MATH3014-6027 Design of Experiments
SEMESTER 2 EXAMINATION 2021/22
1. [25 marks]
Table 1 shows the results of an experiment conducted to study the influence of three different types of glass on the light output of an oscilloscope tube.
Table 1: Oscillioscope experiment: light output (lumens) from three different glass types.
|
Glass 1
|
Glass 2
|
Glass 3
|
|
580
|
1550
|
546
|
|
1090
|
1070
|
1045
|
|
1392
|
1328
|
867
|
|
568
|
1530
|
575
|
|
1087
|
1035
|
1053
|
|
1380
|
1312
|
904
|
|
570
|
1579
|
599
|
|
1085
|
1000
|
1066
|
|
1386
|
1299
|
889
|
The data could be read into R using the following code.
oscil <- data. frame(
Glass = rep(factor(c(1 , 2 , 3)), rep(9 , 3)),
Light = c(580 , 1090 , 1392 , 568 , 1087 , 1380 , 570 , 1085 , 1386 ,
1550 , 1070 , 1328 , 1530 , 1035 , 1312 , 1579 , 1000 , 1299 , 546 , 1045 , 867 , 575 , 1053 , 904 , 599 , 1066 , 889)
)
(a) [10 marks] Complete the top row of ANOVA Table 2, comparing a unit-treatment model to the null model. Using the table, test the hypothesis
H0 : τ1 = τ2 = τ3 = 0 at the 5% level, where τi is the effect of the ith type of glass.
Table 2: Oscillioscope experiment: ANOVA table.
|
|
Df
|
Sum Sq
|
Mean Sq
|
F value
|
|
Glass
|
|
|
|
|
|
Residuals
|
24
|
1783466
|
74311
|
|
|
Total
|
26
|
2761886
|
|
|
(b) [10 marks] Test all pairwise differences between the three treatments at an exact experiment-wise level of 5%.
(c) [5 marks] Find the allocation of n = 27 units to three treatments that leads to the minimum average variance of the estimators of the three pairwise differences.
How efficient is the design given above?
You may find the following quantities from R useful.
qf(0.95 , 2 , 24)
## [1] 3 .4
qf(0.975 , 2 , 24)
## [1] 4.32
qtukey(0.95 , 2 , 24) ## [1] 2 . 92
qtukey(0.95 , 3 , 24) ## [1] 3 . 53
2. [25 marks]
Table 1 shows the results of an experiment conducted to study the influence of three different temperatures on the light output of an oscilloscope tube.
Table 3: Oscillioscope experiment: light output (lumens) from three different temperatures (Fahrenheit).
|
Temp. 100
|
Temp. 125
|
Temp. 150
|
|
580
|
1090
|
1392
|
|
568
|
1087
|
1380
|
|
570
|
1085
|
1386
|
|
550
|
1070
|
1328
|
|
530
|
1035
|
1312
|
|
579
|
1000
|
1299
|
|
546
|
1045
|
867
|
|
575
|
1053
|
904
|
|
599
|
1066
|
889
|
The data could be read into R using the following code.
oscil <- data. frame(
Temperature = rep(factor(c(100 , 125 , 150)), 9),
Light = c(580 , 1090 , 1392 , 568 , 1087 , 1380 , 570 , 1085 , 1386 , 550 , 1070 , 1328 , 530 , 1035 , 1312 , 579 , 1000 , 1299 ,
546 , 1045 , 867 , 575 , 1053 , 904 , 599 , 1066 , 889) )
(a) [10 marks] Complete the top row of ANOVA Table 2, comparing a unit-treatment model to the null model. Using the table, test the hypothesis
H0 : τ1 = τ2 = τ3 = 0 at the 5% level, where τi is the effect of the ith temperature.
Table 4: Oscillioscope experiment: ANOVA table.
|
|
Df
|
Sum Sq
|
Mean Sq
|
F value
|
|
Temperature
|
|
|
|
|
|
Residuals
|
24
|
447996
|
18666
|
|
|
Total
|
26
|
2418330
|
|
|
(b) [15 marks] The three levels of temperature are equally spaced. Define suitable contrasts in the treatments to compare the expected responses from the first and third temperatures, and from the average of the first and third temperature and the second temperature. Using these contrasts, test if there is a linear or quadratic effect of temperature on light output at an approximate experiment-wise rate of 5%.
3. [25 marks]
Block designs with blocks of common size k = 2 are important in a variety of settings, especially when the experimental units are people and there are natural pairs (two hands, two eyes etc.). Such designs are often referred to as paired comparison studies.
Assume a paired comparison study is planned to investigate t = 2 treatments using b blocks. Each treatment will be run in each block.
(a) [4 marks] Write down a unit-block-treatment model appropriate for such a
design, describing the form of the block (X1) and treatment (X2) matrices. Give the form of the incidence matrix N.
(b) [6] Show that the reduced normal equations for the treatment effects τ(ˆ) under this model can be expressed as
where In is the n × n identity matrix, Jn is the n × n matrix with all entries equal to one, 1n is the n-vector with all entries equal to one, with
(c) [3 marks] Prove that for a paired comparison study, the treatment difference can be estimated independently of blocks.
Consider now an equivalent unpaired study, where the same number of units were allocated one-half to treatment 1 and one-half to treatment 2.
(d) [5 marks] Why would we anticipate the variance, σ 2 , of the random error term in the linear model for a paired study to be smaller than that for the unpaired study? How many degrees of freedom does each of the paired and unpaired studies provide for estimating σ 2 ?
(e) [7 marks] Label the error variance from the paired study as σ 1(2), and from the
unpaired study as σ2(2). What is the ratio of the variances of the estimators of the treatment difference from the paired and unpaired studies in terms of σ 1(2)/σ2(2)? In which situation will this variance ratio be less than one, i.e. when will the paired design give smaller variance?
4. [25 marks]
(a) [9 marks] Describe how you could construct a balanced incomplete block design that arranges t = 8 treatments into blocks of size k = 6. Give the values of b, r and λ for your design. Write down one block.
Now assume the t = 8 treatments are formed as the combinations of three two-level factors. Up to a constant σ 2 , what is the variance of any factorial effect estimator using this design?
(b) [4 marks] Now arrange the 23 treatments into b = 2 blocks of size k = 4 by
confounding one factorial effect with blocks. Which factorial effect do you choose and why? Give one block of this design.
Up to a constant σ 2 , what is the variance of any estimable factorial effect using this design?
(c) [12 marks] Assuming σ 2 from the two designs is the same, how many replicates of the confounded design from part (b)would be needed to make the variance of an estimable factorial effect the same as that produced in part (a)? Comment on the relative merits of the BIBD and the replicated confounded design in terms of precision of the factorial effect estimators and size of the designs.
5. [25 marks]
(a) [3 marks] Suppose a factorial experiment is planned for the purpose of
estimating the main effects of, and interactions between, f factors. What are the circumstances that would lead you to use both confounding and fractional replication in the design of such an experiment?
(b) [15 marks] Design an experiment involving f = 8 factors each at two levels, in which four blocks of sixteen treatment combinations are used. Use the generators ABCDG and ABCEFH for the fractional factorial, and confound ACD and AFG with blocks. Write down the full defining relation and all the factorial effects which are confounded with blocks. What is the resolution of this design?
Write down one block only from the design and indicate how you would construct the other blocks.
(c) [7 marks] How would the degrees of freedom be partitioned in this design?
Assuming that interactions between three of more factors are neglible, how many degrees of freedom would be available to provide an estimate of σ2 ?
6. [25 marks]
(a) [3 marks] Suppose a factorial experiment is planned for the purpose of
estimating the main effects of, and interactions between, f factors. What are the circumstances that would lead you to use both confounding and fractional replication in the design of such an experiment?
(b) [15 marks] Consider f = 7 two-level factors, labelled A - G. An investigation is to be carried out into their effects on a particular system. It is thought that factors A - C form a set which operate on one aspect of the system, while the other factors D - G are thought to operate on a different aspect. It is believed that the factors in one set will not interact in any way with the factors in the other set.
Interactions between pairs of factors within each set may be important but interactions between three or more factors may be ignored.
Design an experiment to investigate these seven factors, using the information above, in which only b = 4 blocks of k = 8 treatments are used. Give the defining relation for your design, and specify which factorial effects are aliased with blocks.
Write down one block only from the design and indicate how you would construct the other blocks.
(c) [7 marks] How would the degrees of freedom be partitioned in this design?
Given the statements given above regarding which interactions may be regarded as negligible, how many degrees of freedom could be available to provide an estimate of σ2 ?
Learning objectives:
LO1 Apply theory and methods to a variety of examples.
LO2 Evaluate designs using common optimality criteria and use them to critically compare competing designs.
LO3 Explore the general theory of factorial and block designs and understand this theory sufficiently to find appropriate designs for specific applications.
LO4 Use the R statistical programming language to design and analyse common forms of experiments.
LO5 Encounter the principles of randomisation, replication and stratification, and under- stand how they apply to practical examples.
LO4 is primarily assessed via coursework.