Constructing Orthogonal Arrays Robust System Design 16.881 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 Robust System Design 16.881 MIT What is orthogonality? ? Geometry v v ? Vector algebra x ? y = 0 ? Robust design –Form contrasts for the columns (i) w i1 + w i 2 + w i 3 L + w i 9 = 0 – Inner product of contrasts must be zero w <i> ? w < j> = 0 Robust System Design 16.881 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 Robust System Design 16.881 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 -- 2 1 x3 7 Robust System Design 16.881 MIT Breakdown of DOF Robust System Design MIT16.881 n 1 SS due to mean n-1 (# levels) -1 factor A n = Number of η values (# levels) -1 factor B DOF for error etc. DOF and Modeling Equations ? Additive model 0 η( A i , B j , C k , D i ) =μ+ a i + b j + c k + d i e + ? How many parameters are there? ? How many additional equations constrain the parameters? Robust System Design 16.881 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? γ y z (X,Y,Z) y z α β x x Rotation Translation Robust System Design 16.881 MIT Notation for Matrix Experiments Number of experiments L 9 (3 4 ) Number of levels Number of factors 9=(3-1)x4+1 Robust System Design 16.881 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! Robust System Design 16.881 MIT Standard Orthogonal Arrays Orthogonal Array Number of Rows Maximum Number of Factors Maximum Number of Columns at These Levels 2 3 4 5 L 4 4 3 3 - - - L 8 8 7 7 - - - L 9 9 4 - 4 - - L 12 12 11 11 - - - L 16 16 15 15 - - - L’ 16 16 5 - - 5 - L 18 18 8 1 7 - - L 25 25 6 - - - 6 L 27 27 13 1 13 - - L 32 32 31 31 - - - L’ 32 32 10 1 - 9 - L 36 36 23 11 12 - - L’ 36 36 16 3 13 - - L 50 50 12 1 - - 11 L 54 54 26 1 25 - - L 64 64 63 63 - - - L’ 64 64 21 - - 21 - L 81 81 40 - 40 - - Robust System Design 16.881 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 Robust System Design 16.881 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? Robust System Design 16.881 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? Robust System Design 16.881 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? Robust System Design 16.881 MIT Dummy Levels Preserve Orthogonality ? Let’s demonstrate this for Example 2 ? But only if we assign the dummy level consistently Robust System Design 16.881 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? Robust System Design 16.881 MIT Compounding Factors ? Assigns two factors to a single column by merging two factors into one Before After A 1 B 1 C 1 =A 1 B 1 C 2 =A 1 B 2 A 2 B 2 C 3 =A 2 B 2 Robust System Design 16.881 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? Robust System Design 16.881 MIT Considerations in Compounding ? Balancing property not preserved between compounded factors C 1 =A 1 B 1 C 2 =A 1 B 2 C 3 =A 2 B 2 ? Main effects confounded to some degree ? ANOVA becomes more difficult Robust System Design 16.881 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 Robust System Design 16.881 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? Robust System Design 16.881 MIT Two Level Interactions in L 4 ? AxB Interaction = ( y A 2 B 2 ? y A 1 B 2 ) ? ( y A 2 B 1 ? y A 1 B 1 ) ? As you learned from the noise experiment Interaction Plot 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 B1 B2 A 1 A 2 } AxB Robust System Design 16.881 MIT Interactions in Larger Matrices ? AxB Interaction = ( y A 2 B 2 ? y A 1 B 2 ) ? ( y A 2 B 1 ? y A 1 B 1 ) ? Average the rows with the treatment levels listed above Interaction Plot 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 B1 B2 A 1 A 2 } AxB Robust System Design 16.881 MIT 1 2 3 4 5 6 7 8 Two Factor Interaction Numerical Example ? 4x6 = ( y A 2 B 2 ? y A 1 B 2 ) ? ( y A 2 B 1 ? y A 1 B 1 ) Run c1 4x6 c3 A c5 B c7 N 1 1.2 1 1.7 1 2.1 1 2.6 2 4.9 2 3.9 2 0.9 2 1.1 1 1 1 1 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 2 1 2 1 2 1 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 Robust System Design 16.881 MIT Three Level Interactions ? AxB has 4 DOF ? Each CF has 2DOF ? Requires two unassigned columns (the right ones) 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 B1 B2 B3 A 1 A 2 A3 Robust System Design 16.881 MIT Linear Graphs ? To study interaction between CF dot and CF dot, leave CF on connecting line unassigned 1 3 5 2 6 4 e.g., L 8 Robust System Design 16.881 MIT Column Merging ? Can turn 2 two level factors into a 4 level factor ? Can turn 2 three level factors into a six level factor ? Need to strike out interaction column (account for the right number of DOF!) ? Example on an L 8 Robust System Design 16.881 MIT Branching Design ? One control factor determines the appropriate choice of other control factors ? Strike out the parent x child column to preserve the balancing property Baking Method Conv IR C1 C2 D Temp F Light Int E Time G Belt Speed Robust System Design 16.881 MIT 1 2 3 4 5 6 7 8 Branching Design Oven Type Temp / Light Int Is the balancing Run c1 c2 c3 c4 c5 c6 c7 1 1 1 1 2 2 2 2 T I 1 1 1 1 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 2 1 2 1 2 1 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 property preserved? How can we recover balance? Robust System Design 16.881 MIT Next Steps ? Homework #7 due on Lecture 10 ? Next session tomorrow – Read Phadke Ch. 10 – Read “Planning Efficient Software Tests” – Tought questions: ? What does software do? ? How is software different from hardware? ? How does this affect the application of RD? ? Quiz on Constructing Arrays Robust System Design 16.881 MIT