Stat/For/Hort 572 Larget March 23, 2007
Assignment #8 — Due Friday, March 30, 2007, by 4:00 P.M.
Turn in homework in lecture, discussion, or your TA’s mailbox. Please indicate 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: Wed. 1:00–2:15
1. Ten needles were randomly selected from a large branch of a loblolly pine tree. The stomata (microscopic
breathing holes) on loblolly pine needles are arranged in rows. On each needle, 4 rows were randomly chosen
and the number of stomata per centimeter were determined for each row. The resulting data are shown
below and are in the file needle.txt.
Needle # 1 2 3 4 5 6 7 8 9 10
149 136 143 121 148 129 127 134 117 129
143 139 142 133 121 134 130 137 128 132
138 129 124 126 124 127 123 119 117 131
131 143 134 130 128 113 125 130 118 137
(a) Write down the random effects model appropriate to this problem identifying all terms used. State
the distributional assumptions. Why is a random effects model more appropriate than a fixed effects
model?
(b) Draw a nesting diagram for the model variables as in the notes.
(c) Examine a plot of the stomata counts versus needle. Are the random effects model assumptions rea-
sonable?
(d) Estimate all relevant variance components defined in (a) using both lmer and from computations with
sums of squares using an ANOVA table from an analysis using lm (or aov) based on the expected mean
square error (EMS) expressions of σ2
e+nσ2
afor “treatment” and σ2
efor error. (See end of notes for
Random Effects in R.) Are the estimates similar?
2. Suppose it is of interest to estimate the mean number (µ) of parasitic insects per unit leaf weight for a
particular tree. Eight leaves were randomly selected from the tree. From each leaf, four small disks were
cut. For each disk the number of insects per unit leaf weight was determined. The data presented below are
also in the file leaf.txt.
Leaf # I II III IV V VI VII VIII
11.4 20.2 14.3 23.6 8.4 18.3 21.6 12.8
19.3 17.0 11.1 23.1 10.7 16.2 15.8 9.3
16.2 15.8 12.8 19.9 12.3 23.0 17.2 11.5
13.6 18.9 8.9 21.0 9.8 19.4 19.8 10.1
(a) Examine a dotplot of the data that shows the insect data plotted against each leaf. Do the mean counts
look similar for each leaf? Is the spread of counts similar for each leaf? (You do not need to include
the plot in your solution, but you may if it makes you happy.)
(b) Give a suitable model for describing these data, identifying all terms in the model and identifying any
distribution assumptions. Draw a nesting diagram for the model variables as in the notes, indicating
which variables should be modeled as random effects and which as fixed.
(c) Find a 95% CI for µassuming a tdistribution with 7 degrees of freedom.
(d) Use the mcmcsamp function with a sample size of 10,000 to find a 95% credible region for µ. How does
this region differ from that found in (c)? Is its width much larger, much smaller, or about the same?
