W11INSE6220 - 1 3 INSE 6220 Week 11 A y A y A Advanced...

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1 INSE 6220 -- Week 11 Advanced Statistical Approaches to Quality Design of Experiments 2 k Design Fractional Replication of 2 k Design Dr. A. Ben Hamza Concordia University 2 Analysis Procedure for Factorial Experiments The 2 k Design for k>=3 factors Cube plots are often useful visual displays of experimental results Analysis Procedure for Factorial Designs 1. Estimate the factor effects 4. Analyze residuals 2. Form preliminary model 5. Refine model, if necessary 3. Test for significance of factor effects 6. Interpret results 3 etc, etc, ... A A B B C C A y y B y y C y y Analysis done via computer 4 Except for column the identity I , every column has an equal number of + and signs The sum of the product of signs in any two columns is zero Multiplying any column by column I leaves that column unchanged (identity element) The product of any two columns yields a column in the table: 2 A B AB AB BC AB C AC 2 2 2 (Contrast) (Effect) 2 2 k k SS n n 1 Contrast Effect 2 k n 2 k Design for k=3 factors
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5 The 2 k Design for k>=3 factors Linear model: y A B C AB AC BC ABC 1 [ (1)] 4 A A A y y a ab ac abc b c bc n 1 [ (1)] 4 B B B y y b ab bc abc a c ac n 1 [ (1)] 4 C C C y y c ac bc abc a b ab n 1 [ (1) ] 4 AB ab abc c b a bc ac n 1 [ (1) ] 4 AC ac abc b a c ab bc n 1 [ (1) ] 4 BC bc abc a b c ab ac n 1 [ (1)] 4 ABC abc bc ac c ab b a n 6 A 1 3 1 Contrast 2 1 (1) (2)(2 ) 1 22 27 23 30 20 21 18 16 (2)(4) 1 27 3.375 8 k A n a ab ac abc b c bc Design Factors Run A B C Surface Finish Totals 1 (1) -1 -1 -1 9, 7 16 2 a 1 -1 -1 10, 12 22 3 b -1 1 -1 9, 11 20 4 ab 1 1 -1 12, 15 27 5 c -1 -1 1 11, 10 21 6 ac 1 -1 1 10, 13 23 7 bc -1 1 1 10, 8 18 8 abc 1 1 1 16, 14 30 Example 1: An experiment was performed to investigate the surface finish of a metal part. The experiment is a 2 3 factorial design in the factors feed rate ( A ), depth of cut ( B ), and tool angle ( C ), with n =2 replicated. The Table below presents the observed surface- finish data for this experiment, and the design is shown graphically in Figure 12-25.
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