Two-Level Fractional Factorial Design
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1 Two-Level Fractional Factorial Design Reference DeVor, Statistical Quality Design and Control, Ch. 19, 0 1 Andy Guo Types of Experimental Design Parallel-type approach Sequential-type approach One-factor design ( levels) Hypothesis testing, confidence interval (randomized design) Paired comparison (block design) One-factor design (k levels) Completely randomized design Randomized complete block design Two block design (Latin square) Two-factor design Full factorial design ( level) Fractional factorial design ( level) Robust design Nested design Split-plot design Response surface method design Central composite design Box-Behnken design Computer-aided design (D, G optimal design) EVOP Steepest ascent Andy Guo
2 Redundancy in Two-Level Factorials A 10-variable experiment require 10 = 104 tests. Such a test plan is simply prohibitive in size. In theory, the following effects are estimated from the test plan: 1 Mean response 10 Main effects 45 Two-factor interaction effects 10 Three-factor interaction effects 10 Four-factor interaction effects 5 Five-factor interaction effects 10 Six-factor interaction effects 10 Seven-factor interaction effects 45 Eight-factor interaction effects 10 Nine-factor interaction effects 1 Ten-factor interaction effects Variable effects. In reality, interaction effects involving three factors or more are small and can be simply ignored. This fact provides the opportunity for fractional factorial designs. 3 Andy Guo Why do Fractional Factorial Designs Work? The sparsity of effects principle There may be lots of factors, but few are important System is dominated by main effects, low-order interactions The projection property Every fractional factorial contains full factorials in fewer factors Sequential experimentation Can add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation 4 Andy Guo
3 A 3-1 Fractional Factorial Design Test 1 3= variables are studied. in 3-1 = 4 tests 3. p = 1 of the variables is introduced into a full factorial 4. by assigning it to the interaction 1 (i.e., let 3 = 1) (-,-,+) (+,+,+) 3 (-,+,-) 1 (+,-,-) 5 Andy Guo A 4-1 Fractional Factorial Design Test 1 3 4= variables are studied. in 4-1 = 8 tests 3. p = 1 of the variables is introduced into a 3 full factorial 4. by assigning it to the interaction 13 (i.e., let 4 = 13) 6 Andy Guo
4 A 4-1 Fractional Factorial Design (-,+,+) (+,-,+) (-,-,+) (+,+,+) (+,+,-) (-,+,-) (-,-,-) (+,-,-) Andy Guo Calculation Matrix and Confounding Patterns Test I The following pairs of variables are confounded 1 and 34 1 and 34 and and 4 3 and 14 3 and 14 4 and 13 average(i) and Andy Guo
5 Effect Estimation l 0 estimates mean + (1/)(134) l 1 estimates l estimates l 3 estimates l 1 estimates l 13 estimates l 3 estimates l 13 estimates If three-factor and four-factor interactions can be neglected, we have: l 0 estimates mean l 1 estimates 1 l estimates l 3 estimates 3 l 1 estimates l 13 estimates l 3 estimates l 13 estimates 4 9 Andy Guo Procedure for Fractional Factorial Designs Example: 6 variables, only 8 tests are allowed. Step 1: Define the base design Base design: 3 full factorial design Test I y y y y y y y y y 8 Divisor Andy Guo
6 Step : Introduction of Additional Variables 4 = 1 5 = 13 6 = 3 Design matrix: 6-3 fractional factorial design Test Andy Guo Step 3: Obtain the Defining Relation 4 = 1 4 x 4 = 1 x 4, but since 4 x 4 = I, a column of (+) signs, we have I = 14. The defining relation is given by I = 14, I = 135, I = 36 (the generators) plus two-at-a-time products: (14)(135) = 345 (14)(36) = 1346 (135)(36) = 156 plus the three-at-a-time products: (14)(135)(36) = 456. The complete defining relation I is therefore I = 14 = 135 = 36 = 345 = 1346 = 156 = Andy Guo
7 Step 4: Reveal the Complete Confounding Structure With (1)I = (1)14 = (1)135 = (1)36 = (1)345 = (1)1346 = (1)156 = (1)456 We have 1 = 4 = 35 = 136 = 1345 = 346 = 56 = Assuming that third-and higher-order interactions can be neglected, l 1 estimates Summary of the confounding structure: l 0 estimates mean l 1 estimates l 1 estimates l 13 estimates l estimates l 3 estimates l 3 estimates l 13 estimates Andy Guo Concept of Resolution Resolution = the number of letters (numbers) in the shortest length word (term) in the defining relation, excluding I. I = 14 = 135 = 345 => Resolution III I = 135 = 346 = 1456 => Resolution IV Resolution III => some main effects are confounded with two-factor interactions. Resolution IV => some main effects are confounded with three-factor interactions, and some two-factor interactions are confounded with other two-factor interactions. Resolution V => some main effects are confounded with four-factor interactions and some two-factor interactions are confounded with threefactor interactions. 14 Andy Guo
8 Concept of Design Resolution Higher-resolution designs seem more desirable since they provide the opportunity for low-order effect estimates to be determined in an un-confounded state, assuming that higher-order interaction effects can be neglected. There is a limit to the number of variables that can be considered in a fixed number of tests while maintaining a pre-specified resolution requirement. No more than (n-1) variables can be examined in n tests (n is a power of ) to maintain a design resolution of at least III. Such designs are commonly referred to as saturated designs. Examples are 3-1, 7-4, 15-11, 31-6 For saturated designs all interactions in the base design variables are used to introduce additional variables. 15 Andy Guo Two-Level Fractional Factorial Designs with Maximum Resolution Number of Factors Fraction 3 1 III Number of Runs 3 4 I= IV 4 8 I= V 5 16 I= III 6 1 VI 6 IV 8 I=14= I= III 16 I=135=346 Defining Relation (omitting generalized interactions) 8 I=14=135=36 16 Andy Guo
9 Two-Level Fractional Factorial Designs with Maximum Resolution Number of Factors Fraction 7 1 VII Number of Runs 7 64 I= Defining Relation (omitting generalized interactions) 7 IV 7 3 IV 7 4 III 3 I=1346= I=135=346= I=14=135=36=137 8 V 8 64 I=1347= IV 8 4 IV 9 VI 3 I=136=147= I=345=1346=137= I=134678= IV 64 I=1347=13568= IV 3 I=3456=13457=1458= III 16 I=135=346=1347=148= Andy Guo Orthogonal Arrays and Two-Level Fractional Factorial Designs L8 ( 7 ) Orthogonal Array Factor A B C D E F G Test Result y y y y y y y y 8 III 7-4 Fractional Factorial Design Test D B A F E C G (4) () (1) (6) (5) (3) (7) 18 Andy Guo
10 Orthogonal Arrays and Two-Level Fractional Factorial Designs L16 ( 15 ) Orthogonal Array III 8-4 Fractional Factorial Design A B A x x x F A e B e B E C H e D e D G D Test Test C B A F E H G D 19 Andy Guo Sequential Experimentation A 7-4 fractional factorial design Resolution = III I = 14 = 135 = 36 = 137 Test Overrun (%) Andy Guo
11 Estimation of Effects and Results Cumulative probability (%) Estimate of linear combination of effects The main effects of variables 1,, 6 alone are important. The main effects of variables 1and 6, as well as the interactions 14 and/or 36 are important. We might conclude that l 3 is large because interactions 17 and/or 3 are important instead of variable 6. We might conclude that l 1 is large because interactions 4 and/or 67 are important instead of variable 1. 1 Andy Guo Mirror Image Design Switch the signs for all of the columns in the original design A 7-4 fractional factorial design Resolution = III I = -14 = -135 = -36 = 137 Test Overrun (%) Andy Guo
12 Comparison of Original and Mirror Image Tests Mirror Image Design Original Eight Tests Additional Eight Tests l 0 = and estimates mean l' 0 = and estimates mean l 1 = and estimates l' 1 = and estimates l = and estimates l' = and estimates l 3 = and estimates l' 3 = and estimates l 1 = and estimates l' 1 = and estimates l 13 = and estimates l' 13 =.500 and estimates l 3 = and estimates l' 3 = and estimates l 13 = and estimates l' 13 = and estimates Andy Guo l1 Unconfounding the Main Effects 1 1 = = estimate 1. 1 [( ) + ( ) ] estimates [( ) + ( ) ] Similarly, l = estimate l3 3 = estimate 3 l1 1 = estimates l13 13 = estimates l3 3 = estimates l3 3 = 6.65 estimate 7. 4 Andy Guo
13 Unconfounding the Main Effects l1 l' 1 1 = =.875 estimates [( ) ( ) ] estimates [( ) ( ) ] l l' = estimates l3 l' 3 = estimates l1 l' 1 = 5.15 estimates 4 l13 l' 13 = estimates 5 l3 l' 3 = estimates 6 l13 l' 13 = 9.65 estimates Andy Guo Unconfounding the Main Effects l0 0 = estimates mean. By taking the difference 0 0 between l 0 and l' 0, the following result is obtained l l' = 3.15 estimates : 6 Andy Guo
14 Results of the Combined Designs The combined designs are the same as a 7-3 design. Resolution = IV I = 137 = 345 = 1346 Estimate of 1 = Estimate of = Estimate of 3 = Estimate of = Estimate of = Estimate of = Estimate of 7 = 6.65 Estimate of error = 3.15 Estimate of =.875 Estimate of = Estimate of = Estimate of 4 = Estimate of 5 = Estimate of 6 = Estimate of = Andy Guo Results Based on Tests 1 to 8 and 9 to 16 Cumulative probability (%) Estimate of linear combination of effects The main effects of variables 1,, 4, 6 are important. The interaction 1 is important. 8 Andy Guo
15 Alternative Experimental Strategies Principal fraction Original 7-4 design: I = 14 = 135 = 36 = 137 Alternate fractions Mirror image: I = -14 = -135 = -36 = 137 Family of fractional factorials I = ±14 = ± 135 = ± 36 = ± 137 Depending on the interpretation of the results of the principal fraction, we may choose any one of several other alternate fractions to achieve a particular result when two fractions are combined. 9 Andy Guo One Alternative Design Example Suppose that the larger linear combinations from the first experiment, l 1 estimates = l estimates = l 3 estimates = Suppose that our knowledge suspects that variable 1 was important. => folding only the first column Test Overrun (%) Generators:I = -14, I = -135, I = 36, I = Andy Guo
16 Comparison of Test Results Principal Fraction Design Tests 1-8 Alternative Fraction Design Tests 17-4 l 0 = and estimates mean l'' 0 = and estimates mean l 1 = and estimates l'' 1 = and estimates l = and estimates l'' = and estimates l 3 = and estimates l'' 3 = and estimates l 1 = and estimates l'' 1 = and estimates l 13 = and estimates l'' 13 = and estimates l 3 = and estimates l'' 3 = and estimates l 13 = and estimates l'' 13 = and estimates Andy Guo Results Based on Tests 1 to 8 and 17 to 4 Variable 1 and the interactions 1 and 15 are important. By folding a single column, we can estimate the main effect of that variable and its two-factor interactions better. Cumulative probability (%) Estimate of linear combination of effects 3 Andy Guo
17 7-4 III Family of Fractional Factorials Fraction Generators Combined with Principal Fraction Gives Estimates of:* Principal I=+14 I=+135 I=+36 I= A 1 I=-14 I=+135 I=+36 I=+137 4,14,4,34,45,46,47 A I=+14 I=-135 I=+36 I=+137 5,15,5,35,45,56,57 A 3 I=-14 I=-135 I=+36 I=+137 A 4 I=+14 I=+135 I=-36 I=+137 6,16,6,36,46,56,67 A 5 I=-14 I=+135 I=-36 I=+137 A 6 I=+14 I=-135 I=-36 I=+137 A 7 I=-14 I=-135 I=-36 I=+137 All main effects A 8 I=+14 I=+135 I=+36 I=-137 7,17,7,37,47,57,67 A 9 I=-14 I=+135 I=+36 I=-137 A 10 I=+14 I=-135 I=+36 I=-137 A 11 I=-14 I=-135 I=+36 I=-137 1,1,13,14,15,16,17 A 1 I=+14 I=+135 I=-36 I=-137 A 13 I=-14 I=+135 I=-36 I=-137,1,3,4,5,6,7 A 14 I=+14 I=-135 I=-36 I=-137 3,13,3,34,46,56,67 A 15 I=-14 I=-135 I=-36 I=-137 * Assuming that third- and higher-order interactions are negligible. 33 Andy Guo Summary of Results Principal Fraction + Mirror Image Design (A 7 ) Principal Fraction + Alternative Design (A 11 ) 1 estimated as estimated as estimated as estimated as estimated as estimated as estimated as estimated as estimated as estimated as estimated as estimated as Andy Guo
18 Interpretation Both combined designs produce an unconfounded estimates of the main effect of variable 1. Both combined designs show that interaction 1 is important. Both combined design show that main effect 4 is important. Both combined design show that main effect 6 is important. Based on (principal + A11), interaction 15 is important. Both combined design show that main effect is important. The interactions (36+57) and (6+47) are important. l l l l l 1 13 l 3 ' ' ' = 1.50 estimates = estimates = estimates l l' l l'' = estimates l3 l' 3 l3 l'' 3 = 7.50 estimates l13 l' 13 l13 l'' 13 = 1.50 estimates Andy Guo Possible Strategies for Sequential Experimentation (a) Move to new location to explore an apparent trend in response Initial design (b) Add another fraction to resolve ambiguities from the original fraction (f) Augment to model apparent curvature Pressure Temperature Catalyst feed rate (c) Rescale some factors because they may have been varied over inappropriate ranges (e) Replicate to improve estimates of effects or because some runs were incorrectly made Pressure Temperature (d) Drop and add factors because the original factor catalyst feed rate is negligible Time 36 Andy Guo
19 Case Study: Epitaxy Process The objective is to study how the control factors affect the thickness of the the epitaxial layer. Eight factors were studied: A: rotation method B: the code of the wafers C: deposition temperature D: deposition time E: arsenic gas flow rate F: HCl etch temperature G: HCl flow rate H: nozzle position 37 Andy Guo Experimental Design and Results 8-4 fractional factorial design A full 4 factorial in A, B, C, E D=-ABC, F=ABE, G=ACE, H=CBE A B C D E F G H y log(s ) Andy Guo
20 Analysis Results of the Main Effects Mean thickness A = , B = 0.056, C = , D = E = , F = 0.060, G = , H = 0.14 Variability of thickness A = 0.35, B = 0.1, C = 0.105, D = 0.49 E = -0.01, F = -0.07, G = , H = Factor D has the largest impact on the mean level. Factors A and H affect the variability. 39 Andy Guo Case Study: Plasma Etching A nitride etch process on a single-wafer plasma etcher Output: etch rate Inputs: gap, pressure, C F 6 flow rate, power Design Factor Gap Pressure C F 6 Flow Power A B C D Level (cm) (m Torr) (SCCM) (W) Low (-) High (+) Andy Guo
21 4-1 Fractional Factorial Design Run A B C D=ABC Etch Rate Andy Guo Estimated Effects l A = A + BCD = l B = B + ACD = 4.00 l C = C + ABD = l D = D + ABC = l AB = AB + CD = l AC = AC + BD = l AD = AD + BD = A, D, AD are significant 4 Andy Guo
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