2. Constrained Optimization 3-8 Continuous &/or Discrete Linear + Cross-products (interactions) Good predictions of effects and interactions 2-Level Factorial (+ Center Points)

3. S C U Relative Importance of Three Stages of Experimentation U-Unconstrained Optimization (Response Surfaces Chapter 10) C-Constrained Optimization (Main effects and interactions) S-Screening Experiments (What Factors are important)

5. A Poor Solution is to Use One-at-a-Time Experiments Run A B C D E F G H 1 - - - - - - - - 2 + - - - - - - - 3 - + - - - - - - 4 - - + - - - - - 5 - - - + - - - - 6 - - - - + - - - 7 - - - - - + - - 8 - - - - - - + - 9 - - - - - - - +

6. Fractional Factorial Experiments Method for Strategically Picking a Subset of a 2 k Design Used for Screening purposes Has much Higher Power for detecting Effects through hidden replication Can be used to estimate some interactions and limited optimization

7. I = C A = C I = ABC Half-Fraction of 2 3

8. Paradigms That Justify Use of Fractional Factorials

9. Venus – Moon – Jupiter align Jupiter Mars Venus Crescent Moon Hierarchical Ordering Principle ▪Although its possible that three planets may align with the moon, its more often that two planets will align with moon than three ▪Likewise though three factor interactions and higher order interactions are possible, its more likely that large effects will be main effects or two factor interactions

13. Creating a half fraction design in SAS

16. I = ABCDE

23. Would these conclusions have been reached using one at a time experimentation?

24. · In a one half fraction of a 2 k experiment every effect that could be estimated was confounded with one other effect, thus one half the effects had to be assumed negligible in order to interpret or explain the results · In a one quarter fraction of a 2 k experiment every effect that can be estimated is confounded with three other effects, thus three quarters of the effects must be assumed negligible in order to interpret or explain the results · In a one eighth fraction of a 2 k experiment every effect that can be estimated is confounded with seven other effects, thus seven eights of the effects must be assumed negligible in order to interpret or explain the results, etc.

25. Creating a 2 k-p Design 1. Create a full two-level factorial in k-p factors 2. Add each of the remaining p factors by assigning them to a column of signs for an interaction among the first k-p columns

26. These are the generators

27. the generalized interaction the generators the defining relation

28. Defining Relation Confounding Pattern or Alias Structure

29. 2 6-3 design base design in 6-3 = 3Factors A, B, C

30. The three factor generalized interaction is The defining relation is

31. New Two-Level Design ► Define Variables… ► Add> Select Design…

32. Example ¼ Fraction of 2 6 One possible set of generators is: Resulting in the following Alias Structure

33. Another possible set of generators is: Resulting in the following Alias Structure

34. R, the resolution, is the length of the shortest word in the defining relation. Resolution as a criteria for choosing generators Resolution III – main effects confounded with two-factor interactions Resolution IV – main effects confounded with three-factor interactions, and two factor interactions confounded with other two-factor interactions Resolution V – main effects confounded with four-factor interactions, two-factor interactions confounded with three-factor interactions. In this case if you are willing to assume three factor interactions and higher are negligible, you can estimate all main effects and two factor interactions Higher Resolution means main effects are confounded with higher order interactions

35. Minimum Aberration as a criteria for choosing generators I = ABCDF = ABCEG = DEFG d 2 F = ABC, G = ADE d 1 F = ABCD, G = ABCE I = ABCF = ADEG = BCDEFG Which is better? d 1 (0, 1, 2) Word length pattern: length 3length 4 length 5 d 2 (0, 2, 0, 1)

36. Symbolically: (A 3, A 4, A 5, …) A r is number of words of length r

37. Number of clear Effects as a criteria for choosing generators An effect is defined to be clear if none of its aliases are main effects or two factor interactions See Example64.sas

38. Only 56% Eucalyptus used in Brazilian forests Hemicellulose hydrolyzate acid treatment Paecilomyces variolii Fermentation Edible Biomass rich in essential amino acids

40. Generators for minimum aberration

46. BH

47. EG Maximum appears to be with Ammonium Sulfate and Sodium Phosphate both at 2 g/L

48. CH

49. BEG

50. Recap 8 Factors would require 2 8 = 256 for full factorial 16 + 8 = 24 resulted in plausible interpretation and identification of optimal results LabelFactorOptimal Setting BRice Bran30.0 g/L EAmmonium Sulfate2.0 g/L GSodium Phosphate0.0 g/L

52. Reverse signs of coded factor levels for Factor B

53. +

55. + Example

56. Creating Design Augmented by Foldover in SAS Data Step ADX

57. Augmenting a resolution IV by mirror image or foldover does not break strings of confounded two factor interactions Augment by design with signs reversed on Factor A only, H=ABD

59. Augment by design Reversing signs on A

60. High concentration of arsenic reported in ground water in countries such as Bangladesh, Chile, India, Poland, Nepal … causing people to be prone to various forms of cancer Example:

61. Simple IOCS filters have been used in Bangladesh and Nepal to remove arsenic from ground water

62. Simple household filters are effective iron oxide coated sand raw water pourous membrane purified water

63. Coating solution made of ferric nitrate and sodium hydroxide with NAOH added to control pH. IOCS Ramakrishna et. al. (2006) conducted experiments to optimize The coating process. Mix Coating Solution Age Coating Solution Pour over clean sand MixDryFilter Spiked Water Sample repeat noyes

71. What can be done to separate AD+CF

72. AD CF - - + + - - + + - - + + - - + + - 0 - + 0 + 0 - - + 0 - + 0 + 0 + - - 0 + + 0 + 0 - + + 0 + + 0 + 0 + + 17 18 19 20 33333333 No longer orthogonal Fit model Y=A B F AD CF by regression

74. Exchange Algorithm for maximizing det(X’X) Step 1 replace 0 with -1Step 2 replace -.5 with -1 Step 3 replace.5 with 1 Candidate x’s -1, -.5, 0,.5, 1

75. Choose additional runs to maximize the

78. ● Plackett-Burman Designs are Resolution III, but there is no defining relation ● Main Effects are confounded with two-factor interactions, but rather than being completely confounded with a few two-factor interactions, they are partially confounded with many two-factor interactions Alias Matrixshows the alias structure

79. Example

80. Implications of Partial Confounding 1.We can use Alias matrix to determine what two-factor interactions are confounded with large unassigned effects 2. Models involving main effects and some partially confounded can be fit by regression since X‘X matrix is not saingular

81. Creating a Plackett-Burman Design in SAS

83. Read in the data and merge it with the design created earlier Fit the model and output the parameter estimates

86. Create interactions and do all subsets regression

90. Run12345 100000 200101 301011 401110 510011 610110 711001 811100 920010 1020101 1121000 1221111 OA(12, 3 1, 2 4 )

95. Data Similar to Experiment with Teaching Methods in Chapter 2

99. Dummy variables represent effect of chair style