What’s New in Design-Expert version 7 Mixture and Combined Design Pat Whitcomb March 25, 2006

1. What’s New inDesign-Expert version 7Mixture and Combined DesignPat WhitcombMarch 25, 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. Mixture Design • More components • Simplex lattice 2 to 30 components (v6 2 to 24) • Screening 6 to 40 components (v6 6 to 24) • Detect inverted simplex • Upper bounded pseudo values: U_Pseudo and L_Pseudo • New mixture design “Historical Data” • Coordinate exchange

4. Inverted Simplex When component proportions are restricted by upper bounds it can lead to an inverted simplex. For example: x1 ≤ 0.4 x2 ≤ 0.6 x3 ≤ 0.3

5. Inverted Simplex3 component L_Pseudo Using lower bounded L_Pseudo values leads to the following inverted simplex. Open “I-simplex L_P.dx7” andmodel the response. 0.50 in L_Pseudo

6. Inverted Simplex3 component U_Pseudo (page 1 of 2) • Build a new design and say “Yes” to “Use previous design info”. • Change “User-Defined” to “Simplex Centroid”. • When asked say “Yes” to switch to upper bounded pseudo values “U_Pseudo.

7. Inverted Simplex3 component U_Pseudo (page 1 of 3) • Change the replicates from 4 to 6 and • Right click on the “Block”column header and“Display Point Type”

8. 1 in U_Pseudo 0 in U_Pseudo Inverted SimplexUpper Bounded Pseudo Values The high value becomes 0 and the low value becomes 1!

9. Inverted SimplexUpper Bounded Pseudo Values The upper value becomes 0 and the lower value 1! U_Pseudo values:

10. Inverted Simplex3 component U_Pseudo Go to the “Evaluation” and view the design space:

11. Inverted SimplexNote the Improved Values Coding is U_Pseudo. TermStdErr**VIFRi-Sq A 0.69 1.74 0.4255 B 0.69 1.74 0.4255 C 0.69 1.74 0.4255 AB 3.45 1.94 0.4844 AC 3.45 1.94 0.4844 BC 3.45 1.94 0.4844 ABC 27.03 1.75 0.4300 **Basis Std. Dev. = 1.0 Coding is L_Pseudo. TermStdErr**VIFRi-Sq A 26.33 1550.78 0.9994 B 26.33 1550.78 0.9994 C 26.33 1550.78 0.9994 AB 104.19 2686.10 0.9996 AC 104.19 2686.10 0.9996 BC 104.19 2686.10 0.9996 ABC 216.27 455.72 0.9978 **Basis Std. Dev. = 1.0

12. Inverted Simplex3 component U_Pseudo • Simulate the response using “I-simplex U_P.sim” • Model the response.

13. Inverted SimplexUpper vs Lower Bounded Pseudo Values Low becomes high and high becomes low: U_Pseudo L_Psuedo

14. Mixture Design“Historical Data”

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

16. 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.

17. Mixture Analysis • Cox Model; a new mixture parameterization • New screening tools for linear models: • Constraint Region Bounded Component Effects for Piepel Direction • Constraint Region Bounded Component Effects for Cox Direction • Constraint Region Bounded Component Effects for Orthogonal Direction • Range Adjusted Component Effects for Orthogonal Direction (this is the only measure in v6)

18. Mixture AnalysisCox Model • Cox model option for mixtures: May be more informative for formulators when they are interested in a particular composition.

19. Screening DesignsComponent Effects Concepts • Basis for screening is a linear model: • In a mixture it is impossible to change one component while holding the others fixed. • Changes in the component of interest must be offset by changes in the other components (so the components still sum to their total). • Choosing a direction through the mixture space to vary to component of interest defines how the offsetting changes are made.

20. Screening DesignsComponent Effect Directions Three directions in which component effects are assessed: • Orthogonal • Cox • Piepel The most meaningful direction (or directions) to use for computing effects for a particular mixture DOE depends on the shape of the mixture region. In an unconstrained simplex theCox and Piepel directions are the same. In a constrained simplex they’re not!(Remember the ABS Pipe example.)

21. Screening DesignsComponent Effect Directions Example (equation in actuals):

22. Screening DesignsOrthogonal Direction Component Effect X 1 X X 2 3

23. Orthogonal Component EffectsRange Adjusted versus Constraint Bounded BoundedAdjusted Component EffectEffect A-X1 0.601.80 B-X20.000.00 C-X3-0.30-0.30 In constrained mixtures the “Adjusted” effect is almost never realized.

24. Orthogonal Component GradientsConstraint Bounded Gradient Component at Base Pt. A-X1 3.00 B-X2 0.00 C-X3 -3.00 A has a positive slope B has no slope C has a negative slope Slope = 3.0

25. X 1 X X 2 3 Screening DesignsCox Direction Component Effect

26. Cox Component EffectsConstraint Bounded Gradient Component at Base Pt. A-X1 2.50 B-X2 -0.91 C-X3 -2.94 Component Component Effect A-X1 1.00 B-X2 -0.33 C-X3 -0.29 Slope = 2.5

27. X 1 X X 2 3 Screening DesignsPiepel Direction Component Effect

28. Piepel Component EffectsConstraint Bounded Gradient Component at Base Pt. A-X1 2.25 B-X2 -1.43 C-X3 -2.92 Component Component Effect A-X1 1.35 B-X2 -1.00 C-X3 -0.29 Slope = 2.25

29. SummaryComponent Effect Directions • Orthogonal: The direction for the ith component along a line that is orthogonal to space spanned by the other q-1 components. Appropriate only for simplex regions. • Cox: The direction for the ith component along a line joining the reference blend to the ith vertex (in real values). The line is also extended in the opposite direction to its end point. Appropriate for all regions. • Piepel: The same as the Cox direction after applying the pseudo component transformation. Appropriate for all regions.

30. 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

31. Combined Design Design: • Big new feature: combine two mixture designs! Analysis: • New fit summary layout. • New model graphs: • Mix-Process contour plot • Mix-Process 3D plot

32. Combined Design

33. Combined Design: Analysis New Fit Summary Layout Order Abbreviations in Fit Summary Table M = Mean L = Linear Q = Quadratic SC = Special Cubic C = Cubic Combined Model Mixture Process Fit Summary Table Sequential p-value Summary Statistics Mix Process Mix Process Lack of Fit Adjusted Predicted Order Order R-Squared R-Squared M M M L < 0.0001 0.0027 0.3929 0.3393 M 2FI 0.9550 0.0024 0.3630 0.2678 M Q * * 0.0024 0.3630 0.2678 Aliased M C * * 0.6965 0.0023 0.3528 0.2418 Aliased M M L M < 0.0001 0.0032 0.4350 0.3825 L L < 0.0001 < 0.0001 0.1534 0.9042 0.8715 L 2FI < 0.0001 0.5856 0.1415 0.9013 0.8142 L Q * < 0.0001 * 0.1415 0.9013 0.8142 Aliased L C * < 0.0001 * 0.7605 0.1280 0.8966 0.7536 Aliased

34. Combined Design: Analysis Mix-Process Contour Plot

35. Combined Design: Analysis Mix-Process 3D Plot