MATH3014-6027 – Design (and Analysis) of Experiments
Design (and Analysis) of Experiments
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MATH3014-6027 – Design (and Analysis) of Experiments
Coursework
You should upload two solution files, one for each question. For question 1, upload a handwritten or typeset
solution (eg scanned or using LaTeX) as a pdf file. For question 2, upload annotated R code that produces all
requested analyses and plots. Use comments in the code to explain what you are doing, and to provide insight
into the analysis.
Please name your solution files math3014-6027 cwk student number q1.pdf for question 1 and
math3014-6027 cwk student number q2.R for question 2, where student number is replaced by your student
number. Do not put your name anywhere in the files.
1. [20 marks] In the lecture notes, we have defined unique contrast estimators ĉTτ for various designs where
the contrast coefficients satisfy
∑t
j=1 cj = 0. In general, the linear combination a
Tτ is estimable if there
exists a linear combination of the responses, lTy, such that E(lTy) = aTτ , for vectors aT = (a1, . . . , at)
and lT = (l1, . . . , ln).
For a balanced incomplete block design, demonstrate that a contrast cTτ is estimable by showing that
E
(
k
λt
cTq
)
= cTτ
with
q =
(
XT2 −
1
k
NXT1
)
y .
See Section 3.6.2 for the full definition of the vector q.
2. [20 marks] An experiment was performed to investigate the effects of exercise on pulse rate, using a
specially constructed exercise bike. Eight treatments were compared by using three subjects across eight
days, as shown in the below table.
Subjects
Day 1 2 3
1 2 (45) 6 (25) 7 (18)
2 8 (27) 5 (20) 6 (32)
3 6 (40) 8 (23) 2 (28)
4 7 (17) 2 (32) 4 (24)
5 5 (30) 1 (36) 8 (20)
6 4 (29) 7 (13) 3 (20)
7 1 (34) 3 (18) 5 (25)
8 3 (21) 4 (22) 1 (34)
After each treatment, the time taken in seconds for 50 heart beats was recorded, and this is also given
in the above table (in brackets). For example, on Day 1, subject 2 received treatment 6 and a response
of 25 seconds was recorded. For the analysis of this experiment, it was thought necessary to include
both Subjects and Days as blocking variables. Hence, this is a row-column design (see also Chapter 3,
Exercise 5).
The data are available on Blackboard as the file exercise.csv.
1
(a) Investigate the differences between the eight treatments. State an appropriate unit-block-treatment
linear model that includes both blocking factors and the treatment factor. Fit this model, and
produce and interpret the ANOVA table. If appropriate, test pairwise treatment comparisons at a
5% experiment-wise type I error rate (all pairwise comparisons are uniquely estimable in this design).
Produce and comment on residual plots.
In fact, the treatments are all combinations of the three two-level factors “exercise duration” (1 and 3
minutes), “exercise speed” (40 and 60 rpm) and “pedal type” (foot pedal and hand bars), as in the table
below.
Treatment Duration (mins.) Speed (rpm) Pedal
1 1 40 foot
2 1 40 hand
3 1 60 foot
4 1 60 hand
5 3 40 foot
6 3 40 hand
7 3 60 foot
8 3 60 hand
(b) Find estimates of the main effects and two-factor interactions for these factors. Which factorial
effects are significantly different from zero at the 5% level? Discuss why the standard errors of the
contrast estimators are not all equal.