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

What’s New in Design-Expert version 7 Factorial and RSM Design Pat Whitcomb November, 2006. What’s New. General improvements Design evaluation Diagnostics Updated graphics Better help Miscellaneous Cool New Stuff Factorial design and analysis Response surface design

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

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  1. What’s New inDesign-Expert version 7Factorial and RSM DesignPat WhitcombNovember, 2006

  2. 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. Two-Level Factorial Designs • 2k-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 for6 to 31 factors. • Minimum run equireplicated resolution IV designs for 5 to 50 factors.

  4. 2k-p Factorial DesignsMore Choices Need to “check” box to see factor generators

  5. 2k-p Factorial DesignsSpecify Base Factors for Generators

  6. MR5 Designs Motivation Regular fractions (2k-p fractional factorials) of 2k 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 (27-1) 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 2k 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. MR5 DesignsConstruction • 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. MR5 DesignsProvide Considerable Savings

  9. MR4 DesignsMitigate 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 2k-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. MR4 DesignsSuggest using “MR4+2” Designs Problems: • 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. MR4 DesignsProvide Considerable Savings * No savings

  12. 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. Two-Level Factorial AnalysisEffects 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. Two-Level Factorial AnalysisColored Positive and Negative Effects

  15. Two-Level Factorial AnalysisSelect Model Terms by “Boxing” Them.

  16. Two-Level Factorial AnalysisPareto Chart to Select Effects The Pareto chart is useful for showing the relative size of effects, especially to non-statisticians. Problem: If the 2k-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. Pareto Chart C 11.27 8.45 AC A t-Value of |Effect| 5.63 Bonferroni Limit 5.06751 2.82 t-Value Limit 2.77645 0.00 1 2 3 4 5 6 7 Rank Two-Level Factorial AnalysisPareto Chart to Select Effects

  18. Two-Level Factorial AnalysisSelect Aliased terms via Right Click

  19. Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions • Design-Expert version 6 Design-Expert version 7

  20. Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares] Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 38135.17 4 9533.79 130.22 < 0.0001A10561.33110561.33144.25< 0.0001B8.1718.170.110.7482C11285.33111285.33154.14< 0.0001AC14701.50114701.50200.80< 0.0001 Residual 512.50 7 73.21 Cor Total 38647.67 11

  21. Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Factions 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

  22. Two-Level Factorial AnalysisBetter 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. Two-Level Factorial AnalysisBetter Effect Estimates in Irregular Fractions • Irregular fractions – Use the “Recalculate” key when selecting effects.

  24. 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. General Factorial DesignD-optimal Categoric Balance

  26. General Factorial DesignChoice of Nominal or Ordinal Factor Coding

  27. Categoric FactorsNominal 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: • Nominal – coefficients compare each factor level mean to the overall mean. • Ordinal – uses orthogonal polynomials to give coefficients for linear, quadratic, cubic, …, contributions.

  28. Battery LifeInterpreting the coefficients 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.

  29. Battery LifeInterpreting the coefficients 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

  30. General Factorial AnalysisBackward Stepwise Model Reduction

  31. Select Factor Levels for Interaction Plot

  32. General Factorial Analysis3D Response Plot

  33. 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. 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. 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. MR-5 CCDsResponse 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 (210-3) = 158 • MR-5 = 82 • Small (Draper-Lin) = 71

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

  38. MR-5 CCDs(k=10, a = 1.778)Regular, MR-5 and Small CCDs

  39. MR-5 CCDs(k=10, a = 1.778)Properties of Regular, MR-5 and Small CCDs

  40. MR-5 CCDs(k=10, a = 1.778)Properties closer to regular CCD A-B slice 210-3 CCD MR-5 CCD Small CCD 158 runs 82 runs 71 runs different y-axis scale

  41. MR-5 CCDs(k=10, a = 1.778)Properties closer to regular CCD A-C slice 210-3 CCD MR-5 CCD Small CCD 158 runs 82 runs 71 runs all on the same y-axis scale

  42. MR-5 CCDsConclusion 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. Practical alphaChoosing 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. Standard Error Plots 26-1 CCDSlice with the other four factors = 0 Face Centered Practical Spherical a = 1.000 a = 1.565 a = 2.449

  45. Standard Error Plots 26-1 CCDSlice with two factors = +1 and two = 0 Face Centered Practical Spherical a = 1.000 a = 1.565 a = 2.449

  46. Standard Error Plots MR-5 CCD (k=30) Slice with the other 28 factors = 0 Face Centered Practical Spherical a = 1.000 a = 2.340 a = 5.477

  47. Standard Error Plots MR-5 CCD (k=30) Slice with 14 factors = +1 and 14 = 0 Face Centered Practical Spherical a = 1.000 a = 2.340 a = 5.477

  48. Choosing an alpha value for your CCD

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

  50. D-optimal Coordinate Exchange* • Cyclic Coordinate Exchange Algorithm • Start with a nonsingular set of model points. • 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.) • 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.

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