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### Design and Analysis of Experiments

### 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

Dr. Tai-Yue Wang

Department of Industrial and Information Management

National Cheng Kung University

Tainan, TAIWAN, ROC

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

- 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 ofSquares in Block
- ANOVA

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 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 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 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 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 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 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 2k Factorial Design in Two Blocks

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

Confounding the 2k Factorial Design in Two Blocks

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

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 2k Factorial Design in Two Blocks

- ANOVA
- 32 observations

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 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 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 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

- 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 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 2k design in four blocks

- L1=x1+x4+x5
- L2=x2+x3+x5
- Pairs of L1 and L2 group into four blocks

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 2k design in 2pblocks

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

Partial Confounding

- In Figure 7.3, it is a completely confounded case
- ABC s confounded with blocks in each replicate.

Partial Confounding

- Consider the case below, it is partial confounding.
- ABC is confounded in replicate I and so on.

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

- ANOVA

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-- 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-- example

- How to create partial confounding in Minitab?

Partial Confounding-- example

- Replicate I is confounded with ABC
- STAT>DOE>Factorial >Create Factorial Design

Partial Confounding-- example

- Design >Full Factorial
- Number of blocks 2 OK

Partial Confounding-- example

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

Partial Confounding-- example

- Replicate II is confounded with AB
- STAT>DOE>Factorial >Create Factorial Design
- 2 level factorial (specify generators)

Partial Confounding-- example

- Design >Full Factorial

Partial Confounding-- example

- Generators …> Define blocks by listing … AB
- OK

Partial Confounding-- example

- Result of Replicate II

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-- example

- STAT> DOE> Factorial> Define Custom Factorial Design
- Factors Gap, Gas Flow, RF Power

Partial Confounding-- example

- Low/High > OK
- Designs >Blocks>Specify by column Blocks
- OK

Partial Confounding-- example

- Now you can fill in collected data.

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-- 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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