measured in inches. The
objective of the study was to relate tree height to the mineral composition of the needles.
1. Fit a model for height with predictors an intercept, nitro,phos,potas,ash, and the interaction nitro:phos.
Report the estimated coefficients and the standard errors. Report the estimated error standard deviation.
(Note: The functions coef(),se.coef(), and sigma.hat() will extract the estimated coefficients, their
SEs, and the estimated residual standard deviation from a fitted regression model. The latter two functions
are in the arm library.)
2. This function in Rwill generate a fake data set using a model fitted with lm() using the fitted model from
part (a) as the truth from the same predictors. The first argument, x, should be the original data frame,
while the second argument is the fitted model.
make.fake = function(x,fit) {
# make sure that the arm library is loaded
library(arm)
# n is the number of observations
n = nrow(x)
# sigma.fit() is not in base R, but is in the arm library
# sigma is the estimated standard deviation of the error model
sigma = sigma.hat(fit)
# mu is the fitted value for each observation
mu = fitted(fit)
# this replaces the height variable in the data frame x
# this only modifies the copy of the data in this function;
# the original data frame is left unchanged
x$height = rnorm(n,mu,sigma)
# return the fake data set
return( x )
}
Enter the function into R. Use the function to compute five fake data sets. For each data set, fit a regression
model using the same predictors as the model in Problem 1. Plot residuals versus fitted values for both the
real data and for the fake data sets. Is the pattern or residuals for the real data similar to the pattern for
the fake data? Comment on what similarity or a lack of similarity implies about the goodness of fit of the
model.
