UNIVERSITY OF SOUTHAMPTON MATH6027W1
SEMESTER 2 EXAMINATION 2018/19
MATH6027 Design of Experiments
Duration: 120 min (2 hours)
This paper contains 4 questions.
Question 1 is worth 30 marks, Question 2 is worth 40 marks, and Questions 3 and
4 are worth 15 marks each.
Answer ALL questions.
An outline marking scheme is shown in brackets to the right of each question.
Formula Sheet FS/MATH6027/2018/19 will be available.
Only University approved calculators may be used.
A foreign language direct ‘Word to Word’ translation dictionary (paper version) ONLY is
permitted provided it contains no notes, additions or annotations.
Page 1 of 6 + Appendices (Formula Sheet)
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2 MATH6027W1
1. (a) An incomplete block design was used to investigate four types of resistor
(treatments) using four mountings (blocks), each of which could hold four
resistors. The design and data (current noise ×100) are given below.
Treatment (resistor)
Block 1 2 3 4
(mounting)
1 110 95 80
2 170 120 95
3 160 110 150
4 120 150 120
(i) Write down an appropriate and estimable linear model for this experiment.
Write down the model matrices for treatments (X) and blocks (Z) such that
treatment 4 and block 4 are used as baselines. [4 marks]
(ii) Given that the residual sum of squares from fitting a null model is 8816.67, the
residual sum of squares from fitting the model with just the intercept and block
parameters is 5366.67 and the residual sum of squares from fitting the model
with intercept, block and treatment parameters is 672.92, complete the below
analysis of variance table, and test the null hypothesis of no difference
between treatments at the 5% level. [9 marks]
Source degrees of Sum of squares Mean Square
freedom
Blocks
Extra due to
treatments
Residual
Total
(iii) From these data, the estimates of the treatment parameters are βˆ1 = 50.63,
βˆ2 = 3.75 and βˆ3 = 33.13. Use Bonferroni’s method at an experimentwise
5% level to decide which pairs of treatments produce a significantly different
mean response. [7 marks]
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(b) (i) An engineer would like to compare 5 new materials, A, B, C, D, E, and a
control material F. The machine in which he tests the materials can only hold
2 units at a time, so each run of the machine is considered as a block. He
suggests the following design for the 6 materials in 5 blocks of 2 units:
Materials
1 A F
2 B F
Blocks 3 C F
4 D F
5 E F
Find the degrees of freedom associated with each source of variation from the
ANOVA table. Do you find any problem? Can the problem be solved by using
a different allocation of the 6 treatments to 5 blocks of size 2? [4 marks]
(ii) The engineer receives a research grant, from which he buys a new machine
that can test 12 units simultaneously. He suggests to have 7 units of material
F and one each of A, B, C, D and E (Design 1). His PhD student would prefer
to test 2 units of each of the 6 materials (Design 2). Using material F as the
baseline treatment, the covariance matrices for
βˆ = (βˆ0, βˆA, βˆB, βˆC , βˆD, βˆE) for Design 1 and Design 2 are, respectively,
V ar1(βˆ) =
1
7

1 −1 −1 −1 −1 −1
−1 8 1 1 1 1
−1 1 8 1 1 1
−1 1 1 8 1 1
−1 1 1 1 8 1
−1 1 1 1 1 8
 σ
2,
V ar2(βˆ) =
1
2

1 −1 −1 −1 −1 −1
−1 2 1 1 1 1
−1 1 2 1 1 1
−1 1 1 2 1 1
−1 1 1 1 2 1
−1 1 1 1 1 2
 σ
2.
Explain, without performing explicit calculations, how you would find these
covariance matrices. Which design would you prefer for estimating the
treatment differences? Give your reasons. [6 marks]
[Total: 30 marks]
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2. (a) Two choices of a 27−2 factorial design are being considered for use in an
experiment. One design, d1, aliases F1F2F3F4 and F3F4F5F6F7 with the mean;
the other design, d2, aliases F1F2F3F4 and F2F3F4F5F6F7 with the mean.
(i) Write down the full defining relation for each design. [4 marks]
(ii) State the resolution of each design. [2 marks]
(iii) Which design would you prefer? Give your reasons. [4 marks]
(b) A design is required for five factors, each at two levels, that can estimate all main
effects independently of each other, when higher-order interactions are negligibly
small.
(i) Write down the defining relation for a fractional factorial design with the
smallest number of runs that allows all the required effects to be estimated.
[4 marks]
(ii) What is the resolution of your chosen design? Could there be a design with
smaller or larger resolution, with the design still satisfying the requirements of
part (i)? Explain why (or why not). [4 marks]
(c) A 34 experiment is to be laid out in three blocks with 27 units per block. It has
been decided that certain contrasts from the highest order interaction should be
confounded with blocks.
(i) If the component F 11F
1
2F
1
3F
1
4 is confounded with blocks, which other
component is also confounded? [3 marks]
(ii) Write down a congruence whose solution generates the initial block of the
design. Determine five treatments from this initial block, and indicate how the
other two blocks could be found. [8 marks]
(iii) Give the partition of the degrees of freedom for your experiment. [6 marks]
(iv) If the experiment is carried out as a 1/3 replicate using only one of these
blocks, write down the defining relation of the design. Can all main effects be
estimated if three-factor and four-factor interactions are negligible? Justify
your answer. [5 marks]
[Total: 40 marks]
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3. A chemist performs an experiment to study the yield of a reaction. There are two
factors to investigate, the time of the reaction (A) and the temperature of the reaction
(B). A full factorial design was used, with 5 additional centre points. The results are:
A B Response
-1 -1 58.9
+1 -1 60.5
-1 +1 60.1
+1 +1 61.2
0 0 60.2
0 0 60.4
0 0 60.6
0 0 60.5
0 0 60.3
(a) For these data, use an appropriate test (at the 5% level) to test the null
hypothesis of no curvature in the response surface. [8 marks]
(b) Suppose the chemist’s ultimate aim is to find the combination of time and
temperature where the maximum yield of the reaction is achieved. She is asking
you for advice on what she should do next, in the light of the result of the test in
(a). Describe, without performing any explicit calculations, which strategy she
should pursue. [4 marks]
(c) Assume that, through your recommended strategy, the chemist has found a
region in which the maximum is likely to lie. She now wishes to fit a second-order
model to observations from this region. How could she augment the given first
order design? [3 marks]
[Total: 15 marks]
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4. (a) Define D-, A-, V - and G-optimality and state their respective objective functions
in terms of the information matrix X ′X . State a situation in which you would use
a V -optimal design rather than a D-optimal design. [5 marks]
(b) Consider the following model:
Y = β0 + β1x+ ε ,
where β0 and β1 are unknown constants to be estimated from the data and ε is a
normally distributed random error term. The errors for different observations are
assumed to be independent and have constant variance. Three different designs,
d1, d2 and d3, each having 12 runs, are considered to collect data to estimate
this model. The designs are summarized in the table below.
x
-1 0 1
d1 6 0 6
d2 4 4 4
d3 3 6 3
For example, design d1 takes 6 observations at x = −1, no observations at
x = 0, and 6 observations at x = 1.
(i) Write down the model matrix X and the information matrix X ′X for an
arbitrary design with 12 observations at points x1, x2, . . . , x12. For each
design (d1, d2 and d3), calculate the information matrix X ′X from this
general formula. [5 marks]
(ii) For each design, calculate the value of the D-optimality objective function.
Which design is preferred under D-optimality? Can you think of a situation
where you would recommend the design that comes second in terms of
D-optimality? [5 marks]
[Total: 15 marks]
END OF PAPER
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