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.

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Presentation transcript:

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 Effect Estimates - The Drilling Experiment Model TermEffectSumSqr% Contribution Intercept A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

3 Half-Normal Probability Plot of Effects

4 Residual Plots

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

9 ANOVA Following the Log Transformation 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

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

11 Following the Log Transformation

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 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 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 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 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 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 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 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

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 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 Center Points and Qualitative Factors