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Stat 232

Experimental Design

Spring 2008

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Ching-Shui Cheng

Office:  419 Evans HallPhone:  642-9968Email: 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

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.

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

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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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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

Start with an initial design

Randomly permute (labels of) the experimental units

Complete randomization: Pick one of the 72! Permutationsrandomly

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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

4 treatments

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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Randomized complete block design

After randomization:

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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 ＊ columnLatin 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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A α B β C γ D δ

B γ A δ D α C β

C δ D γ A β B α

D β C α B δ A γ

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 particularcomplex-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, sixfactors 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 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 10 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 10 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 00 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 10 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 00 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 11 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 01 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 01 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 11 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 01 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 [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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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

Outline