Constructing
Orthogonal Arrays
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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
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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
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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
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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
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Breakdown of DOF
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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?
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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
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Notation for Matrix Experiments
Number of
experiments
L
9
(3
4
)
Number of levels
Number of factors
9=(3-1)x4+1
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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!
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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 - -
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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
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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?
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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?
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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?
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Dummy Levels Preserve
Orthogonality
? Let’s demonstrate this for Example 2
? But only if we assign the dummy level
consistently
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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?
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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
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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?
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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
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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
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Interaction Table Example
? We are running an L8
? We believe that CF4 and CF6 have a
significant interaction
? Which column do we leave open?
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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
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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
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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
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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
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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
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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
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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
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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?
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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
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