Constructing Orthogonal Arrays 16 881 Robust System Design MIT Learning Objectives Introduce explore orthogonality Study the standard OAs Practice computing DOF of an experiment Learn how to select a standard OA Introduce means to modify OAs Consider studying interactions in OAs 16 881 Robust System Design MIT What is orthogonality Geometry Vector algebra Robust design v v x y 0 Form contrasts for the columns i wi1 wi 2 wi 3 L wi 9 0 Inner product of contrasts must be zero w i w j 0 16 881 Robust System Design MIT Before Constructing an Array We must define Number of factors to be studied Number of levels for each factor 2 factor interactions to be studied Special difficulties in running experiments 16 881 Robust System Design MIT Counting Degrees of Freedom Grand mean 1 Each control factor e g A of levels of A 1 Each two factor interaction e g AxB DOF for A x DOF for B Example 21x37 16 881 Robust System Design MIT Breakdown of DOF n 1 SS due to mean levels 1 factor A 16 881 n Number of values n 1 levels 1 factor B Robust System Design etc DOF for error MIT DOF and Modeling Equations Additive model Ai B j Ck Di ai b j ck d i e How many parameters are there 0 How many additional equations constrain the parameters 16 881 Robust System Design MIT DOF Analogy with Rigid Body Motion How many parameters define the position and orientation of a rigid body How do we remove these DOF z z y X Y Z x Rotation 16 881 y x Translation Robust System Design MIT Notation for Matrix Experiments L9 34 Number of experiments Number of levels Number of factors 9 3 1 x4 1 16 881 Robust System Design MIT Standard Orthogonal Arrays See table 7 1 on Phadke page 152 Note You can never use an array that has fewer rows than DOF req d Note The number of factors of a given level is a maximum You can put a factor with fewer columns into a column that has more levels But NOT fewer 16 881 Robust System Design MIT Standard Orthogonal Arrays Orthogonal Array L4 L8 L9 L12 L16 L 16 L18 L25 L27 L32 L 32 L36 L 36 L50 L54 L64 L 64 L81 16 881 Number of Rows 4 8 9 12 16 16 18 25 27 32 32 36 36 50 54 64 64 81 Maximum Number of Factors 3 7 4 11 15 5 8 6 13 31 10 23 16 12 26 63 21 40 Maximum Number of Columns at These Levels 2 3 4 5 3 7 4 11 15 5 1 7 6 1 13 31 1 9 11 12 3 13 1 11 1 25 63 21 40 Robust System Design MIT Difficulty in Changing Levels Some factor levels cost money to change Paper airplane Other examples Note All the matrices in Appendix C are arranged in increasing order of number of level changes required left to right Therefore put hard to change levels in the leftmost columns 16 881 Robust System Design MIT Choosing an Array Example 1 1 two level factor 5 three level factors What is the number of DOF What is the smallest standard array that will work 16 881 Robust System Design MIT Choosing an Array Example 2 2 two level factor 3 three level factors What is the number of DOF What is the smallest standard array that will work 16 881 Robust System Design MIT Dummy Levels Turns a 2 level factor into a 3 level factor or a 3 to a 4 etc By creating a new level A3 that is really just A1 or A2 Let s consider example 2 Question What will the factor effect plot look like 16 881 Robust System Design MIT Dummy Levels Preserve Orthogonality Let s demonstrate this for Example 2 But only if we assign the dummy level consistently 16 881 Robust System Design MIT Considerations in Assigning Dummy Levels Desired accuracy of factor level effect Examples Cost of the level assignment Examples Can you assign dummy levels to more than one factor in a matrix experiment Can you assign more than one dummy level to a single factor 16 881 Robust System Design MIT Compounding Factors Assigns two factors to a single column by merging two factors into one After Before A1 A2 16 881 C1 A1B1 B1 C2 A1B2 B2 C3 A2B2 Robust System Design MIT Compounding Factors Example 3 two level factors 6 three level factors What is the smallest array we can use How can compounding reduce the experimental effort 16 881 Robust System Design MIT Considerations in Compounding Balancing property not preserved between compounded factors C A B 1 1 1 C2 A1B2 C3 A2B2 Main effects confounded to some degree ANOVA becomes more difficult 16 881 Robust System Design MIT Interaction Tables To avoid confounding A and B with AxB leave a column unassigned To know which column to leave unassigned use an interaction table 16 881 Robust System Design MIT Interaction Table Example We are running an L8 We believe that CF4 and CF6 have a significant interaction Which column do we leave open 16 881 Robust System Design MIT Two Level Interactions in L4 AxB Interaction y A2 B2 y A1B2 y A2 B1 y A1B1 As you learned from the noise experiment 16 881 Interaction Plot 65 0 AxB 60 0 55 0 50 0 A1 45 0 A2 40 0 35 0 30 0 Robust System Design B1 B2 MIT Interactions in Larger Matrices AxB Interaction y A2 B2 y A1B2 y A2 B1 y A1B1 Average the rows with the treatment levels listed above 16 881 Interaction Plot 65 0 AxB 60 0 55 0 50 0 A1 45 0 A2 40 0 35 0 30 0 Robust System Design B1 B2 MIT Two Factor Interaction Numerical Example 4x6 Run 1 2 3 4 5 6 7 8 16 881 y A2 B2 y A1B2 y A2 B1 y A1B1 c1 4x6 1 1 1 1 2 2 2 2 1 1 2 2 1 1 2 2 c3 A c5 B c7 1 1 2 2 2 2 1 1 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 Robust System Design N 1 2 1 7 2 1 2 6 4 9 3 9 0 9 1 1 MIT Three Level Interactions AxB has 4 DOF Each CF has 2DOF Requires two unassigned columns the right ones 16 881 70 0 60 0 50 0 40 0 A1 A2 A3 30 0 20 0 10 0 0 0 Robust System Design B1 B2 B3 MIT Linear Graphs To study interaction between CF dot and CF dot leave CF on connecting line unassigned 16 881 Robust System Design 1 3 2 5 6 4 e g L8 MIT Column Merging Can turn 2 two level factors into a 4 level factor Can …