MATH6027 Design of Experiments
SEMESTER 2 EXAMINATION 2023/24
1. Consider a pain relief study to compare three drugs. Each patient in the trial is assigned one of the drugs at random. The response is the number of hours of pain relief provided, measured from administration of the drug. The data from this experiment are given in Table 1 below. The three drugs are labelled T1, T2 and T3.
Table 1: The pain relief study.
These data could be loaded into R using the following code.
pain <- data.frame(
y = c(
2, 6, 4, 13, 5, 8, 4, 6, 7, 6, 8, 12, 4, 4, 2, 0, 3,
3, 0, 0, 8, 1, 4, 2, 2, 1, 3, 6, 4, 4, 0, 1, 8, 2, 8,
12, 1, 5, 2, 1, 4, 6, 5
),
drug = factor(rep(c("T1", "T2", "T3"), c(14, 13, 16)))
)
(a) [10 marks] Assuming the usual unit-treatment model, test all pairwise differences between treatments at an experiment-wise level of 5%.
(b) [10 marks] Assuming the relative replication of the three treatments is the same as in the original study above, find an appropriate sample size n for the experiment to make T = 2 for a comparison of treatment 1 and treatment 2, where
for an assumed signal-to-noise ratio d = 3, with τi
the effect of the ith treatment. Describe how this choice relates to inference for the experiment.
(c) [5 marks] Is the relative replication of the treatments in the original study optimal for testing all pairwise comparisons? Give a reason for your answer.
2. An experiment has been conducted to compare the effect of four materials for the construction of a radon detector. Each detector, made from one material, is tested in a laboratory chamber that is large enough to test four units. A randomised complete block design (RCBD) was run with b = 4 blocks, corresponding to the batches. The data, in percentages reacted, are given in Table 2 below.
Table 2: The radon experiment.
These data could be loaded into R using the following code.
radon <- data.frame(
material = factor(rep(1:4, each = 4)),
block = factor(rep(c(1, 2, 3, 4), 4)),
y = c(
6.60, 6.70, 6.11, 6.39, 6.11, 6.22, 6.07, 6.22, 6.52, 6.32,
5.95, 6.54, 6.20, 5.97, 5.82, 6.18
)
)
(a) [6 marks] Assuming the standard unit-block-treatment model, produce the analysis of variance for this experiment. Test if there is a significant difference between materials at the 5% level.
(b) [4 marks] Now assume that blocks are ignored, i.e. the experiment is treated as a completely randomised design (CRD). Do you now reject the null hypothesis of no difference between materials? Use only the analysis of variance from part (a), i.e., do not fit a new linear model. Explain your result and relate it to the conclusion drawn in part (a).
(c) [4 marks] What is the standard error of a pairwise treatment difference under both the RCBD and the CRD? Interpret the difference between the two standard errors.
(d) [5 marks] Now assume that each block was actually only large enough to test three treatments. Find a balanced incomplete block design (BIBD) for this experiment. Show that your design is indeed a BIBD.
(e) [6 marks] Now consider investigating t treatments in b blocks using either a BIBD with block size k or an RCBD. Prove that 2k/λt > 2/b, and hence compare the precision of a pairwise treatment comparison from the two designs assuming within-block variance σ
2
is the same for both experiments.
3. (a) [10 marks] Find the parameters (b, r, λ) for a balanced incomplete block design for a 2
3
factorial experiment using as few blocks as possible of size k = 7.
(b) [5 marks] Write down the design from part (a); that is, indicate which treatments are in each block.
(c) [10 marks] Compare this design to a replicated confounded block design with blocks of size 4, replicated to be of equal size to the BIBD in part (a), in terms of the variance of an estimated factorial effect.
4. An experiment was conducted to study the effect of five factors, each studied at two levels, on the percentage yield from a chemical reaction. The factors and their levels are given below in Table 3.
Table 3: Factors for the reaction experiment.
A full factorial design was used, and the factorial effect estimates in Table 4 (see page 9) were obtained.
(a) [5 marks] The half-normal plot in Figure 1 (see page 10) was produced using the estimated effects. Identify which effects are likely to be important (different from zero).
(b) [6 marks] Produce rough sketch plots of any interactions identified as non-zero in part (a).
(c) Now assume that a fractional factorial design was used which aliases F = CT and A = CT S.
(i) [6 marks] Write down the set of treatments for this design in the original units of the factors.
(ii) [8 marks] Find the estimates from the fractional factorial design of any interaction effects found to be important in part (a).
Table 4: Estimated factorial effects from the reaction experiment, ordered by absolute value. F = Feed rate, C = Catalyst, A = Addition rate, T = Temperature, S = Concentration.
Half−normal quantiles
ME = 3.39 SME = 6.44 Reference SD = 1.643 (Lenth method)
Figure 1: Half-normal plot for the reaction experiment.
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.