ISE 426-526 - Lecture 7 - Single Replicates of 2^k Designs

# ISE 426-526 - Lecture 7 - Single Replicates of 2^k Designs...

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1 ISE 426/526 Lecture 7 - 2007 Single Replicate of a 2 k Design A Single Replicate of the 2 k Design • The Problem? – No internal estimate of error (pure error) • The Question – How do we analyze this type of an experiment?

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2 Strategy 1 – Assume higher order interactions are negligible and combine their mean squares (MS) to estimate the error. – Based on the sparsity of effects principle • Most systems dominated by some of the main effects and lower order interactions Strategy 2 • Normal probability plot of the effects estimates • Daniel * concluded that the negligible effects will be N(0, σ 2 ) • Significant effects will not lie on the line • Combine negligible effects and use as estimate of error. * Daniel, C. (1959). “Use of Half-Normal Plots in Interpreting Factorial Two Level Experiments,” Technometrics , Vol. 1, pp. 311-342.
3 Example 6-2 A Single Replicate of the 2 4 Design • Objectives – Primary - Increase Filtration Rate – Secondary - Reduce Formaldehyde Concentration • Design – CRD w/ 2 4 Factorial (Single Replicate) – Temperature [A] – Pressure [B] – Concentration of Formaldehyde [C] – Stirring Rate [D] • Response – Filtration Rate Effects Estimates & SS Effect Percent 20 10 0 -10 -20 99 95 90 80 70 60 50 40 30 20 10 5 1 Factor D Name AA BB CC D Effect Type Not Significant Significant AD AC D C A Normal Probability Plot of the Effects (response is FR, Alpha = .10) Lenth's PSE = 2.625 Figure 6-11, pg. 230 & Table 6-12, pg. 229 Term Effect Sum of Squares A 21.625 1870.56 B 3.125 39.06 C 9.875 390.06 D 14.625 855.56 A*B 0.125 0.06 A*C -18.125 1314.06 A*D 16.625 1105.56 B*C 2.375 22.56 B*D -0.375 0.56 C*D -1.125 5.06 A*B*C 1.875 14.06 A*B*D 4.125 58.06 A*C*D -1.625 10.56 B*C*D -2.625 27.56 A*B*C*D 1.375 7.56 5720.9

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4 Cube Plot 1 -1 1 -1 1 -1 1 -1 D C B A 96 86 75 70 104 100 43 45 65 60 68 80 65 71 45 48 Cube Plot (data means) for Filtration Rate Main Effect & Interaction Plots A 1 -1 1 -1 1 -1 100 75 50 B 100 75 50 C 100 75 50 D A -1 1 B -1 1 C -1 1 Interaction Plot (data means) for FR Mean of FR 1 -1 80 75 70 65 60 1 -1 1 -1 80 75 70 65 60 1 -1 A B C D Main Effects Plot (data means) for FR Figure 6-12, pg. 230
5 Selecting Terms in Minitab • Stat/DOE/Analyze Factorial Design/ Terms Analysis of Variance Estimated Effects and Coefficients for FR (coded units) Term Effect Coef SE Coef T P Constant 70.063 1.184 59.16 0.000 A 21.625 10.812 1.184 9.13 0.000 C 9.875 4.938 1.184 4.17 0.003 D 14.625 7.312 1.184 6.18 0.000 A*C -18.125 -9.062 1.184 -7.65 0.000 A*D 16.625 8.313 1.184 7.02 0.000 C*D -1.125 -0.562 1.184 -0.48 0.647 A*C*D -1.625 -0.813 1.184 -0.69 0.512 S = 4.73682 R-Sq = 96.87% R-Sq(adj) = 94.13% Analysis of Variance for FR (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 3116.19 3116.19 1038.73 46.29 0.000 2-Way Interactions 3 2424.69 2424.69 808.23 36.02 0.000 3-Way Interactions 1 10.56 10.56 10.56 0.47 0.512 Residual Error 8 179.50 179.50 22.44 Pure Error 8 179.50 179.50 22.44 Total 15 5730.94

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6 Reduced Model Calculation of Residuals
7 Residual Plots Residual Percent 10 5 0 -5 -10 99 90 50 10 1 N1 6 AD 0.254 P-Value 0.686 Fitted Value Residual 100 80 60 40 5.0 2.5 0.0 -2.5 -5.0 Residual Frequency 6 4 2 0 -2 -4 -6 4.8 3.6 2.4 1.2 0.0 Observation Order 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5.0 2.5 0.0 -2.5 -5.0

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ISE 426-526 - Lecture 7 - Single Replicates of 2^k Designs...

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