1. Short answer.
(a) [2 points] (True or False; if False explain why) In a model with both xed and random
eects, one should use the Bonferroni multiple comparisons adjustment to compare means
for each pair of levels of the xed eect.
(b) [2 points] (True or False; if False explain why) Inference using REML works well when
the number of levels of the random factors is small.
(c) [3 points] Give one benet of randomization in an experiment.
(d) [3 points] In the context of mixed eects models, explain the dierence between marginal
and conditional residuals.
(e) [3 points] Suppose in an experiment one had 4 machines and 5 operators per machine. Is
operator nested within or crossed with machine? Why?
2. Experimental design scenarios.
(a) [6 points] Consider an experiment to study the loss of pigmentation in cells in the presence
or absence of four compounds. That is, there are four factors each at two levels for a total
of 16 treatments. The 16 treatment combinations were mixed up and from these mixtures,
each treatment was applied to ve containers of cells for a total of 80 containers.
i. Identify the the experimental unit(s) and measurement unit in this experiment.
ii. What kind of design is this?
(b) [6 points] Suppose that an experiment is conducted to study the strength of an object
constructed using an additive manufacturing device. Two factors of interest are needle
dispensing speed and needle diameter. The dispensing speed can easily be changed via a
setting on the machine, but to change the needle diameter the needle assembly must be
taken apart and cleaned in a time-consuming process.
i. Identify the experimental unit(s) and the measurement unit for the type of experiment
most appropriate here.
ii. What type of experiment is most appropriate?
(c) [6 points] Consider an experimental comparison of seven treatments aimed to reduce itch-
ing. Ten randomly selected subjects each receive the seven treatments in random order,
one treatment per day. After each treatment, an itch stimulus is applied and the response
is the number of seconds the itching lasts. The treatments are: (1) no drug; (2) placebo; (3)
papaverine; (4) morphine; (5) aminophylline; (6) pentobarbital; and (7) tripelennamine.
Identify each of the following: (a) the factors; (b) whether each factor is xed or random;
and (c) the type of design.
3. Consider a one-way experiment which studied the laser-assisted manufacturing of a thermo-
plastic composite. The experimental factor was laser power (3 levels: 40, 50, and 60 watts)
and there were 3 replicates. The eects model is
Yij = + i + ij ;
i=1 i = 0 and ij N(0; 2), i = 1; 2; 3; j = 1; 2; 3.
(a) [4 points] Is 1 + 2 estimable? Justify your answer.
(b) [4 points] Explain why the constraint
i=1 i = 0 is necessary.
(c) Using this experiment, write down regression models for the following two situations,
including an explanation of the model terms:
i. [4 points] Consider the factor as categorical with three levels.
ii. [4 points] Consider the factor as numeric and the model should capture possible cur-
4. Consider a study to measure the perception of the color of frozen peas. There were 10 randomly
chosen food samples measured by ve randomly chosen judges. Each judge will measure each
food sample three times, so each judge will measure 30 randomly ordered food samples. The
data, code, and output accompany this exam.
(a) [4 points] The error term for the sample row in the ANOVA table is MS(sample*judge).
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