Stat 232

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• Stat 232
• Experimental Design
• Spring 2008

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• Ching-Shui Cheng
• Office  419 Evans HallPhone  642-9968Email
  cheng_at_stat.berkeley.edu
• Office Hours Tu Th 200-300 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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Fishers solution Randomization

• In the same field and same year, apply fertilizer
  to randomly spaced plots within the field.
• This averages out the effect of variation within
  the field in drainage and soil composition on
  yield, as well as controlling for weather, etc.

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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
Randomly permute (labels of) the experimental
units Complete randomization Pick one of the
72! Permutations randomly
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4 treatments
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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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 companys 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 washersa,b,c,d
configurations of dryers

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• Block structure2 blocks/(4 washers4 dryers)

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Block 1 Block 2

• 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0
• 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1
• 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 0
• 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1
• 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0
• 0 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1
• 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0
• 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 1
• 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0
• 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1
• 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0
• 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1
• 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0
• 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 1
  0 1 0 1
• 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1
  1 0 1 0
• 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1
  1 0 0 1

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• GenStat code
• factor nvalue32levels2 block,A,B,C,D,E,F,a,b,
  c,d
• levels4 wash, dryer
• generate block,wash,dryer
• blockstructure block/(washdryer)
• treatmentstructure
• (ABCDEF)(ABCDEF)
• (abcd)(abcd)
• (ABCDEF)(abcd)

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matrix rows10 columns5 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 blockfactorsblock,wash,dryer KeyMkey
• rowprimes!(10(2))colprimes!(5(2))
  colmappings!(1,2,2,3,3)
• Pdesign
• Arandom blocksblock/(washdryer)seed12345
• 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