1. 1 Design of Engineering Experiments – The 2 k Factorial Design Text reference, Chapter 6 Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative) Very widely used in industrial experimentation Form a basic “building block” for other very useful experimental designs Special (short-cut) methods for analysis

2. 2 Design of Engineering Experiments – The 2 k Factorial Design Assumptions The factors are fixed The designs are completely randomized Usual normality assumptions are satisfied It provides the smallest number of runs can be studied in a complete factorial design – used as factor screening experiments Linear response in the specified range is assumed

3. 3 The Simplest Case: The 2 2 “-” 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

4. 4 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery

5. 5 Analysis Procedure for a Factorial Design Estimate factor effects Formulate model –With replication, use full model –With an unreplicated design, use normal probability plots Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results

6. 6 Estimation of Factor Effects See textbook, pg. 206 For manual calculations The effect estimates are: A = 8.33, B = -5.00, AB = 1.67 Practical interpretation

7. 7 Estimation of Factor Effects Effects(1)abab A+1+1 B +1 AB+1 +1 “(1), a, b, ab” – standard order Used to determine the proper sign for each treatment combination TreatmentFactorialEffect combinationIABAB (1)+--+ a++-- b+-+- ab++++

8. 8 Statistical Testing - ANOVA Response:Conversion ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model291.67397.2224.820.0002 A208.331208.3353.19< 0.0001 B75.00175.0019.150.0024 AB8.3318.332.130.1828 Pure Error31.3383.92 Cor Total323.0011 Std. Dev.1.98R-Squared 0.9030 Mean27.50Adj R-Squared0.8666 C.V.7.20Pred R-Squared0.7817 PRESS70.50Adeq Precision11.669 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?

9. 9 Estimation of Factor Effects Form Tentative Model Term Effect SumSqr% Contribution Model Intercept Model A 8.33333 208.333 64.4995 Model B -5 75 23.2198 Model AB 1.66667 8.33333 2.57998 Error Lack Of Fit 0 0 Error P Error 31.3333

10. 10 Regression Model y =  o +  1 x 1 +  2 x 2 +  12 x 1 x 2 +  or let x 3 = x 1 x 2,  3 =  12 y =  o +  1 x 1 +  2 x 2 +  3 x 3 +  A linear regression model. Coded variables are related to natural variables by Therefore,

11. 11 Statistical Testing - ANOVA Coefficient Standard95% CI95% CI Factor Estimate DFErrorLowHighVIF Intercept 27.50 10.5726.1828.82 A-Concent 4.17 10.572.855.481.00 B-Catalyst -2.50 10.57 -3.82 -1.181.00 AB 0.83 10.57 -0.482.151.00 General formulas for the standard errors of the model coefficients and the confidence intervals are available. They will be given later.

12. 12 Refined/reduced Model y =  o +  1 x 1 +  2 x 2 +  Response:Conversion ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model283.332141.6732.14< 0.0001 A208.331208.3347.27< 0.0001 B75.00175.0017.020.0026 Residual39.6794.41 Lack of Fit8.3318.332.130.1828 Pure Error31.3383.92 Cor Total323.0011 Std. Dev.2.10R-Squared0.8772 Mean27.50Adj R-Squared0.8499 C.V.7.63Pred R-Squared0.7817 PRESS70.52Adeq Precision12.702 There is now a residual sum of squares, partitioned into a “lack of fit” component (the AB interaction) and a “pure error” component

13. 13 Regression Model for the Process

14. 14 Residuals and Diagnostic Checking

15. 15 The Response Surface Direction of potential improvement for a process (method of steepest ascent)