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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 1/33 Blocking and Confounding in TwoLevel Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC 2/33 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 Bi2 y...2 12 i 1 4 6.50 3 SS Blocks 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. 9 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. 10 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. 11 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 12 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 13 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 14 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. 15 Confounding the 2k Factorial Design in Two Blocks We could confound any 2k design in two blocks. Three factors example 16 Confounding the 2k Factorial Design in Two Blocks ABC is confounded with blocks It is a random order within one block. 17 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 18 Confounding the 2k Factorial Design in Two Blocks ANOVA 32 observations 19 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. 20 Confounding the 2k Factorial Design in Two Blocks --example 21 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 = * 22 Confounding the 2k Factorial Design in Two Blocks --example Factorial Fit: Filtration versus Block, Temperature, Pressure, ... Analysis of Variance for Filtration (coded units) Source Blocks Main Effects 2-Way Interactions 3-Way Interactions Residual Error Total DF 1 4 6 4 0 15 Seq SS 1387.6 3155.2 2447.9 120.2 * 7110.9 Adj SS 1387.6 3155.2 2447.9 120.2 * Adj MS 1387.56 788.81 407.98 30.06 * F * * * * P * * * * 23 Confounding the 2k Factorial Design in Two Blocks --example 24 Confounding the 2k Factorial Design in Two Blocks --example 25 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 Constant 60.063 1.141 52.63 Block -9.313 1.141 -8.16 Temperature 21.625 10.812 1.141 9.47 Conc. 9.875 4.938 1.141 4.33 Stir rate 14.625 7.313 1.141 6.41 Temperature*Conc. -18.125 -9.062 1.141 -7.94 Temperature*Stir rate 16.625 8.312 1.141 7.28 P 0.000 0.000 0.000 0.002 0.000 0.000 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 Blocks 1 1387.6 Main Effects 3 3116.2 2-Way Interactions 2 2419.6 Residual Error 9 187.6 Total 15 7110.9 Adj SS 1387.6 3116.2 2419.6 187.6 Adj MS 1387.56 1038.73 1209.81 20.84 F 66.58 49.84 58.05 P 0.000 0.000 0.000 26 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 27 Another Illustration 28 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. 29 Confounding the 2k design in four blocks L1=x1+x4+x5 L2=x2+x3+x5 Pairs of L1 and L2 group into four blocks 30 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 31 Confounding the 2k design in 2p blocks 2k factorial design confounded in 2p blocks of 2k-p observations. 32 Confounding the 2k design in 2p blocks 33 Partial Confounding In Figure 7.3, it is a completely confounded case ABC s confounded with blocks in each replicate. 34 Partial Confounding Consider the case below, it is partial confounding. ABC is confounded in replicate I and so on. 35 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 36 Partial Confounding ANOVA 37 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. 38 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. 39 Partial Confounding-- example How to create partial confounding in Minitab? 40 Partial Confounding-- example Replicate I is confounded with ABC STAT>DOE>Factorial >Create Factorial Design 41 Partial Confounding-- example Design >Full Factorial Number of blocks 2 OK 42 Partial Confounding-- example Factors > Fill in appropriate information OK OK 43 Partial Confounding-- example Result of Replicate I (default is to confound with ABC) 44 Partial Confounding-- example Replicate II is confounded with AB STAT>DOE>Factorial >Create Factorial Design 2 level factorial (specify generators) 45 Partial Confounding-- example Design >Full Factorial 46 Partial Confounding-- example Generators …> Define blocks by listing … AB OK 47 Partial Confounding-- example Result of Replicate II 48 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. 49 Partial Confounding-- example 50 Partial Confounding-- example STAT> DOE> Factorial> Define Custom Factorial Design Factors Gap, Gas Flow, RF Power 51 Partial Confounding-- example Low/High > OK Designs >Blocks>Specify by column Blocks OK 52 Partial Confounding-- example Now you can fill in collected data. 53 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 Constant 776.06 12.63 Block 1 -22.94 28.23 Block 2 -8.19 28.23 Block 3 32.69 28.23 Gap -101.62 -50.81 12.63 Gas Flow 7.38 3.69 12.63 RF 306.13 153.06 12.63 Gap*Gas Flow -42.00 -21.00 17.86 Gap*RF -153.63 -76.81 12.63 Gas Flow*RF -2.13 -1.06 12.63 Gap*Gas Flow*RF -1.75 -0.87 17.86 T 61.46 -0.81 -0.29 1.16 -4.02 0.29 12.12 -1.18 -6.08 -0.08 -0.05 P 0.000 0.453 0.783 0.299 0.010 0.782 0.000 0.293 0.002 0.936 0.963 S = 50.5071 PRESS = 130609 R-Sq = 97.60% R-Sq(pred) = 75.42% R-Sq(adj) = 92.80% 54 Partial Confounding-- example ANOVA Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF Analysis of Variance for Etch Rate (coded units) Source Blocks Main Effects 2-Way Interactions 3-Way Interactions Residual Error Total DF 3 3 3 1 5 15 Seq SS 4333 416378 97949 6 12755 531421 Adj SS 5266 416378 97949 6 12755 Adj MS 1755 138793 32650 6 0.00 2551 F 0.69 54.41 12.80 0.963 P 0.597 0.000 0.009 * NOTE * There is partial confounding, no alias table was printed. 55 Partial Confounding-- example ANOVA 56