1. Design-Expert version 71 What’s New in Design-Expert version 7 Factorial and RSM Design Pat Whitcomb November, 2006

2. Design-Expert version 72 What’s New  General improvements  Design evaluation  Diagnostics  Updated graphics  Better help  Miscellaneous Cool New Stuff  Factorial design and analysis  Response surface design  Mixture design and analysis  Combined design and analysis

3. Design-Expert version 73 Two-Level Factorial Designs  2 k-p factorials for up to 512 runs (256 in v6) and 21 factors (15 in v6).  Design screen now shows resolution and updates with blocking choices.  Generators are hidden by default.  User can specify base factors for generators.  Block names are entered during build.  Minimum run equireplicated resolution V designs for 6 to 31 factors.  Minimum run equireplicated resolution IV designs for 5 to 50 factors.

4. Design-Expert version 74 2 k-p Factorial Designs More Choices Need to “check” box to see factor generators

5. Design-Expert version 75 2 k-p Factorial Designs Specify Base Factors for Generators

6. Design-Expert version 76 MR5 Designs Motivation Regular fractions (2 k-p fractional factorials) of 2 k designs often contain considerably more runs than necessary to estimate the [1+k+k(k-1)/2] effects in the 2FI model.  For example, the smallest regular resolution V design for k=7 uses 64 runs (2 7-1 ) to estimate 29 coefficients.  Our balanced minimum run resolution V design for k=7 has 30 runs, a savings of 34 runs. “Small, Efficient, Equireplicated Resolution V Fractions of 2 k designs and their Application to Central Composite Designs”, Gary Oehlert and Pat Whitcomb, 46th Annual Fall Technical Conference, Friday, October 18, 2002. Available as PDF at: http:// www.statease.com/pubs/small5.pdf

7. Design-Expert version 77 MR5 Designs Construction  Designs have equireplication, so each column contains the same number of +1s and −1s.  Used the columnwise-pairwise of Li and Wu (1997) with the D-optimality criterion to find designs.  Overall our CP-type designs have better properties than the algebraically derived irregular fractions.  Efficiencies tend to be higher.  Correlations among the effects tend be lower.

8. Design-Expert version 78 MR5 Designs Provide Considerable Savings k2 k-p MR5k2 k-p MR5 6322215256122 7643016256138 8643817256154 91284618512172 101285619512192 111286820512212 122568021512232 1325692251024326 14256106301024466

9. Design-Expert version 79 MR4 Designs Mitigate the use of Resolution III Designs The minimum number of runs for resolution IV designs is only two times the number of factors (runs = 2k). This can offer quite a savings when compared to a regular resolution IV 2 k-p fraction.  32 runs are required for 9 through 16 factors to obtain a resolution IV regular fraction.  The minimum-run resolution IV designs require 18 to 32 runs, depending on the number of factors. A savings of (32 – 18) 14 runs for 9 factors. No savings for 16 factors. “Screening Process Factors In The Presence of Interactions”, Mark Anderson and Pat Whitcomb, presented at AQC 2004 Toronto. May 2004. Available as PDF at: http://www.statease.com/pubs/aqc2004.pdf.

10. Design-Expert version 710 MR4 Designs Suggest using “MR4+2” Designs Problems: III  If even 1 run lost, design becomes resolution III – main effects become badly aliased.  Reduction in runs causes power loss – may miss significant effects.  Evaluate power before doing experiment. Solution:  To reduce chance of resolution loss and increase power, consider adding some padding:  New Whitcomb & Oehlert “MR4+2” designs

11. Design-Expert version 711 MR4 Designs Provide Considerable Savings k2 k-p MR4+2k2 k-p MR4+2 61614163234* 71616*176436 81618*186438 93220196440 103222206442 113224216444 123226226446 133228236448 143230246450 153232*256452 * No savings

12. Design-Expert version 712 Two-Level Factorial Analysis  Effects Tool bar for model section tools.  Colored positive and negative effects and Shapiro-Wilk test statistic add to probability plots.  Select model terms by “boxing” them.  Pareto chart of t-effects.  Select aliased terms for model with right click.  Better initial estimates of effects in irregular factions by using “Design Model”.  Recalculate and clear buttons.

13. Design-Expert version 713 Two-Level Factorial Analysis Effects Tool Bar  New – Effects Tool on the factorial effects screen makes all the options obvious.  New – Pareto Chart  New – Clear Selection button  New – Recalculate button (discuss later in respect to irregular fractions)

14. Design-Expert version 714 Two-Level Factorial Analysis Colored Positive and Negative Effects

15. Design-Expert version 715 Two-Level Factorial Analysis Select Model Terms by “Boxing” Them.

16. Design-Expert version 716 Two-Level Factorial Analysis Pareto Chart to Select Effects The Pareto chart is useful for showing the relative size of effects, especially to non-statisticians. Problem: If the 2 k-p factorial design is not orthogonal and balanced the effects have differing standard errors, so the size of an effect may not reflect its statistical significance. Solution: Plotting the t-values of the effects addresses the standard error problems for non-orthogonal and/or unbalanced designs. Problem: The largest effects always look large, but what is statistically significant? Solution: Put the t-value and the Bonferroni corrected t-value on the Pareto chart as guidelines.

17. Design-Expert version 717 Two-Level Factorial Analysis Pareto Chart to Select Effects Pareto Chart t-Value of |Effect| Rank 0.00 2.82 5.63 8.45 11.27 Bonferroni Limit 5.06751 t-Value Limit 2.77645 1234567 C AC A

18. Design-Expert version 718 Two-Level Factorial Analysis Select Aliased terms via Right Click

19. Design-Expert version 719 Two-Level Factorial Analysis Better Effect Estimates in Irregular Factions Design-Expert version 6 Design-Expert version 7

20. Design-Expert version 720 Two-Level Factorial Analysis Better Effect Estimates in Irregular Factions ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model38135.1749533.79130.22< 0.0001 A10561.33110561.33144.25< 0.0001 B8.1718.170.110.7482 C11285.33111285.33154.14< 0.0001 AC14701.50114701.50200.80< 0.0001 Residual512.50773.21 Cor Total38647.6711

21. Design-Expert version 721 Main effects only model: [Intercept] = Intercept - 0.333*CD - 0.333*ABC - 0.333*ABD [A] = A - 0.333*BC - 0.333*BD - 0.333*ACD [B] = B - 0.333*AC - 0.333*AD - 0.333*BCD [C] = C - 0.5*AB [D] = D - 0.5*AB Main effects & 2fi model: [Intercept] = Intercept - 0.5*ABC - 0.5*ABD [A] = A - ACD [B] = B - BCD [C] = C [D] = D [AB] = AB [AC] = AC - BCD [AD] = AD - BCD [BC] = BC - ACD [BD] = BD - ACD [CD] = CD - 0.5*ABC - 0.5*ABD Two-Level Factorial Analysis Better Effect Estimates in Irregular Factions

22. Design-Expert version 722 Two-Level Factorial Analysis Better Effect Estimates in Irregular Factions  Design-Expert version 6 calculates the initial effects using sequential SS via hierarchy.  Design-Expert version 7 calculates the initial effects using partial SS for the “Base model for the design”.  The recalculate button (next slide) calculates the chosen (model) effects using partial SS and then remaining effects using sequential SS via hierarchy.

23. Design-Expert version 723 Two-Level Factorial Analysis Better Effect Estimates in Irregular Fractions  Irregular fractions – Use the “Recalculate” key when selecting effects.

24. Design-Expert version 724 General Factorials Design:  Bigger designs than possible in v6.  D-optimal now can force categoric balance (or impose a balance penalty).  Choice of nominal or ordinal factor coding. Analysis:  Backward stepwise model reduction.  Select factor levels for interaction plot.  3D response plot.

25. Design-Expert version 725 General Factorial Design D-optimal Categoric Balance

26. Design-Expert version 726 General Factorial Design Choice of Nominal or Ordinal Factor Coding

27. Design-Expert version 727 Categoric Factors Nominal versus Ordinal The choice of nominal or ordinal for coding categoric factors has no effect on the ANOVA or the model graphs. It only affects the coefficients and their interpretation: 1.Nominal – coefficients compare each factor level mean to the overall mean. 2.Ordinal – uses orthogonal polynomials to give coefficients for linear, quadratic, cubic, …, contributions.

28. Design-Expert version 728 Nominal contrasts – coefficients compare each factor level mean to the overall mean. Name A[1] A[2] A1 1 0 A2 0 1 A3 -1 -1  The first coefficient is the difference between the overall mean and the mean for the first level of the treatment.  The second coefficient is the difference between the overall mean and the mean for the second level of the treatment.  The negative sum of all the coefficients is the difference between the overall mean and the mean for the last level of the treatment. Battery Life Interpreting the coefficients

29. Design-Expert version 729 Ordinal contrasts – using orthogonal polynomials the first coefficient gives the linear contribution and the second the quadratic: Name B[1] B[2] 15 -1 1 70 0 -2 125 1 1 B[1] = linear B[2] = quadratic Battery Life Interpreting the coefficients

30. Design-Expert version 730 General Factorial Analysis Backward Stepwise Model Reduction

31. Design-Expert version 731 Select Factor Levels for Interaction Plot

32. Design-Expert version 732 General Factorial Analysis 3D Response Plot

33. Design-Expert version 733 Factorial Design Augmentation  Semifold: Use to augment 2k-p resolution IV; usually as many additional two-factor interactions can be estimated with half the runs as required for a full foldover.  Add Center Points.  Replicate Design.  Add Blocks.

34. Design-Expert version 734 What’s New  General improvements  Design evaluation  Diagnostics  Updated graphics  Better help  Miscellaneous Cool New Stuff  Factorial design and analysis  Response surface design  Mixture design and analysis  Combined design and analysis

35. Design-Expert version 735 Response Surface Designs  More “canned” designs; more factors and choices.  CCDs for ≤ 30 factors (v6 ≤ 10 factors) New CCD designs based on MR5 factorials. New choices for alpha “practical”, “orthogonal quadratic” and “spherical”.  Box-Behnken for 3–30 factors (v6 3, 4, 5, 6, 7, 9 & 10)  “Odd” designs moved to “Miscellaneous”.  Improved D-optimal design.  for ≤ 30 factors (v6 ≤ 10 factors)  Coordinate exchange

36. Design-Expert version 736 MR-5 CCDs Response Surface Design  Minimum run resolution V (MR-5) CCDs:  Add six center points to the MR-5 factorial design.  Add 2(k) axial points.  For k=10 the quadratic model has 66 coefficients. The number of runs for various CCDs: Regular (2 10-3 ) = 158 MR-5 = 82 Small (Draper-Lin) = 71

37. Design-Expert version 737 MR-5 CCDs (k = 6 to 30) Number of runs closer to small CCD

38. Design-Expert version 738 MR-5 CCDs (k=10,  = 1.778) Regular, MR-5 and Small CCDs 2 10-3 CCD 158 runs MR-5 CCD 82 runs Small CCD 71 runs Model65 Residuals92165 Lack of Fit83111 Pure Error954 Corr Total1578170

39. Design-Expert version 739 MR-5 CCDs (k=10,  = 1.778) Properties of Regular, MR-5 and Small CCDs 2 10-3 CCD 158 runs MR-5 CCD 82 runs Small CCD 71 runs Max coefficient SE0.2140.22716.514 Max VIF1.5432.89212,529 Max leverage0.4980.9911.000 Ave leverage0.4180.8050.930 Scaled D-optimality1.5682.0763.824

40. Design-Expert version 740 MR-5 CCDs (k=10,  = 1.778) Properties closer to regular CCD 2 10-3 CCDMR-5 CCDSmall CCD 158 runs82 runs71 runs different y-axis scale A-B slice

41. Design-Expert version 741 2 10-3 CCDMR-5 CCDSmall CCD 158 runs82 runs71 runs all on the same y-axis scale A-C slice MR-5 CCDs (k=10,  = 1.778) Properties closer to regular CCD

42. Design-Expert version 742 MR-5 CCDs Conclusion Best of both worlds:  The number of runs are closer to the number in the small than in the regular CCDs.  Properties of the MR-5 designs are closer to those of the regular than the small CCDs. The standard errors of prediction are higher than regular CCDs, but not extremely so. Blocking options are limited to 1 or 2 blocks.

43. Design-Expert version 743 Practical alpha Choosing an alpha value for your CCD Problems arise as the number of factors increase:  The standard error of prediction for the face centered CCD (alpha = 1) increases rapidly. We feel that an alpha > 1 should be used when k > 5.  The rotatable and spherical alpha values become too large to be practical. Solution:  Use an in between value for alpha, i.e. use a practical alpha value. practical alpha = (k) ¼

44. Design-Expert version 744 Standard Error Plots 2 6-1 CCD Slice with the other four factors = 0 Face CenteredPracticalSpherical  = 1.000  = 1.565  = 2.449

45. Design-Expert version 745 Standard Error Plots 2 6-1 CCD Slice with two factors = +1 and two = 0 Face CenteredPracticalSpherical  = 1.000  = 1.565  = 2.449

46. Design-Expert version 746 Standard Error Plots MR-5 CCD (k=30) Slice with the other 28 factors = 0 Face CenteredPracticalSpherical  = 1.000  = 2.340  = 5.477

47. Design-Expert version 747 Standard Error Plots MR-5 CCD (k=30) Slice with 14 factors = +1 and 14 = 0 Face CenteredPractical Spherical  = 1.000  = 2.340  = 5.477

48. Design-Expert version 748 Choosing an alpha value for your CCD

49. Design-Expert version 749 D-optimal Design Coordinate versus Point Exchange There are two algorithms to select “optimal” points for estimating model coefficients: Point exchange Coordinate exchange

50. Design-Expert version 750 D-optimal Coordinate Exchange* Cyclic Coordinate Exchange Algorithm 1.Start with a nonsingular set of model points. 2.Step through the coordinates of each design point determining if replacing the current value increases the optimality criterion. If the criterion is improved, the new coordinate replaces the old. (The default number of steps is twelve. Therefore 13 levels are tested between the low and high factor constraints; usually ±1.) 3.The exchanges continue and cycle through the model points until there is no further improvement in the optimality criterion. *R.K. Meyer, C.J. Nachtsheim (1995), “The Coordinate-Exchange Algorithm for Constructing Exact Optimal Experimental Designs”, Technometrics, 37, 60-69.