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Design and Analysis of Experiments. Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC. Blocking and Confounding in Two-Level Factorial Designs. Dr. Tai- Yue Wang Department of Industrial and Information Management

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Design and analysis of experiments

Design and Analysis of Experiments

Dr. Tai-Yue Wang

Department of Industrial and Information Management

National Cheng Kung University

Tainan, TAIWAN, ROC


Blocking and confounding in two level factorial designs

Blocking and Confounding in Two-Level Factorial Designs

Dr. Tai-Yue Wang

Department of Industrial and Information Management

National Cheng Kung University

Tainan, TAIWAN, ROC


Outline
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 2pBlocks

  • Partial Confounding


Introduction
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


Introduction1
Introduction

  • Two cases are considered

    • Replicated designs

    • Unreplicated designs


Blocking a replicated 2 k factorial design
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 2 k factorial design example
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 2 k factorial design example1
Blocking a Replicated 2k Factorial Design -- example

  • Sum ofSquares in Block

  • ANOVA


Confounding in the 2 k factorial design
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.


Confounding in the 2 k factorial design1
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.


Confounding the 2 k factorial design in two blocks
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.


Confounding the 2 k factorial design in two blocks1
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


Confounding the 2 k factorial design in two blocks2
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


Confounding the 2 k factorial design in two blocks3
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


Confounding the 2 k factorial design in two blocks4
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.


Confounding the 2 k factorial design in two blocks5
Confounding the 2k Factorial Design in Two Blocks

  • We could confound any 2k design in two blocks.

  • Three factors example


Confounding the 2 k factorial design in two blocks6
Confounding the 2k Factorial Design in Two Blocks

  • ABC is confounded with blocks

  • It is a random order within one block.


Confounding the 2 k factorial design in two blocks7
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


Confounding the 2 k factorial design in two blocks8
Confounding the 2k Factorial Design in Two Blocks

  • ANOVA

  • 32 observations


Confounding the 2 k factorial design in two blocks example
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.


Confounding the 2 k factorial design in two blocks example1
Confounding the 2k Factorial Design in Two Blocks --example


Confounding the 2 k factorial design in two blocks example2
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 Coef

Constant 60.063

Block -9.313

Temperature 21.625 10.812

Pressure 3.125 1.563

Conc. 9.875 4.938

Stir rate 14.625 7.313

Temperature*Pressure 0.125 0.063

Temperature*Conc. -18.125 -9.063

Temperature*Stir rate 16.625 8.313

Pressure*Conc. 2.375 1.188

Pressure*Stir rate -0.375 -0.188

Conc.*Stir rate -1.125 -0.562

Temperature*Pressure*Conc. 1.875 0.938

Temperature*Pressure*Stir rate 4.125 2.063

Temperature*Conc.*Stir rate -1.625 -0.812

Pressure*Conc.*Stir rate -2.625 -1.312

S = * PRESS = *


Confounding the 2 k factorial design in two blocks example3
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 P

Blocks 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


Confounding the 2 k factorial design in two blocks example4
Confounding the 2k Factorial Design in Two Blocks --example


Confounding the 2 k factorial design in two blocks example5
Confounding the 2k Factorial Design in Two Blocks --example


Confounding the 2 k factorial design in two blocks example adj
Confounding the 2k Factorial Design in Two Blocks –example(Adj)

ABCD

Factorial Fit: Filtration versus Block, Temperature, Conc., Stir rate

Estimated Effects and Coefficients for Filtration (coded units)

Term Effect Coef SE Coef T P

Constant 60.063 1.141 52.63 0.000

Block -9.313 1.141 -8.16 0.000

Temperature 21.625 10.812 1.141 9.47 0.000

Conc. 9.875 4.938 1.141 4.33 0.002

Stir rate 14.625 7.313 1.141 6.41 0.000

Temperature*Conc. -18.125 -9.062 1.141 -7.94 0.000

Temperature*Stir rate 16.625 8.312 1.141 7.28 0.000

S = 4.56512 PRESS = 592.790

R-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 P

Blocks 1 1387.6 1387.6 1387.56 66.58 0.000

Main Effects 3 3116.2 3116.2 1038.73 49.84 0.000

2-Way Interactions 2 2419.6 2419.6 1209.81 58.05 0.000

Residual Error 9 187.6 187.6 20.84

Total 15 7110.9


Another illustration
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



Confounding the 2 k design in four blocks
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.


Confounding the 2 k design in four blocks1
Confounding the 2k design in four blocks

  • L1=x1+x4+x5

  • L2=x2+x3+x5

  • Pairs of L1 and L2 group into four blocks


Confounding the 2 k design in four blocks2
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


Confounding the 2 k design in 2 p blocks
Confounding the 2k design in 2pblocks

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


Confounding the 2 k design in 2 p blocks1
Confounding the 2k design in 2pblocks


Partial confounding
Partial Confounding

  • In Figure 7.3, it is a completely confounded case

  • ABC s confounded with blocks in each replicate.


Partial confounding1
Partial Confounding

  • Consider the case below, it is partial confounding.

  • ABC is confounded in replicate I and so on.


Partial confounding2
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



Partial confounding example
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.


Partial confounding example1
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.


Partial confounding example2
Partial Confounding-- example

  • How to create partial confounding in Minitab?


Partial confounding example3
Partial Confounding-- example

  • Replicate I is confounded with ABC

  • STAT>DOE>Factorial >Create Factorial Design


Partial confounding example4
Partial Confounding-- example

  • Design >Full Factorial

  • Number of blocks  2  OK


Partial confounding example5
Partial Confounding-- example

  • Factors > Fill in appropriate information

     OK  OK


Partial confounding example6
Partial Confounding-- example

  • Result of Replicate I (default is to confound with ABC)


Partial confounding example7
Partial Confounding-- example

  • Replicate II is confounded with AB

  • STAT>DOE>Factorial >Create Factorial Design

  • 2 level factorial (specify generators)


Partial confounding example8
Partial Confounding-- example

  • Design >Full Factorial


Partial confounding example9
Partial Confounding-- example

  • Generators …> Define blocks by listing …  AB

  • OK


Partial confounding example10
Partial Confounding-- example

  • Result of Replicate II


Partial confounding example11
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.


Partial confounding example12
Partial Confounding-- example


Partial confounding example13
Partial Confounding-- example

  • STAT> DOE> Factorial> Define Custom Factorial Design

  • Factors  Gap, Gas Flow, RF Power


Partial confounding example14
Partial Confounding-- example

  • Low/High > OK

  • Designs >Blocks>Specify by column  Blocks

  • OK


Partial confounding example15
Partial Confounding-- example

  • Now you can fill in collected data.


Partial confounding example16
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 P

Constant 776.06 12.63 61.46 0.000

Block 1 -22.94 28.23 -0.81 0.453

Block 2 -8.19 28.23 -0.29 0.783

Block 3 32.69 28.23 1.16 0.299

Gap -101.62 -50.81 12.63 -4.02 0.010

Gas Flow 7.38 3.69 12.63 0.29 0.782

RF 306.13 153.06 12.63 12.12 0.000

Gap*Gas Flow -42.00 -21.00 17.86 -1.18 0.293

Gap*RF -153.63 -76.81 12.63 -6.08 0.002

Gas Flow*RF -2.13 -1.06 12.63 -0.08 0.936

Gap*Gas Flow*RF -1.75 -0.87 17.86 -0.05 0.963

S = 50.5071 PRESS = 130609

R-Sq = 97.60% R-Sq(pred) = 75.42% R-Sq(adj) = 92.80%


Partial confounding example17
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 P

Blocks 3 4333 5266 1755 0.69 0.597

Main Effects 3 416378 416378 138793 54.41 0.000

2-Way Interactions 3 97949 97949 32650 12.80 0.009

3-Way Interactions 1 6 6 6 0.00 0.963

Residual Error 5 12755 12755 2551

Total 15 531421

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



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