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Design and Analysis of Experiments

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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

National Cheng Kung University

Tainan, TAIWAN, ROC

- 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

- 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

- Two cases are considered
- Replicated designs
- Unreplicated designs

- 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.

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

- Sum ofSquares in Block
- ANOVA

- 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.

- 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.

- 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.

- One can treat batches as blocks
- One needs assign two of the four treatment combinations to each 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

- For the effects, AB:
AB=1/2[ab-a-b+(1)]

is identical to block effect

ïƒ AB is confounded with blocks

- We could assign the block effects to confounded with A or B.
- However we usually want to confound with higher order interaction effects.

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

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

- Multiple replicates are required to obtain the estimate error when k is small.
- For example, 23 design with four replicate in two blocks

- ANOVA
- 32 observations

- 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.

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 = *

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

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

- Assuming we donâ€™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

- 2k factorial design confounded in four blocks of 2k-2 observations.
- Useful if k â‰§ 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.

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

- Example: L1=1, L2=1 ïƒ block 4
- abcde: L1=x1+x4+x5=1+1+1=3(mod 2)=1L2=x2+x3+x5=1+1+1=3(mod 2)=1

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

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

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

- As a result, information on ABC can be obtained from data in replicate II, II, IV, and so on.
- We say Â¾ of information can be obtained on the interactions because they are unconfounded in only three replicates.
- Â¾ is the relative information for the confounded effects

- ANOVA

- 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.

- 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.

- How to create partial confounding in Minitab?

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

- Design >Full Factorial
- Number of blocks ïƒ 2 ïƒ OK

- Factors > Fill in appropriate information
ïƒ OK ïƒ OK

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

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

- Design >Full Factorial

- Generators â€¦> Define blocks by listing â€¦ ïƒ AB
- OK

- Result of Replicate II

- To combine the two design in one worksheet
- Change block number 3 -> 1, 2 -> 4 in Replicate II
- Copy columns of CenterPt, Gap, â€¦RF Power from Replicate II to below the corresponding columns in Replicate I.

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

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

- Now you can fill in collected data.

- 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%

- 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 MSF 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.

- ANOVA