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1 Chapter 6 The 2 k Factorial Design

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2 6.1 Introduction The special cases of the general factorial design (Chapter 5) k factors and each factor has only two levels Levels: –quantitative (temperature, pressure,…), or qualitative (machine, operator,…) –High and low –Each replicate has 2 2 = 2 k observations

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3 Assumptions: (1) the factor is fixed, (2) the design is completely randomized and (3) the usual normality assumptions are satisfied Wildly used in factor screening experiments

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4 6.2 The 2 2 Factorial Design Two factors, A and B, and each factor has two levels, low and high. Example: the concentration of reactant v.s. the amount of the catalyst (Page 219)

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5 “-” And “+” denote the low and high levels of a factor, respectively Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative, although their treatment in the final model will be different

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6 Average effect of a factor = the change in response produced by a change in the level of that factor averaged over the levels if the other factors. (1), a, b and ab: the total of n replicates taken at the treatment combination. The main effects:

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7 The interaction effect: In that example, A = 8.33, B = and AB = 1.67 Analysis of Variance The total effects:

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8 Sum of squares:

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9 Response:Conversion ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model A < B AB Pure Error Cor Total Std. Dev.1.98R-Squared Mean27.50Adj R-Squared C.V.7.20Pred R-Squared PRESS70.50Adeq Precision The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?

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10 Table of plus and minus signs: IABAB (1)+––+ a++–– b+–+– ab++++

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11 The regression model: –x 1 and x 2 are coded variables that represent the two factors, i.e. x 1 (or x 2 ) only take values on – 1 and 1. –Use least square method to get the estimations of the coefficients –For that example, –Model adequacy: residuals (Pages 224~225) and normal probability plot (Figure 6.2)

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12 Response surface plot: –Figure 6.3

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The 2 3 Design Three factors, A, B and C, and each factor has two levels. (Figure 6.4 (a)) Design matrix (Figure 6.4 (b)) (1), a, b, ab, c, ac, bc, abc 7 degree of freedom: main effect = 1, and interaction = 1

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15 Estimate main effect: Estimate two-factor interaction: the difference between the average A effects at the two levels of B

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16 Three-factor interaction: Contrast: Table 6.3 –Equal number of plus and minus –The inner product of any two columns = 0 –I is an identity element –The product of any two columns yields another column –Orthogonal design Sum of squares: SS = (Contrast) 2 /8n

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17 Factorial Effect Treatment Combination I ABABCACBC ABC (1)+ –– + – ++ – a ++ –––– ++ b + – + –– + – + ab++++ –––– c+ –– ++ –– + ac++ –– ++ –– bc+ – + – + – + – abc Contrast Effect Table of – and + Signs for the 2 3 Factorial Design (pg. 231)

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18 Example 6.1 A = carbonation, B = pressure, C = speed, y = fill deviation

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19 TermEffectSumSqr% Contribution Model Intercept Error A Error B Error C Error AB Error AC Error BC Error ABC Error LOF0 Error P Error Lenth's ME Lenth's SME Estimation of Factor Effects

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20 ANOVA Summary – Full Model Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model A < B C AB AC BC ABC Pure Error Cor Total Std. Dev.0.79R-Squared Mean1.00Adj R-Squared C.V.79.06Pred R-Squared PRESS20.00Adeq Precision13.416

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21 The regression model and response surface: –The regression model: –Response surface and contour plot (Figure 6.7) Coefficient Standard 95% CI 95% CI Factor EstimateDFErrorLowHigh Intercept A-Carbonation B-Pressure C-Speed AB

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22 Contour & Response Surface Plots – Speed at the High Level

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23 Refine Model – Remove Nonsignificant Factors Response:Fill-deviation ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model < A < B C AB Residual LOF Pure E C Total Std. Dev.0.81R-Squared Mean1.00Adj R-Squared C.V.81.18Pred R-Squared PRESS15.34Adeq Precision15.424

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The General 2 k Design k factors and each factor has two levels Interactions The standard order for a 2 4 design: (1), a, b, ab, c, ac, bc, abc, d, ad, bd, abd, cd, acd, bcd, abcd

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25 The general approach for the statistical analysis: –Estimate factor effects –Form initial model (full model) –Perform analysis of variance (Table 6.9) –Refine the model –Analyze residual –Interpret results

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A Single Replicate of the 2 k Design These are 2 k factorial designs with one observation at each corner of the “cube” An unreplicated 2 k factorial design is also sometimes called a “single replicate” of the 2 k If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data

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27 Lack of replication causes potential problems in statistical testing –Replication admits an estimate of “pure error” (a better phrase is an internal estimate of error) –With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem –Pooling high-order interactions to estimate error (sparsity of effects principle) –Normal probability plotting of effects (Daniels, 1959)

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28 Example 6.2 (A single replicate of the 2 4 design) –A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin –The factors are A = temperature, B = pressure, C = concentration of formaldehyde, D= stirring rate

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30 Estimates of the effects TermEffectSumSqr% Contribution Model Intercept Error A Error B Error C Error D Error AB Error AC Error AD Error BC Error BD Error CD Error ABC Error ABD Error ACD Error BCD Error ABCD Lenth's ME Lenth's SME13.699

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31 The normal probability plot of the effects

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33 B is not significant and all interactions involving B are negligible Design projection: 2 4 design => 2 3 design in A,C and D ANOVA table (Table 6.13)

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34 Response:Filtration Rate ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb >F Model < A < C D < AC < AD < Residual Cor Total Std. Dev.4.42R-Squared Mean70.06Adj R-Squared C.V.6.30Pred R-Squared PRESS499.52Adeq Precision20.841

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35 The regression model: Residual Analysis (P. 251) Response surface (P. 252) Final Equation in Terms of Coded Factors: Filtration Rate = * Temperature * Concentration * Stirring Rate * Temperature * Concentration * Temperature * Stirring Rate

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37 Half-normal plot: the absolute value of the effect estimates against the cumulative normal probabilities.

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38 Example 6.3 (Data transformation in a Factorial Design) A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill

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39 The normal probability plot of the effect estimates

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40 Residual analysis

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41 The residual plots indicate that there are problems with the equality of variance assumption The usual approach to this problem is to employ a transformation on the response In this example,

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42 Three main effects are large No indication of large interaction effects What happened to the interactions?

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43 Response:adv._rate Transform: Natural log Constant: ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model < B < C < D Residual Cor Total Std. Dev.0.12R-Squared Mean1.60Adj R-Squared C.V.7.51Pred R-Squared PRESS0.31Adeq Precision34.391

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44 Following Log transformation Final Equation in Terms of Coded Factors: Ln(adv._rate) = * B * C * D

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46 Example 6.4: –Two factors (A and D) affect the mean number of defects –A third factor (B) affects variability –Residual plots were useful in identifying the dispersion effect –The magnitude of the dispersion effects: –When variance of positive and negative are equal, this statistic has an approximate normal distribution

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The Addition of Center Points to the 2 k Design Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models:

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48 The hypotheses are: This sum of squares has a single degree of freedom

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49 Example 6.6 Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature

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50 Response:yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model A B AB2.500E E Curvature2.722E E Pure Error Cor Total3.008 Std. Dev.0.21R-Squared Mean40.44Adj R-Squared C.V.0.51Pred R-SquaredN/A PRESSN/AAdeq Precision14.234

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51 If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model

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