The Drilling Experiment Example 6-3, pg. 237 A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill 1 Effect Estimates - The Drilling Experiment Model Term Intercept A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD Effect SumSqr % Contribution 0.9175 6.4375 3.2925

2.29 0.59 0.155 0.8375 1.51 1.5925 0.4475 0.1625 0.76 0.585 0.175 0.5425 3.36722 165.766 43.3622 20.9764 1.3924 0.0961 2.80563 9.1204 10.1442 0.801025 0.105625 2.3104 1.3689 0.1225 1.17722 1.28072 63.0489 16.4928 7.97837 0.529599 0.0365516

1.06712 3.46894 3.85835 0.30467 0.0401744 0.87876 0.520661 0.0465928 0.447757 2 Half-Normal Probability Plot of Effects DESIGN-EXPERT Plo t ad v. _ra te A: loa d B: flow C: sp ee d D: mu d Half Normal plot 99 Half Norm al % probability 97 B 95 90 C 85

D 80 BD BC 70 60 40 20 0 0.00 1.61 3.22 4.83 6.44 |Effect| 3 DE SI GN-E XPERT Plo t ad v._r a te Residual Plots Residuals vs. Predicted Normal plot of residuals 2.58625

99 1.44875 90 80 Res iduals 70 50 0.31125 Residuals vs. Predicted 2. 58625 1. 44875 R e s id u a l s Norm al % probability 95 30 0. 31125 - 0. 82625 - 1. 96375 1.6 9

4. 70 7. 70 10. 71 13. 71 Pre d i c te d 20 10 -0.82625 5 1 -1.96375 -1.96375 -0.82625 0.31125 Residual 1.44875 2.58625 1.69 4.70

7.70 10.71 13.71 Predicted 4 Residual Plots 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 Power family transformations are widely used * y y Transformations are typically performed to Stabilize variance Induce normality Simplify the model/improve the fit of the model to the data 5 Selecting a Transformation Empirical selection of lambda Prior (theoretical) knowledge or experience can often suggest the form of a transformation Analytical selection of lambdathe Box-Cox (1964) method (simultaneously estimates the

model parameters and the transformation parameter lambda) Box-Cox method implemented in Design-Expert 6 DESIG N-EXPERT Plo t ad v._ra te The Box-Cox Method (Chapter 14) La mb da Curre nt = 1 Best = -0 .23 Lo w C. I. = -0 .7 9 High C.I. = 0 .3 2 Recom m en d tran sf orm : Lo g (L am bd a = 0 ) Box-Cox Plot for Power Transforms A log transformation is recommended 6.85 The procedure provides a confidence interval on the transformation parameter lambda Ln(Res idualSS) 5.40

3.95 If unity is included in the confidence interval, no transformation would be needed 2.50 1.05 -3 -2 -1 0 1 2 3 Lam bda 7 Effect Estimates Following the Log Transformation DESIGN-EXPERT Plo t Ln (adv. _ra te) A: loa d B: flow

C: sp ee d D: mu d Half Normal plot 99 Half Norm al % probability 97 B 95 90 No indication of large interaction effects C 85 D 80 Three main effects are large 70 60 40 20 0

0.00 0.29 0.58 0.87 1.16 |Effect| 8 ANOVA Following the Log Transformation Response: adv._rate Transform: Natural log Constant: 0.000 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 7.11 3 2.37 164.82 < 0.0001 B 5.35 1

5.35 381.79 < 0.0001 C 1.34 1 1.34 95.64 < 0.0001 D 0.43 1 0.43 30.79 0.0001 Residual 0.17 12 0.014 Cor Total 7.29 15 Std. Dev. 0.12 Mean 1.60 C.V. 7.51 R-Squared Adj R-Squared Pred R-Squared 0.9763 0.9704 0.9579 PRESS

Adeq Precision 34.391 0.31 9 Following the Log Transformation Final Equation in Terms of Coded Factors: Ln(adv._rate) = +1.60 +0.58 * B +0.29 * C +0.16 * D 10 Following the Log Transformation DESIGN- EXPERT Plot Ln(ad v. _rate) DESIGN- EXPERT Plot Ln(ad v. _rate) Normal plot of residuals Residuals vs. Predicted 0.194177 99 0.104087 90

80 Res iduals Norm al % probability 95 70 50 0.0139965 30 20 10 -0.0760939 5 1 -0.166184 -0.166184 -0.0760939 0.0139965 Res idual 0.104087 0.194177

0.57 1.08 1.60 2.11 2.63 Predicted 11 The Log Advance Rate Model Is the log model better? We would generally prefer a simpler model in a transformed scale to a more complicated model in the original metric What happened to the interactions? Sometimes transformations provide insight into the underlying mechanism 12 Other Examples of Unreplicated 2k Designs The sidewall panel experiment (Example 6-4, pg. 239) Two factors affect the mean number of defects A third factor affects variability Residual plots were useful in identifying the dispersion effect The oxidation furnace experiment (Example 6-5, pg. 242)

Replicates versus repeat (or duplicate) observations? Modeling within-run variability 13 Addition of Center Points to 2k Designs 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: k k k First-order model (interaction) y 0 i xi ij xi x j i 1 k k i 1 j i k k Second-order model y 0 i xi ij xi x j ii xi2 i 1 i 1 j i

i 1 14 Addition of Center Points to 2k Designs of quantitative factors Conduct nc runs at the center (xi = 0, i=1,,k) y F : the average of the four runs at the four factorial points y C: the average of the nc runs at the center point 15 yF yC no "curvature" k H 0 : ii 0 The hypotheses are: i 1 k H1 : ii 0 SS Pure Quad nF nC ( yF yC )

nF nC 2 i 1 This sum of squares has a single degree of freedom This quantity is compared to the error mean square to test for pure quadratic curvature If the factorial points are not replicated, the nC points can be used to construct an estimate of error with nC1 degrees of freedom 16 Example with Center Points nC 5 Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature 17 ANOVA for Example w/ Center Points Response: yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Source Model A B

AB Curvature Pure Error Cor Total Sum of Squares 2.83 2.40 0.42 2.500E-003 2.722E-003 0.17 3.00 Std. Dev. Mean 0.21 40.44 R-Squared Adj R-Squared C.V. 0.51 Pred R-Squared N/A PRESS N/A

Adeq Precision 14.234 DF 3 1 1 1 1 4 8 Mean Square 0.94 2.40 0.42 2.500E-003 2.722E-003 0.043 F Value 21.92 55.87 9.83 0.058 0.063 Prob > F 0.0060 0.0017 0.0350

0.8213 0.8137 0.9427 0.8996 18 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 (more in Chapter 11) 19 Practical Use of Center Points (pg. 250) Use current operating conditions as the center point Check for abnormal conditions during the time the experiment was conducted Check for time trends Use center points as the first few runs when there is little or no information available about the magnitude of error Center points and qualitative factors? 20 Center Points and Qualitative Factors 21