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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 13 Fractional Factorials Confounding, Aliases, Design Resolution,. Pilot Plant Experiment. 45. 80. C2. Catalyst. 52. 83. 54. 68. C1. 40. Concentration. 60. 72. 20. 160. 180. Temperature.

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Statistical Analysis

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  1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 13 Fractional Factorials Confounding, Aliases, Design Resolution,

  2. Pilot Plant Experiment 45 80 C2 Catalyst 52 83 54 68 C1 40 Concentration 60 72 20 160 180 Temperature

  3. Pilot Plant Experiment :Aliasing/Confounding with Operators Complete Factorial : 1/2 Replicate for Each of 2 Operators 45 80 C2 Catalyst 52 83 Operator 1 Operator 2 54 68 C1 40 Concentration 60 72 20 160 180 Temperature

  4. Aliases : Main Effect for Temperatures and Main Effect for Operators Main Effect for Operator Aliased with Main Effect for Temperature Aliasing/Confounding of Effects :Pilot Plant Experiment y = Constant + Main Effects + Interaction Effects + Operator Effect + Error M(Temp) = {180 Temp + Operator 2} - {160 Temp + Operator 1} = 75.75 - 52.75 = 23.0 Does 23.0 Measure the Effect of Temperatures, Operators, or Both ?

  5. Aliasing/Confounding of Effects :Pilot Plant Experiment y = Constant + Main Effects + Interaction Effects + Operator Effect + Error M(Temp) = {180 Temp + Operator 2} - {160 Temp + Operator 1} = 75.75 - 52.75 = 23.0 M(Cat) = {Cat C2 + (Operator 1 + Operator 2)/2} - {Cat C1 + (Operator 1 + Operator 2)/2} = {Cat C2 – Cat C1} = 65.0 - 63.5 = 1.5 Operator Effect Not Aliased with the Main Effect for Catalyst

  6. Main Effect for Temperature Catalyst Effect Effects Representations Overall Average Includes Average Influences From All Sources

  7. Overall Average c’cD = 0 Not Aliased Main Effect for Temperature cT’cD = 2 Aliased Catalyst Effect cC’cD = 0 Not Aliased Pilot Plant Experiment :Aliased Effects Operator Effect

  8. Aliasing / Confounding of Factor Effects Factor effects are Aliased or Confounded when differences in average responses cannot uniquely be attributed to a single effect • Unplanned confounding can result in loss of ability to evaluate important main effects and interactions • Planned aliasing of unimportant interactions can enable the size of the experiment to be reduced while still enabling the estimation of important effects Factor effects are Aliased or Confounded when they are estimated by the same linear combination of response values Factor effects are Partially Aliased or Partially Confounded when they are estimated by nonorthogonal linear combinations of response values

  9. General Confounding Principle for 2k Balanced Factoral Experiments Effects Representations Effect 1 = c1’y Effect 2 = c2’y Two Effects are Confounded or Aliased if Aliases : c1 = const x c2 Partial Aliases :

  10. Effects Representation for a Complete 23 Factorial Lower Level = -1 Upper level = +1 Effect = c’y / Divisor y = Vector of Responses or Average Responses

  11. Aliasing with Operator Same Alias if All Signs Reversed

  12. Aliasing with Operators Better design for operator aliasing?

  13. Aliasing with Operators Note: Operator effect is unconfounded with all effects except ABC; Good choice of contrast for aliasing with operators

  14. Summary • Some designs have one or more factors aliased with one another • Sums of squares measure the same effect or partially measure the same effect • The sums of squares are not statistically independent • Determining Aliases • If two-level factors, multiply effect contrasts • If nonzero, the effects are partially aliased • If one is a multiple of another, the effects are aliased

  15. Summary (con’t) • Accommodation • Eliminate one of the aliased effects • Leave all In but properly interpret analysis of variance results (to be discussed in subsequent classes)

  16. Two Types of Aliasing Fractional Factorials in Completely Randomized Designs: Can’t Run All Combinations Distinguish Randomized Incomplete Block Designs : Insufficient Homogeneous Experimental Units or Homogeneous Test Conditions in Each Block – Must Include Combinations in Two or More Blocks

  17. Fractional Factorials • Pilot Plant Chemical Yield Study • Temperature: 160, 180 oC • Concentration: 20, 40 % • Catalysts: 1, 2 • Too costly to run all 8 combinations • Must run fewer combinations

  18. Fractional Factorial Effect Partial Aliases Mean A, B, AB A Mean, B, AB C AC, BC, ABC Ad-Hoc Fraction

  19. Half-Fraction Fractional Factorial Half Fraction # Possible Combinations # Combinations in Design

  20. Poor Choice for a Fractional Factorial

  21. Poor Choice for a Fractional Factorial

  22. Good Choice for a Fractional Factorial Notation Defining Equation (Contrast) The effect(s) aliased with the mean I = ABC Convention Designate the mean by I (Identity)

  23. Confounding Pattern Main effects only aliased with interactions Defining Contrast I = ABC

  24. Resolution III (R = 3) Main Effects (s = 1) are unconfounded with other main effects (R - s = 2) Example : Half-Fraction of 23 (23-1) Design Resolution Resolution R Effects involving s factors are unconfounded with effects involving fewer than R-s factors

  25. Design Resolution Resolution R Effects involving s factors are unconfounded with effects involving fewer than R-s factors Resolution IV (R = 4) Main Effects (s = 1) are unconfounded with other main effects & two-factor interactions(R - s = 3) Two-factor interactions (s = 2) are unconfounded with main effects (R - s = 2); confounded with other two-factor interactions

  26. Confounding Pattern Resolution III Main Effects (s = 1) unaliased with other main effects (R - s = 2)

  27. Importance of Design Resolution • Quickly identifies the overall structure of the confounding pattern • A design of resolution R is a complete factorial in any R-1 or fewer factors

  28. B A C C B B A C A Figure 7.3 Projections of a half fraction of a three-factor complete factorial experiment (I=ABC).

  29. Pilot Plant Experiment :Half Fraction 45 80 C2 Catalyst 52 83 54 68 C1 I = ABC 40 Concentration 60 72 20 160 180 Temperature

  30. Pilot Plant Experiment : RIII is a Complete Factorial in any R-1 = 2 Factors 80 52 80 52 54 80 54 54 72 52 72 72 Catalyst Concentration Temperature

  31. Importance of Fractional Factorial Experiments Design Efficiency Reduce the size of the experiment through intentional aliasing of relatively unimportant effects

  32. Effects Representation for a Complete 23 Factorial Lower Level = -1 Upper level = +1 Effect = c’y / Divisor y = Vector of responses or average responses for the run numbers

  33. Designing a 1/2 Fraction of a 2k Complete Factorial Resolution = k • Write the effects representation for the main effects and the highest-order interaction for a complete factorial in k factors • Randomly choose the +1 or -1 level for the highest-order interaction (defining contrast, defining equation) • Eliminate all rows except those of the chosen level (+1 or -1) in the highest-order interaction • Add randomly chosen repeat tests, if possible • Randomize the test order or assignment to experimental units

  34. Resolution III Fractional Factorial I = +ABC Defining Contrast

  35. Aliasing Pattern • Write the defining equation (contrast) (I = Highest-order interaction) • Symbolically multiply both sides of the defining equation by each of the other effects • Reduce the right side of the equations: X x I = X X x X = X2 = I (powers mod(2) ) Resolution = III (# factors in the defining contrast) Defining Equation: I = ABC Aliases : A = AABC = BC B = ABBC = AC C = ABCC = AB

  36. Acid Plant Corrosion Rate Study 64 Combinations Cannot Test All Possible Combinations

  37. Acid Plant Corrosion Rate Study: Half Fraction (I = - ABCDEF) RVI

  38. F E C B A D Figure 7.4 Half Fraction (RVI) of a 26 Experiment: I = -ABCDEF.

  39. Designing Higher-Order Fractions • Total number of factor-level combinations = 2k • Experiment size desired = 2k/2p = 2k-p • Choose p defining contrasts (equations) • For each defining contrast randomly decide which level will be included in the design • Select those combinations which simultaneously satisfy all the selected levels • Add randomly selected repeat test runs • Randomize

  40. Acid Plant Corrosion Rate Study: Half Fraction (I = - ABCDEF) Half Fraction 26-1 RVI

  41. Acid Plant Corrosion Rate Study: Quarter Fractions I = - ABCDEF & I = ABC Quarter Fraction 26-2

  42. Acid Plant Corrosion Rate Study: Quarter Fraction (I = - ABCDEF = +ABC) Quarter Fraction 26-2

  43. F E C B A D Figure 7.5 Quarter fraction (RIII) of a 26 experiment: I = -ABCDEF = ABC (= -DEF).

  44. Acid Plant Corrosion Rate Study: Half Fraction (I = - ABCDEF = +ABC = -DEF) Implicit Contrast -ABCDEF x ABC = -AABBCCDEF = -DEF

  45. Design Resolution for Fractional Factorials • Determine the p defining equations • Determine the 2p - p - 1 implicit defining equations: symbolically multiply all of the defining equationsResolution = Smallest ‘Word’ length in the defining & implicit equations • Each effect has 2p aliases

  46. 26-2 Fractional Factorials :Confounding Pattern Build From 1/4 Fraction RIII I = ABCDEF = ABC = DEF A = BCDEF = BC = ADEF B = ACDEF = AC = BDEF . . . (I + ABCDEF)(I + ABC) = I + ABCDEF + ABC + DEF Defining Contrasts Implicit Contrast

  47. Optimal 1/4 Fraction RIV I = ABCD = CDEF = ABEF A = BCD = ACDEF = BEF B = ACD = BCDEF = AEF . . . 26-2 Fractional Factorials :Confounding Pattern Build From 1/2 Fraction RIII I = ABCDEF = ABC = DEF A = BCDEF = BC = ADEF B = ACDEF = AC = BDEF . . .

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