1. 1 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

2. 2 Effect Estimates - The Drilling Experiment Model TermEffectSumSqr% Contribution Intercept A0.91753.367221.28072 B6.4375165.76663.0489 C3.292543.362216.4928 D2.2920.97647.97837 AB0.591.39240.529599 AC0.1550.09610.0365516 AD0.83752.805631.06712 BC1.519.12043.46894 BD1.592510.14423.85835 CD0.44750.8010250.30467 ABC0.16250.1056250.0401744 ABD0.762.31040.87876 ACD0.5851.36890.520661 BCD0.1750.12250.0465928 ABCD0.54251.177220.447757

3. 3 Half-Normal Probability Plot of Effects

4. 4 Residual Plots

5. 5 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 Transformations are typically performed to –Stabilize variance –Induce normality –Simplify the model/improve the fit of the model to the data Residual Plots

6. 6 Selecting a Transformation Empirical selection of lambda Prior (theoretical) knowledge or experience can often suggest the form of a transformation Analytical selection of lambda…the Box-Cox (1964) method (simultaneously estimates the model parameters and the transformation parameter lambda) Box-Cox method implemented in Design-Expert

7. 7 The Box-Cox Method (Chapter 14) A log transformation is recommended The procedure provides a confidence interval on the transformation parameter lambda If unity is included in the confidence interval, no transformation would be needed

8. 8 Effect Estimates Following the Log Transformation Three main effects are large No indication of large interaction effects

9. 9 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 ofMeanF SourceSquaresDFSquareValueProb > F Model7.1132.37164.82< 0.0001 B5.3515.35381.79< 0.0001 C1.3411.3495.64< 0.0001 D0.4310.4330.790.0001 Residual0.17120.014 Cor Total7.2915 Std. Dev.0.12R-Squared 0.9763 Mean1.60Adj R-Squared0.9704 C.V.7.51Pred R-Squared0.9579 PRESS0.31Adeq Precision34.391

10. 10 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

11. 11 Following the Log Transformation

12. 12 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

13. 13 Other Examples of Unreplicated 2 k 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

14. 14 Addition of Center Points to 2 k 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:

15. 15 Addition of Center Points to 2 k Designs of quantitative factors Conduct n c runs at the center (x i = 0, i=1,…,k) : the average of the four runs at the four factorial points : the average of the n c runs at the center point

16. 16 The hypotheses are: 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 n C points can be used to construct an estimate of error with n C –1 degrees of freedom

17. 17 Example with Center Points Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature

18. 18 ANOVA for Example w/ Center Points Response:yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model2.8330.9421.920.0060 A2.4012.4055.870.0017 B0.4210.429.830.0350 AB2.500E-00312.500E-0030.0580.8213 Curvature2.722E-00312.722E-0030.0630.8137 Pure Error0.1740.043 Cor Total3.008 Std. Dev.0.21R-Squared 0.9427 Mean40.44Adj R-Squared0.8996 C.V.0.51Pred R-SquaredN/A PRESSN/AAdeq Precision14.234

19. 19 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)

20. 20 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?

21. 21 Center Points and Qualitative Factors