Stat 232
Experimental Design
Spring 2008
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Ching-Shui Cheng
Office: 419 Evans Hall
Phone: 642-9968
Email: cheng@stat.berkeley.edu
Office Hours: Tu Th 2:00-3:00 and by
appointment
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Course webpage:
http://www.stat.berkeley.edu/~cheng/232.htm
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No textbook
Recommended (for first half of the course):
Design of Comparative Exeperiments by R. A. Bailey, to appear in
2008
http://www.maths.qmul.ac.uk/~rab/DOEbook/
Experiments: Planning, Analysis, and Parameter Design
Optimization by C. F. J. Wu and M. Hamada
Statistics for Experimenters: Design, Innovation and Discovery by
Box, Hunter and Hunter
A useful software: GenStat
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Experimental Design
Planning of experiments to produce valid
information as efficiently as possible
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Comparative Experiments


Treatments (varieties)
Varieties of grain, fertilizers, drugs, ….
Experimental units (plots): smallest division of the
experimental material so that different units can receive
different treatments
Plots, patients, ….
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Design: How to assign the treatments to the
experimental units
Fundamental difficulty: variability among the units; no two units are
exactly the same.
Each unit can be assigned only one treatment.
Different responses may be observed even if the same treatment is
assigned to the units.
Systematic assignments may lead to bias.
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R. A. Fisher worked at the Rothamsted Experimental
Station in the United Kingdom to evaluate the success of
various fertilizer treatments.
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Fisher found the data from experiments going on for decades to be
basically worthless because of poor experimental design.

Fertilizer had been applied to a field one year and not in another in
order to compare the yield of grain produced in the two years.
BUT



It may have rained more, or been sunnier, in different years.
The seeds used may have differed between years as well.
Or fertilizer was applied to one field and not to a nearby field in the
same year.
BUT
 The fields might have different soil, water, drainage, and history
of previous use.
 Too many factors affecting the results were “uncontrolled.”
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Fisher’s solution: Randomization

In the same field and same year,
apply fertilizer to randomly spaced
F
F
plots within the field.

This averages out the effect of
F
F F FF
F
F
F F
F
F F
F
F F F
F
F F
F
variation within the field in drainage
and soil composition on yield, as
well as controlling for weather, etc.
F
F
F
F F
F F F F
F F F
F
F
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Randomization prevents any particular treatment from
receiving more than its fair share of better units, thereby
eliminating potential systematic bias. Some treatments
may still get lucky, but if we assign many units to each
treatment, then the effects of chance will average out.
Replications
In addition to guarding against potential systematic
biases, randomization also provides a basis for doing
statistical inference.
(Randomization model)
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Start with an initial design
F F F F F F F F F F F F
F F F F F F F F F F F F
F F F F F F F F F F F F
Randomly permute (labels of) the experimental units
Complete randomization: Pick one of the 72! Permutations
randomly
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4 treatments
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4
Pick one of the 72! Permutations randomly
Completely randomized design
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blocking
A disadvantage of complete randomization is
that when variations among the experimental
units are large, the treatment comparisons do
not have good precision. Blocking is an effective
way to reduce experimental error. The
experimental units are divided into more
homogeneous groups called blocks. Better
precision can be achieved by comparing the
treatments within blocks.
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After randomization:
Randomized complete block design
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Wine tasting
Four wines are tasted and evaluated by each of
eight judges.
A unit is one tasting by one judge; judges are
blocks. So there are eight blocks and 32 units.
Units within each judge are identified by order of
tasting.
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Block what you can and randomize what
you cannot.
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


Randomization
Blocking
Replication
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Incomplete block design
7 treatments
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Each of ten housewives does four washloads
in an experiment to compare five new
detergents.
5 treatments and 10 blocks of size 4.
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Incomplete block design
7 treatments
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Incomplete block design
Balanced incomplete block design
Randomize by randomly permuting the block labels and
independently permuting the unit labels within each
block.
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Two simple block (unit) structures

Nesting
block/unit

Crossing
row ＊ column
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Two simple block structures

Nesting
block/unit

Crossing
row ＊ column
Latin square
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Wine tasting
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Simple block structures
Iterated crossing and nesting

cover most, though not all block structures
encountered in practice
Nelder (1965)
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Consumer testing
A consumer organization wishes to compare 8 brands of
vacuum cleaner. There is one sample for each brand.
Each of four housewives tests two cleaners in her home
for a week. To allow for housewife effects, each housewife
tests each cleaner and therefore takes part in the trial for 4
weeks.
8 treatments
Block structure:
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Aα
Bβ
Cγ
Dδ
Bγ
Aδ
Dα
Cβ
Cδ
Dγ
Aβ
Bα
Dβ
Cα
Bδ
Aγ
Trojan square
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Treatment structures

No structure

Treatments vs. control

Factorial structure
A fertilizer may be a combination of three factors
(variables) N (nitrogen), P (Phosphate), K (Potassium)
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
Treatment structure

Block structure (unit structure)

Design

Randomization

Analysis
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Choice of design



Efficiency
Combinatorial considerations
Practical considerations
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McLeod and Brewster (2004) Technometrics
A company was experiencing problems with one of its
chrome-plating processes in that when a particular
complex-shaped part was being plated, excessive pitting
and cracking, as well as poor adhesion and uneven
deposition of chrome across the part, were observed. With
the goal being the identification of key factors affecting the
quality of the process, a screening experiment was
planned.
In collaboration with the company’s process engineers, six
factors were identified for consideration in the experiment.
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Hard-to-vary treatment factors



A: chrome concentration
B: Chrome to sulfate ratio
C: bath temperature
Easy-to-vary treatment factors



p: etching current density
q: plating current density
r: part geometry
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The responses included the numbers of pits
and cracks, in addition to hardness and
thickness readings at various locations on the
part.
Suppose each of the six factors have two
levels, then there are 64 treatments.
A complete factorial design needs 64
experimental runs
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Block structure: 4 weeks/4 days/2 runs
Treatment structure: A ＊ B ＊ C ＊ p ＊ q ＊ r
Each of the six factors has two levels
Fractional factorial design
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Miller (1997) Technometrics
Experimental objective: Investigate methods of
reducing the wrinkling of clothes being
laundered
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Miller (1997)
The experiment is run in 2 blocks and employs
4 washers and 4 driers. Sets of cloth samples
are run through the washers and the samples
are divided into groups such that each group
contains exactly one sample from each washer.
Each group of samples is then assigned to one
of the driers. Once dried, the extent of wrinkling
on each sample is evaluated.
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Treatment structure:
A, B, C, D, E, F: configurations of washers
a,b,c,d: configurations of dryers
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Block structure:
2 blocks/(4 washers＊4 dryers)
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Block 1
0000000000
0000000011
0000001100
0000001111
0110110000
0110110011
0110111100
0110111111
1011010000
1011010011
1011011100
1011011111
1101100000
1101100011
1101101100
1101101111
Block 2
0001110110
0001110101
0001111010
0001111001
0111000110
0111000101
0111001010
0111001001
1010100110
1010100101
1010101010
1010101001
1100010110
1100010101
1100011010
1100011001
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GenStat code
factor [nvalue=32;levels=2] block,A,B,C,D,E,F,a,b,c,d
& [levels=4] wash, dryer
generate block,wash,dryer
blockstructure block/(wash*dryer)
treatmentstructure
(A+B+C+D+E+F)*(A+B+C+D+E+F)
+(a+b+c+d)*(a+b+c+d)
+(A+B+C+D+E+F)*(a+b+c+d)
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matrix [rows=10;
columns=5; values=“
b r1 r2 c1 c2"
0, 0, 1, 0, 0,
0, 1, 0, 0, 0,
0, 1, 1, 0, 0,
1, 0, 1, 0, 0,
1, 1, 0, 0, 0,
1, 1, 1, 0, 0,
0, 0, 0, 0, 1,
1, 0, 0, 0, 1,
1, 0, 0, 1, 0,
0, 0, 0, 1, 0] Mkey
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Akey [blockfactors=block,wash,dryer; Key=Mkey;
rowprimes=!(10(2));colprimes=!(5(2));
colmappings=!(1,2,2,3,3)]
Pdesign
Arandom [blocks=block/(wash*dryer);seed=12345]
PDESIGN
ANOVA
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Outline










Introduction; randomization and blocking
Some mathematical preliminaries
Linear models
Block structures; strata, null ANOVA
Computation of estimates; ANOVA table
Orthogonal designs
Non-orthogonal designs
Factorial designs
Response surface methodology
Other topics as time permits
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