1. Design-Expert version 71 What’s New in Design-Expert version 7 Mixture and Combined Design Pat Whitcomb March 25, 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 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. Design-Expert version 74 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. Design-Expert version 75 Inverted Simplex 3 component L_Pseudo Using lower bounded L_Pseudo values leads to the following inverted simplex. Open “I-simplex L_P.dx7” and model the response. 0.50 in L_Pseudo

6. Design-Expert version 76 Inverted Simplex 3 component U_Pseudo (page 1 of 2) 1.Build a new design and say “Yes” to “Use previous design info”. 2.Change “User-Defined” to “Simplex Centroid”. 3.When asked say “Yes” to switch to upper bounded pseudo values “U_Pseudo.

7. Design-Expert version 77 Inverted Simplex 3 component U_Pseudo (page 1 of 3) 4.Change the replicates from 4 to 6 and 5.Right click on the “Block” column header and “Display Point Type”

8. Design-Expert version 78 Inverted Simplex Upper Bounded Pseudo Values The high value becomes 0 and the low value becomes 1! 0 in U_Pseudo1 in U_Pseudo

9. Design-Expert version 79 Inverted Simplex Upper Bounded Pseudo Values The upper value becomes 0 and the lower value 1! U_Pseudo values: RealPseudo LiLi UiUi LiLi UiUi x10.10.410 x20.30.610 x30.00.310

10. Design-Expert version 710 Inverted Simplex 3 component U_Pseudo Go to the “Evaluation” and view the design space:

11. Design-Expert version 711 Inverted Simplex Note the Improved Values Coding is U_Pseudo. TermStdErr**VIFRi-Sq A0.691.740.4255 B0.691.740.4255 C0.691.740.4255 AB3.451.940.4844 AC3.451.940.4844 BC3.451.940.4844 ABC27.031.750.4300 **Basis Std. Dev. = 1.0 Coding is L_Pseudo. TermStdErr**VIFRi-Sq A26.331550.780.9994 B26.331550.780.9994 C26.331550.780.9994 AB104.192686.100.9996 AC104.192686.100.9996 BC104.192686.100.9996 ABC216.27455.720.9978 **Basis Std. Dev. = 1.0

12. Design-Expert version 712 Inverted Simplex 3 component U_Pseudo 1.Simulate the response using “I-simplex U_P.sim” 2.Model the response.

13. Design-Expert version 713 Inverted Simplex Upper vs Lower Bounded Pseudo Values Low becomes high and high becomes low: U_PseudoL_Psuedo

14. Design-Expert version 714 Mixture Design “Historical Data”

15. Design-Expert version 715 D-optimal Design Coordinate versus Point Exchange There are two algorithms to select “optimal” points for estimating model coefficients: Point exchange Coordinate exchange

16. Design-Expert version 716 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.

17. Design-Expert version 717 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. Design-Expert version 718 Mixture Analysis Cox Model  Cox model option for mixtures: May be more informative for formulators when they are interested in a particular composition.

19. Design-Expert version 719 Screening Designs Component 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. Design-Expert version 720 Screening Designs Component Effect Directions Three directions in which component effects are assessed: 1.Orthogonal 2.Cox 3.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 the Cox and Piepel directions are the same. In a constrained simplex they’re not! (Remember the ABS Pipe example.)

21. Design-Expert version 721 Screening Designs Component Effect Directions Example (equation in actuals) :

22. Design-Expert version 722 Screening Designs Orthogonal Direction Component Effect 1 2 X XX 3

23. Design-Expert version 723 Orthogonal Component Effects Range Adjusted versus Constraint Bounded BoundedAdjusted ComponentEffectEffect A-X10.601.80 B-X20.000.00 C-X3-0.30-0.30 In constrained mixtures the “Adjusted” effect is almost never realized.

24. Design-Expert version 724 Orthogonal Component Gradients Constraint Bounded Gradient Componentat Base Pt. A-X13.00 B-X20.00 C-X3-3.00 A has a positive slope B has no slope C has a negative slope Slope = 3.0

25. Design-Expert version 725 Screening Designs Cox Direction Component Effect

26. Design-Expert version 726 Cox Component Effects Constraint Bounded Gradient Componentat Base Pt. A-X12.50 B-X2-0.91 C-X3-2.94 Component ComponentEffect A-X11.00 B-X2-0.33 C-X3-0.29 Slope = 2.5

27. Design-Expert version 727 Screening Designs Piepel Direction Component Effect 1 2 X XX 3

28. Design-Expert version 728 Piepel Component Effects Constraint Bounded Gradient Componentat Base Pt. A-X12.25 B-X2-1.43 C-X3-2.92 Component ComponentEffect A-X11.35 B-X2-1.00 C-X3-0.29 Slope = 2.25

29. Design-Expert version 729 Summary Component Effect Directions 1.Orthogonal: The direction for the i th component along a line that is orthogonal to space spanned by the other q-1 components. Appropriate only for simplex regions. 2.Cox: The direction for the i th component along a line joining the reference blend to the i th vertex (in real values). The line is also extended in the opposite direction to its end point. Appropriate for all regions. 3.Piepel: The same as the Cox direction after applying the pseudo component transformation. Appropriate for all regions.

30. Design-Expert version 730 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. Design-Expert version 731 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. Design-Expert version 732 Combined Design

33. Design-Expert version 733 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 MixProcessMixProcessLack of FitAdjustedPredicted OrderOrderR-SquaredR-SquaredM ML< 0.00010.00270.39290.3393 M2FI0.95500.00240.36300.2678 MQ* * 0.00240.36300.2678Aliased MC* * 0.69650.00230.35280.2418Aliased MM LM< 0.00010.00320.43500.3825 LL< 0.0001< 0.00010.15340.90420.8715 L2FI< 0.00010.58560.14150.90130.8142 LQ* < 0.0001* 0.14150.90130.8142Aliased LC* < 0.0001* 0.76050.12800.89660.7536Aliased

34. Design-Expert version 734 Combined Design: Analysis Mix-Process Contour Plot

35. Design-Expert version 735 Combined Design: Analysis Mix-Process 3D Plot