department of mathematics and computer science 1212 1 6BV04 Screening Designs.

department of mathematics and computer science

Screening Designs

Contents• regression analysis and effects• 2p-experiments• blocks• 2p-k-experiments

(fractional factorial experiments)• software• literature

Three factors: exampleThree factors: example

Response: deviation filling height bottles

Factors: carbon dioxide level (%) Apressure (psi) Bspeed (bottles/min) C

Effects

How do we determine whether an individual factor is of importance?

Measure the outcome at 2 different settings of that factor.

Scale the settings such that they become the values +1 and -1.

setting factor A

measurement

setting factor A

measurement

setting factor A

measurement

effect

setting factor A

measurement

effect

N.B. effect = 2 * slope

setting factor A

measurement

35Effect factor A = 50 – 35 = 15

More factorsWe denote factors with capitals: A, B,…

Each factor only attains two settings: -1 and +1

The joint settings of all factors in one measurement is called a level combination.

More factors

LevelCombination

NotationA level combination consists of small letters. The small letters denote which factors are set at +1; the letters that do not appear are set at -1.

Example: ac means: A and C at 1, the remaining factors at -1

N.B. (1) means that all factors are set at -1.

An experiment consists of performing measurements at different level combinations.A run is a measurement at one level combination.

Suppose that there are 2 factors, A and B.

We perform 4 measurements with the following settings:• A -1 and B -1 (short: (1) )• A +1 and B -1 (short: a )

• A -1 and B +1 (short: b )• A +1 and B +1 (short: ab )

A 22 Experiment with 4 runs

A B yield

(1) -1 -1

b -1 1

a 1 -1

ab 1 1

Note:Note:

CAPITALS for factors and effects

small letters for level combinations ( = settings of the experiments)

(A, BC, CDEF)

(a, bc, cde, (1))

Graphical display

A-1 +1

2 estimates for effect A:

A-1 +1

2 estimates for effect A: 50 - 35 = 15

A-1 +1

60 - 40 = 20

50 - 35 = 15

A-1 +1

60 - 40 = 20

50 - 35 = 15

Which estimate is superior?

A-1 +1

60 - 40 = 20

50 - 35 = 15

Combine both estimates: ½(50-35) + ½(60-40) = 17.5

A-1 +1

In the same way we estimate the effect B(note that all 4 measurements are used!):

½(40-35) ½(60-50)+ = 7.5

A-1 +1

The interaction effect AB is the difference between the estimates for the effect A:

½(60-40) ½(50-35)- = 2.5

Interaction effects

Cross terms in linear regression models cause interaction effects:

Y = 3 + 2 xA + 4 xB + 7 xA xB

xA xA +1 YY + 2 + 7 xB,

so increase depends on xB. Likewise for xB xB+1

This explains the notation AB .

No interaction

Factor A

low high

B high

Interaction I

Factor A

low high

B high

Interaction II

Factor A

low high

B low55 50

B high

Interaction III

Factor A

low high

B high

Trick to Compute Effects

A B yield

(1) -1 -1 35

b -1 1 40

a 1 -1 50

ab 1 1 60

(coded)measurement settings

A B yield

(1) -1 -1 35

b -1 1 40

a 1 -1 50

ab 1 1 60

Effect estimates

A B yield

(1) -1 -1 35

b -1 1 40

a 1 -1 50

ab 1 1 60

Effect estimates

Effect A = ½(-35 - 40 + 50 + 60) = 17.5Effect B = ½(-35 + 40 – 50 + 60) = 7.5

A B AB yield

(1) -1 -1 ? 35

b -1 1 ? 40

a 1 -1 ? 50

ab 1 1 ? 60

Effect AB = ½(60-40) - ½(50-35) = 2.5

A B AB yield

(1) -1 -1 1 35

b -1 1 -1 40

a 1 -1 -1 50

ab 1 1 1 60

Effect AB = ½(60-40) - ½(50-35) = 2.5

× =× =× =× =

AB equals the product of the columns A and B

I A B AB yield

(1) + - - + 35

b + - + - 40

a + + - - 50

ab + + + + 60

Computational rules: I×A = A, I×B = B, A×B=AB etc.This holds true in general (i.e., also for more factors).

I A B AB C AC BC ABC(1)a

3 Factors: a 23 Design

A B C Yield

(1) - - - 5

a + - - 2

b - + - 7

ab + + - 1

c - - + 7

ac + - + 6

bc - + + 9

abc + + + 7

(1)=5 a=2

ab=1b=7

abc=7bc=9

effect A =

¼(+16-28)=-3

scheme 23 design

effect AB =

¼(+20-24)=-1

scheme 23 design

(1)=5 a=2

ab=1b=7

abc=7bc=9

Back to 2 factors – Blocking

I A B AB(1) + - - +b + - + -a + + - -

ab + + + +

Suppose that we cannot perform all measurements at the same day. We are not interested in the difference between 2 days, but we must take the effect of this into account. How do we accomplish that?

I A B AB day(1) + - - + 1b + - + - 1a + + - - 2

ab + + + + 2

Suppose that we cannot perform all measurements at the same day. We are not interested in the difference between 2 days, but we must take the effect of this into account. How do we accomplish that?

“hidden”blockeffect

I A B AB day(1) + - - + -b + - + - -a + + - - +

ab + + + + +

We note that the columns A and day are the same.

Consequence: the effect of A and the day effect cannot be distinguished. This is called confounding or aliasing).

I A B AB day(1) + - - + ?b + - + - ?a + + - - ?

ab + + + + ?

A general guide-line is to confound the day effect with an interaction of highest possible order. How can we accomplish that here?

Solution:day 1: a, b day 2: (1), abor interchange the days!

I A B AB day(1) + - - + +b + - + - -a + + - - -

ab + + + + +

Solution:day 1: a, b day 2: (1), abor interchange the days!

I A B AB day(1) + - - + +b + - + - -a + + - - -

ab + + + + +

Choose within the days by drawing lots which experiment must be performed first. In general, the order of experiments must be determined by drawing lots. This is called randomisation.

Here is a scheme for 3 factors. Interactions of order 3 or higher can be neglected in practice. How should we divide the experiments over 2 days?

I A B AB C AC BC ABC

Fractional experiments

Often the number of parameters is too large to allow a complete 2p design (i.e, all 2p possible settings -1 and 1 of the p factors).

By performing only a subset of the 2p

experiments in a smart way, we can arrange that by performing relatively few, it is possible to estimate the main effects and (possibly) 2nd order interactions.

I A B AB C AC BCABC

(1) + - - + - + + -

a + + - - - - + +

b + - + - - + - +

ab + + + + - - - -

c + - - + + - - +

ac + + - - + + - -

bc + - + - + - + -

abc + + + + + + + +

I A B AB C AC BCABC

(1) + - - + - + + -

a + + - - - - + +

b + - + - - + - +

ab + + + + - - - -

c + - - + + - - +

ac + + - - + + - -

bc + - + - + - + -

abc + + + + + + + +

I A B AB C AC BCABC

(1) + - - + - + + -

a + + - - - - + +

b + - + - - + - +

ab + + + + - - - -

I A B AB C AC BCABC

(1) + - - + - + + -

a + + - - - - + +

b + - + - - + - +

ab + + + + - - - -With this half fraction (only 4 = ½×8 experiments) we see that a number of columns are the same (apart from a minus sign):

I = -C, A = -AC, B = -BC, AB = -ABC

I A B AB C AC BCABC

(1) + - - + - + + -

a + + - - - - + +

b + - + - - + - +

ab + + + + - - - -We say that these factors are confounded or aliased. In this particular case we have an ill-chosen fraction, because I and C are confounded.

I = -C, A = -AC, B = -BC, AB = -ABC

Fractional experiments – Better Choice: I = ABC

I A B AB C AC BCABC

(1) + - - + - + + -

a + + - - - - + +

b + - + - - + - +

ab + + + + - - - -

c + - - + + - - +

ac + + - - + + - -

bc + - + - + - + -

abc + + + + + + + +

Fractional experiments – Better Choice: I = ABC

I A B AB C AC BCABC

a + + - - - - + +

b + - + - - + - +

c + - - + + - - +

abc + + + + + + + +

The other “best choice” would be: I = -ABC

Aliasing structure: I = ABC, A = BC, B = AC, C = AB

I A B AB C AC BCABC

a + + - - - - + +

b + - + - - + - +

c + - - + + - - +

abc + + + + + + + +

In the case of 3 factors further reducing the number of experiments is not possible in practice, because this leads to undesired confounding, e.g. : I = A = BC = ABC, B = C = AB = AC,

I A B AB C AC BCABC

a + + - - - - + +

abc + + + + + + + +

Other quarter fractions also have confounded main effects, which is unacceptable.

Further remarks on fractions

• there exist computational rules for aliases. E.g., it follows from A=C that AB = BC. Note that I = A2 = B2 = C2 etc. always holds (see the next lecture)

•tables and software are available for choosing a suitable fraction . The extent of confounding is indicated by the resolution. Resolution III is a minimal ; designs with a higher resolution are very much preferred.

Plackett-Burman designs

So far we discussed fractional designs for screening. This is sensible if one cannot exclude the possibility of interactions.

If one knows based on foreknowledge that there are no interactions or if one is for some reason is only interested in main effects, than Plackett-Burman designs are preferred. They are able to detect significant main effects using only very few runs. A disadvantage of these designs is their complicated aliasing structure.

Number of measurements

For every main or interaction effect that has to estimated separately, at least one measurement is necessary. If there are k blocks, then this requires additional k - 1 measurements. The remaining measurements are used for estimation of the variance. It is important to have sufficient measurements for the variance.

Choice of design

After a design has been chosen, the factors A, B, … must be assigned to the factors of the experiment. It is recommended to combine any foreknowledge on the factors with the alias structure. The individual measurements must be performed in a random order.

• never confound two effects that might both be significant

• if you know that a certain effect will not be significant,you can confound it with an effect that might be significant.

Centre points and Replications

If there are not enough measurements to obtain a good estimate of the variance, then one can perform replications. Another possibility is to add centre points .

A-1 +1

bAdding centre points serves two purposes:• better variance estimate• allow to test curvature using a lack-of-fit test

Centre point

Curvature

A design in which each factor is only allowed to attain the levels -1 and 1, is implicitly assuming a linear model. This is because knowing only the functions values at -1 and +1, then 1 and x2 cannot be distinguished. We can distinguish them by adding the level 0. This is the idea behind adding centre points.

Analysis of a Design

A B C Yield

(1) - - - 5

a + - - 2

b - + - 7

ab + + - 1

c - - + 7

ac + - + 6

bc - + + 9

abc + + + 7

Analysis of a Design – With 2-way InteractionsAnalysis Summary----------------File name: <Untitled>

Estimated effects for Yield----------------------------------------------------------------------average = 5.5 +/- 0.25A:A = -3.0 +/- 0.5B:B = 1.0 +/- 0.5C:C = 3.5 +/- 0.5AB = -1.0 +/- 0.5AC = 1.5 +/- 0.5BC = 0.5 +/- 0.5----------------------------------------------------------------------Standard errors are based on total error with 1 d.f.

Analysis of a Design – With 2-way InteractionsAnalysis of Variance for Yield--------------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value--------------------------------------------------------------------------------A:A 18.0 1 18.0 36.00 0.1051B:B 2.0 1 2.0 4.00 0.2952C:C 24.5 1 24.5 49.00 0.0903AB 2.0 1 2.0 4.00 0.2952AC 4.5 1 4.5 9.00 0.2048BC 0.5 1 0.5 1.00 0.5000Total error 0.5 1 0.5--------------------------------------------------------------------------------Total (corr.) 52.0 7

R-squared = 99.0385 percentR-squared (adjusted for d.f.) = 93.2692 percentStandard Error of Est. = 0.707107Mean absolute error = 0.25Durbin-Watson statistic = 2.5Lag 1 residual autocorrelation = -0.375

Analysis of a Design – Only Main EffectsAnalysis Summary----------------File name: <Untitled>

Estimated effects for Yield----------------------------------------------------------------------average = 5.5 +/- 0.484123A:A = -3.0 +/- 0.968246B:B = 1.0 +/- 0.968246C:C = 3.5 +/- 0.968246----------------------------------------------------------------------Standard errors are based on total error with 4 d.f.

Effect estimates remain the same!

Analysis of a Design – Only Main EffectsAnalysis of Variance for Yield--------------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value--------------------------------------------------------------------------------A:A 18.0 1 18.0 9.60 0.0363B:B 2.0 1 2.0 1.07 0.3601C:C 24.5 1 24.5 13.07 0.0225Total error 7.5 4 1.875--------------------------------------------------------------------------------Total (corr.) 52.0 7

R-squared = 85.5769 percentR-squared (adjusted for d.f.) = 74.7596 percentStandard Error of Est. = 1.36931Mean absolute error = 0.8125Durbin-Watson statistic = 2.16667 (P=0.3180)Lag 1 residual autocorrelation = -0.125

Analysis of a Design with Blocks

Block A B C Yield

(1) 1 - - - 5

ab 1 + + - 1

ac 1 + - + 6

bc 1 - + + 9

a 2 + - - 2

b 2 - + - 7

c 2 - - + 7

abc 2 + + + 7

Analysis of a Design with Blocks – With 2-way Interactions

Analysis of Variance for Yield--------------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value--------------------------------------------------------------------------------A:A 18.0 1 18.0B:B 2.0 1 2.0C:C 24.5 1 24.5AB 2.0 1 2.0AC 4.5 1 4.5BC 0.5 1 0.5blocks 0.5 1 0.5Total error 0.0 0--------------------------------------------------------------------------------Total (corr.) 52.0 7

R-squared = 100.0 percentR-squared (adjusted for d.f.) = 100.0 percent

Saturated design: 0 df for the error term → no testing possible

Analysis of a Design with Blocks – Only Main Effects

Analysis of Variance for Yield--------------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value--------------------------------------------------------------------------------A:A 18.0 1 18.0 7.71 0.0691B:B 2.0 1 2.0 0.86 0.4228C:C 24.5 1 24.5 10.50 0.0478blocks 0.5 1 0.5 0.21 0.6749Total error 7.0 3 2.33333--------------------------------------------------------------------------------Total (corr.) 52.0 7

R-squared = 86.5385 percentR-squared (adjusted for d.f.) = 76.4423 percentStandard Error of Est. = 1.52753Mean absolute error = 0.75Durbin-Watson statistic = 3.21429 (P=0.0478)Lag 1 residual autocorrelation = -0.642857

Analysis of a Fractional Design (I = -ABC)

A B C Yield

(1) - - - 5

ac + - + 6

bc - + + 9

ab + + - 1

Analysis of a Fractional Design (I = -ABC)Analysis of Variance for Yield--------------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value--------------------------------------------------------------------------------A:A-BC 12.25 1 12.25B:B-AC 0.25 1 0.25C:C-AB 20.25 1 20.25Total error 0.0 0--------------------------------------------------------------------------------Total (corr.) 32.75 3

R-squared = 100.0 percentR-squared (adjusted for d.f.) = 0.0 percent

Estimated effects for Yield----------------------------------------------------------------------average = 5.25A:A-BC = -3.5B:B-AC = -0.5C:C-AB = 4.5 ----------------------------------------------------------------------No degrees of freedom left to estimate standard errors.

A B Yield

(1) - - 5

a + - 6

b - + 9

ab + + 1

Pure Error =3

1 1( )

3 1 3ii

Analysis of a Design with Centre Points

Analysis of Variance for Yield--------------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value--------------------------------------------------------------------------------A:A 12.25 1 12.25 36.75 0.0261B:B 0.25 1 0.25 0.75 0.4778AB 20.25 1 20.25 60.75 0.0161Lack-of-fit 10.0119 1 10.0119 30.04 0.0317Pure error 0.666667 2 0.333333--------------------------------------------------------------------------------Total (corr.) 43.4286 6

R-squared = 75.4112 percentR-squared (adjusted for d.f.) = 50.8224 percentStandard Error of Est. = 0.57735Mean absolute error = 1.18367Durbin-Watson statistic = 0.801839 (P=0.1157)Lag 1 residual autocorrelation = 0.524964

P-Value < 0.05→

Lack-of-fit!

Software• Statgraphics: menu Special -> Experimental Design

• StatLab: http://www.win.tue.nl/statlab2/

• Design Wizard (illustrates blocks and fractions): http://www.win.tue.nl/statlab2/designApplet.html

• Box (simple optimization illustration): http://www.win.tue.nl/~marko/box/box.html

Literature• J. Trygg and S. Wold, Introduction to Experimental

Design – What is it? Why and Where is it Useful?, homepage of chemometrics, editorial August 2002: www.acc.umu.se/~tnkjtg/Chemometrics/editorial/aug2002.html

• Introduction from moresteam.com: www.moresteam.com/toolbox/t408.cfm

• V. Czitrom, One-Factor-at-a-Time Versus Designed Experiments, American Statistician 53 (1999), 126-131

• Thumbnail Handbook for Factorial DOE, StatEase

department of mathematics and computer science 1212 1 6BV04 Screening Designs.

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