Design and Analysis of Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Blocking and Confounding in Two-

Level Factorial Designs

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Outline Introduction Blocking Replicated 2k factorial Design Confounding in 2k factorial Design Confounding the 2k factorial Design in Two

Blocks Why Blocking is Important Confounding the 2k factorial Design in Four

Blocks Confounding the 2k factorial Design in 2p Blocks Partial Confounding

Introduction Sometimes it is impossible to perform all of

runs in one batch of material Or to ensure the robustness, one might

deliberately vary the experimental conditions to ensure the treatment are equally effective.

Blocking is a technique for dealing with controllable nuisance variables

Introduction Two cases are considered

Replicated designs Unreplicated designs

Blocking a Replicated 2k Factorial Design

A 2k design has been replicated n times. Each set of nonhomogeneous conditions

defines a block Each replicate is run in one of the block The runs in each block would be made in

random order.

Blocking a Replicated 2k Factorial Design -- example

Only four experiment trials can be made from a single batch. Three batch of raw material are required.

Blocking a Replicated 2k Factorial Design -- example

Sum of Squares in Block

ANOVA

2 23...

1 4 126.50

iBlocks

i

B ySS

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Confounding in The 2k Factorial Design

Problem: Impossible to perform a complete replicate of a factorial design in one block

Confounding is a design technique for arranging a complete factorial design in blocks, where block size is smaller than the number of treatment combinations in one replicate.

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Confounding in The 2k Factorial Design

Short comings: Cause information about certain treatment effects (usually high order interactions ) to be indistinguishable from, or confounded with, blocks.

If the case is to analyze a 2k factorial design in 2p incomplete blocks, where p<k, one can use runs in two blocks (p=1), four blocks (p=2), and so on.

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Confounding the 2k Factorial Design in Two Blocks

Suppose we want to run a single replicate of the 22 design. Each of the 22=4 treatment combinations requires a quantity of raw material, for example, and each batch of raw material is only large enough for two treatment combinations to be tested.

Two batches are required.

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Confounding the 2k Factorial Design in Two Blocks

One can treat batches as blocks One needs assign two of the four treatment

combinations to each blocks

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Confounding the 2k Factorial Design in Two Blocks

The order of the treatment combinations are run within one block is randomly selected.

For the effects, A and B:

A=1/2[ab+a-b-(1)]B=1/2[ab-a+b-(1)]Are unaffected

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Confounding the 2k Factorial Design in Two Blocks

For the effects, AB:AB=1/2[ab-a-b+(1)]is identical to block effect AB is confounded with blocks

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Confounding the 2k Factorial Design in Two Blocks

We could assign the block effects to confounded with A or B.

However we usually want to confound with higher order interaction effects.

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Confounding the 2k Factorial Design in Two Blocks

We could confound any 2k design in two blocks. Three factors example

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Confounding the 2k Factorial Design in Two Blocks

ABC is confounded with blocks It is a random order within one block.

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Confounding the 2k Factorial Design in Two Blocks

Multiple replicates are required to obtain the estimate error when k is small.

For example, 23 design with four replicate in two blocks

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Confounding the 2k Factorial Design in Two Blocks

ANOVA 32 observations

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Confounding the 2k Factorial Design in Two Blocks --example

Same as example 6.2 Four factors: Temperature, pressure,

concentration, and stirring rate. Response variable: filtration rate. Each batch of material is nough for 8 treatment

combinations only. This is a 24 design n two blocks.

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Confounding the 2k Factorial Design in Two Blocks --example

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Confounding the 2k Factorial Design in Two Blocks --example

Factorial Fit: Filtration versus Block, Temperature, Pressure, ... Estimated Effects and Coefficients for Filtration (coded units)Term Effect CoefConstant 60.063Block -9.313Temperature 21.625 10.812Pressure 3.125 1.563Conc. 9.875 4.938Stir rate 14.625 7.313Temperature*Pressure 0.125 0.063Temperature*Conc. -18.125 -9.063Temperature*Stir rate 16.625 8.313Pressure*Conc. 2.375 1.188Pressure*Stir rate -0.375 -0.188Conc.*Stir rate -1.125 -0.562Temperature*Pressure*Conc. 1.875 0.938Temperature*Pressure*Stir rate 4.125 2.063Temperature*Conc.*Stir rate -1.625 -0.812Pressure*Conc.*Stir rate -2.625 -1.312

S = * PRESS = *

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Confounding the 2k Factorial Design in Two Blocks --example

Factorial Fit: Filtration versus Block, Temperature, Pressure, ...

Analysis of Variance for Filtration (coded units)

Source DF Seq SS Adj SS Adj MS F PBlocks 1 1387.6 1387.6 1387.56 * *Main Effects 4 3155.2 3155.2 788.81 * *2-Way Interactions 6 2447.9 2447.9 407.98 * *3-Way Interactions 4 120.2 120.2 30.06 * *Residual Error 0 * * *Total 15 7110.9

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Confounding the 2k Factorial Design in Two Blocks --example

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Confounding the 2k Factorial Design in Two Blocks --example

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Confounding the 2k Factorial Design in Two Blocks –example(Adj)

Factorial Fit: Filtration versus Block, Temperature, Conc., Stir rate Estimated Effects and Coefficients for Filtration (coded units)Term Effect Coef SE Coef T PConstant 60.063 1.141 52.63 0.000Block -9.313 1.141 -8.16 0.000Temperature 21.625 10.812 1.141 9.47 0.000Conc. 9.875 4.938 1.141 4.33 0.002Stir rate 14.625 7.313 1.141 6.41 0.000Temperature*Conc. -18.125 -9.062 1.141 -7.94 0.000Temperature*Stir rate 16.625 8.312 1.141 7.28 0.000

S = 4.56512 PRESS = 592.790R-Sq = 97.36% R-Sq(pred) = 91.66% R-Sq(adj) = 95.60%

Analysis of Variance for Filtration (coded units)Source DF Seq SS Adj SS Adj MS F PBlocks 1 1387.6 1387.6 1387.56 66.58 0.000Main Effects 3 3116.2 3116.2 1038.73 49.84 0.0002-Way Interactions 2 2419.6 2419.6 1209.81 58.05 0.000Residual Error 9 187.6 187.6 20.84Total 15 7110.9

ABCD

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Another Illustration Assuming we don’t have blocking in previous

example, we will not be able to notice the effect AD.

Now the first eight runs (in run order) have filtration rate reduced by 20 units

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Another Illustration

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Confounding the 2k design in four blocks

2k factorial design confounded in four blocks of 2k-2 observations.

Useful if k 4 and block sizes are relatively ≧small.

Example 25 design in four blocks, each block with eight runs.

Select two factors to be confound with, say ADE and BCE.

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Confounding the 2k design in four blocks

L1=x1+x4+x5

L2=x2+x3+x5

Pairs of L1 and L2 group into four blocks

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Confounding the 2k design in four blocks

Example: L1=1, L2=1 block 4 abcde: L1=x1+x4+x5=1+1+1=3(mod 2)=1

L2=x2+x3+x5=1+1+1=3(mod 2)=1

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Confounding the 2k design in 2p blocks

2k factorial design confounded in 2p blocks of 2k-p observations.

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Confounding the 2k design in 2p blocks

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Partial Confounding In Figure 7.3, it is a completely confounded

case ABC s confounded with blocks in each

replicate.

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Partial Confounding Consider the case below, it is partial

confounding.

ABC is confounded in replicate I and so on.

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Partial Confounding As a result, information on ABC can be

obtained from data in replicate II, II, IV, and so on.

We say ¾ of information can be obtained on the interactions because they are unconfounded in only three replicates.

¾ is the relative information for the confounded effects

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Partial Confounding ANOVA

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Partial Confounding-- example From Example 6.1 Response variable: etch rate Factors: A=gap, B=gas flow, C=RF power. Only four treatment combinations can be

tested during a shift. There is shift-to-shift difference in etch

performance. The experimenter decide to use shift as a blocking factor.

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Partial Confounding-- example Each replicate of the 23 design must be run

in two blocks. Two replicates are run. ABC is confounded in replicate I and AB is

confounded in replicate II.

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Partial Confounding-- example How to create partial confounding in

Minitab?

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Partial Confounding-- example Replicate I is confounded with ABC STAT>DOE>Factorial >Create Factorial

Design

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Partial Confounding-- example Design >Full Factorial Number of blocks 2 OK

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Partial Confounding-- example Factors > Fill in appropriate information OK OK

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Partial Confounding-- example Result of Replicate I (default is to confound

with ABC)

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Partial Confounding-- example Replicate II is confounded with AB STAT>DOE>Factorial >Create Factorial

Design 2 level factorial (specify generators)

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Partial Confounding-- example Design >Full Factorial

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Partial Confounding-- example Generators …> Define blocks by listing … AB

OK

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Partial Confounding-- example Result of Replicate II

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Partial Confounding-- example To combine the two design in one worksheet

Change block number 3 -> 1, 2 -> 4 in Replicate II Copy columns of CenterPt, Gap, …RF Power from

Replicate II to below the corresponding columns in Replicate I.

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Partial Confounding-- example

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Partial Confounding-- example STAT> DOE> Factorial> Define Custom Factorial

Design Factors Gap, Gas Flow, RF Power

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Partial Confounding-- example Low/High > OK Designs >Blocks>Specify by column Blocks OK

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Partial Confounding-- example Now you can fill in collected data.

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Partial Confounding-- example ANOVA

Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF Estimated Effects and Coefficients for Etch Rate (coded units)Term Effect Coef SE Coef T PConstant 776.06 12.63 61.46 0.000Block 1 -22.94 28.23 -0.81 0.453Block 2 -8.19 28.23 -0.29 0.783Block 3 32.69 28.23 1.16 0.299Gap -101.62 -50.81 12.63 -4.02 0.010Gas Flow 7.38 3.69 12.63 0.29 0.782RF 306.13 153.06 12.63 12.12 0.000Gap*Gas Flow -42.00 -21.00 17.86 -1.18 0.293Gap*RF -153.63 -76.81 12.63 -6.08 0.002Gas Flow*RF -2.13 -1.06 12.63 -0.08 0.936Gap*Gas Flow*RF -1.75 -0.87 17.86 -0.05 0.963

S = 50.5071 PRESS = 130609R-Sq = 97.60% R-Sq(pred) = 75.42% R-Sq(adj) = 92.80%

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Partial Confounding-- example ANOVA

Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF

Analysis of Variance for Etch Rate (coded units)

Source DF Seq SS Adj SS Adj MS F PBlocks 3 4333 5266 1755 0.69 0.597Main Effects 3 416378 416378 138793 54.41 0.0002-Way Interactions 3 97949 97949 32650 12.80 0.0093-Way Interactions 1 6 6 6 0.00 0.963Residual Error 5 12755 12755 2551Total 15 531421

* NOTE * There is partial confounding, no alias table was printed.

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Partial Confounding-- example ANOVA