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 60 72 Concentration 20 160 180 Temperature

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 60 72 Concentration 20 160 180 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 ? Main Effect for Operator Aliased with Main Effect for Temperature Aliases : Main Effect for Temperatures and Main Effect for Operators

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

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

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

Aliasing / Confounding of Factor Effects Factor effects are Aliased or Confounded when differences in average responses cannot uniquely be attributed to a single effect 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 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

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 :

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

Aliasing with Operator Same Alias if All Signs Reversed

Aliasing with Operators Better design for operator aliasing?

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

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

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)

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

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

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

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

Poor Choice for a Fractional Factorial

Poor Choice for a Fractional Factorial

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)

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

Design Resolution Resolution R Effects involving s factors are unconfounded with effects involving fewer than R-s factors 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 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

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

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

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).

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

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

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

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

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

Resolution III Fractional Factorial I = +ABC Defining Contrast

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) ) Defining Equation: I = ABC Aliases : A = AABC = BC B = ABBC = AC C = ABCC = AB Resolution = III (# factors in the defining contrast)

Acid Plant Corrosion Rate Study Factor Levels Raw-material feed rate 3000 pph 6000 pph Gas temperature 100 oC 200 oC Scrubber water 5% 20% Reactor-bed acid 30% Exit temperature 300 oC 360 oC Reactant distribution point East West 64 Combinations Cannot Test All Possible Combinations

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

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

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

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

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

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

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

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

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

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

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