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Orthogonal Array

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The Orthogonal Arrays constructed with a fraction of a Full factorial array but the orthogonality (in-dependency) between the factors is kept. The factors are independent from each other and in balance. Due to this the factors can be evaluated independently of each other. These arrays are often used in an DOE test setup. Interactions are not included in the array but the array can be constructed to make it possible (see Interactions).



The use of Fractional arrays can increase the amount of factors with the same amount of tests. In this case 7 in stead of 3 different factors can be investigated with 8 tests.
Full Factorial Orthogonal
A B C A B C D E F G
1 1 1 1 ==> 1 1 1 1 1 1 1
2 2 1 1 1 1 1 2 2 2 2
3 1 2 1 1 2 2 1 1 2 2
4 2 2 1 1 2 2 2 2 1 1
5 1 1 2 2 1 2 1 2 1 2
6 2 1 2 2 1 2 2 1 2 1
7 1 2 2 2 2 1 1 2 2 1
8 2 2 2 2 2 1 2 1 1 2

The use of Fractional arrays reduces the amount of test in this case 4 in stead of 8.
Full Factorial Orthogonal
A B C A B C
1 1 1 1 ==> 1 1 1
2 2 1 1 1 2 2
3 1 2 1 2 1 2
4 2 2 1 2 2 1
5 1 1 2
6 2 1 2
7 1 2 2
8 2 2 2

Selecting a Orthogonal array

Menu => DOE => Factorial Array
Select the array.

Combining columns and editing arrays

See edit factorial array.

Notation of a Orthogonal Array

formula
L = Latin square
a = Amount of tests (rows)
b = Amount of Levels
c = number of columns

Name Factors 2 Levels 3 Levels 5 Levels
L4 3 3
L8 7 7
L9 4 4
L12 11 11
L16 15 15
L18 8 1 7
L25 6 6
L27 13 13
L32 31 31
L36 23 11 12
L50 12 1 11
L54 26 1 25
When 4 or more levels are needed columns can be combined to achieve this (see edit factorial array) .

L4

formula => 3 Factors on 2 levels
Tests ↓ Factors
A B C
1 -1 -1 -1
2 -1 1 1
3 1 -1 1
4 1 1 -1

L8

formula => 7 Factors on 2 levels
Tests ↓ Factors
A B C D E F G
1 -1 -1 -1 -1 -1 -1 -1
2 -1 -1 -1 1 1 1 1
3 -1 1 1 -1 -1 1 1
4 -1 1 1 1 1 -1 -1
5 1 -1 1 -1 1 -1 1
6 1 -1 1 1 -1 1 -1
7 1 1 -1 -1 1 1 -1
8 1 1 -1 1 -1 -1 1

L9

formula => 4 Factors on 3 levels
Tests ↓ Factors
A B C D
1 -1 -1 -1 -1
2 -1 -1 -1 -1
3 -1 1 1 1
4 -1 -1 -1 1
5 -1 -1 1 -1
6 -1 1 -1 -1
7 1 -1 1 -1
8 1 -1 -1 1
9 1 1 -1 -1

L12

formula => 11 Factors on 2 levels
Tests ↓ Factors
A B C D E F G H I J K
1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
2 -1 -1 -1 -1 -1 1 1 1 1 1 1
3 -1 -1 1 1 1 -1 -1 -1 1 1 1
4 -1 1 -1 1 1 -1 1 1 -1 -1 1
5 -1 1 1 -1 1 1 -1 1 -1 1 -1
6 -1 1 1 1 -1 1 1 -1 1 -1 -1
7 1 -1 1 1 -1 -1 1 1 -1 1 -1
8 1 -1 1 -1 1 1 1 -1 -1 -1 1
9 1 -1 -1 1 1 1 -1 1 1 -1 -1
10 1 1 1 -1 -1 -1 -1 1 1 -1 1
11 1 1 -1 1 -1 1 -1 -1 -1 1 1
12 1 1 -1 -1 1 -1 1 -1 1 1 -1

L16

formula => 15 Factors on 2 levels
Tests ↓ Factors
A B C D E F G H I J K L M N O
1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
2 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1
3 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1
4 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1
5 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1
6 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1
7 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1
8 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1
9 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
10 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1
11 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1
12 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1
13 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1
14 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1
15 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1
16 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1

L18

formula => 1 Factor on 2 levels and 7 on 3 levels
Tests ↓ Factors
A B C D E F G H
1 -1 -1 -1 -1 -1 -1 -1 -1
2 -1 -1 -1 -1 -1 -1 -1 -1
3 -1 -1 1 1 1 1 1 1
4 -1 -1 -1 -1 -1 -1 1 1
5 -1 -1 -1 -1 1 1 -1 -1
6 -1 -1 1 1 -1 -1 -1 -1
7 -1 1 -1 -1 -1 1 -1 1
8 -1 1 -1 1 -1 -1 1 -1
9 -1 1 1 -1 1 -1 -1 -1
1-1 -1 -1 -1 1 1 -1 -1 -1
11 -1 -1 -1 -1 -1 1 1 -1
12 -1 -1 1 -1 -1 -1 -1 1
13 -1 -1 -1 -1 1 -1 1 -1
14 -1 -1 -1 1 -1 -1 -1 1
15 -1 -1 1 -1 -1 1 -1 -1
16 -1 1 -1 1 -1 1 -1 -1
17 -1 1 -1 -1 1 -1 -1 1
18 -1 1 1 -1 -1 -1 1 -1

L25

formula => 6 Factors on 5 Levels
Test ↓ Factors
A B C D E F
1 -2 -2 -2 -2 -2 -2
2 -2 -1 -1 -1 -1 -1
3 -2 0 0 0 0 0
4 -2 1 1 1 1 1
5 -2 2 2 2 2 2
6 -1 -2 -1 0 1 2
7 -1 -1 0 1 2 -2
8 -1 0 1 2 -2 -1
9 -1 1 2 -2 -1 0
10 -1 2 -2 -1 0 1
11 0 -2 0 2 -1 1
12 0 -1 1 -2 0 2
13 0 0 2 -1 1 -2
14 0 1 -2 0 2 -1
15 0 2 -1 1 -2 0
16 1 -2 1 -1 2 0
17 1 -1 2 0 -2 1
18 1 0 -2 1 -1 2
19 1 1 -1 2 0 -2
20 1 2 0 -2 1 -1
21 2 -2 2 1 0 -1
22 2 -1 -2 2 1 0
23 2 0 -1 -2 2 1
24 2 1 0 -1 -2 2
25 2 2 1 0 -1 -2

L27

formula => 13 Factors on 3 Levels
Tests ↓ Factors
A B C D E F G H I J K L M
1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
2 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0
3 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1
4 -1 0 0 0 -1 -1 -1 0 0 0 1 1 1
5 -1 0 0 0 0 0 0 1 1 1 -1 -1 -1
6 -1 0 0 0 1 1 1 -1 -1 -1 0 0 0
7 -1 1 1 1 -1 -1 -1 1 1 1 0 0 0
8 -1 1 1 1 0 0 0 -1 -1 -1 1 1 1
9 -1 1 1 1 1 1 1 0 0 0 -1 -1 -1
10 0 -1 0 1 -1 0 1 -1 0 1 -1 0 1
11 0 -1 0 1 0 1 -1 0 1 -1 0 1 -1
12 0 -1 0 1 1 -1 0 1 -1 0 1 -1 0
13 0 0 1 -1 -1 0 1 0 1 -1 1 -1 0
14 0 0 1 -1 0 1 -1 1 -1 0 -1 0 1
15 0 0 1 -1 1 -1 0 -1 0 1 0 1 -1
16 0 1 -1 0 -1 0 1 1 -1 0 0 1 -1
17 0 1 -1 0 0 1 -1 -1 0 1 1 -1 0
18 0 1 -1 0 1 -1 0 0 1 -1 -1 0 1
19 1 -1 1 0 -1 1 0 -1 1 0 -1 1 0
20 1 -1 1 0 0 -1 1 0 -1 1 0 -1 1
21 1 -1 1 0 1 0 -1 1 0 -1 1 0 -1
22 1 0 -1 1 -1 1 0 0 -1 1 1 0 -1
23 1 0 -1 1 0 -1 1 1 0 -1 -1 1 0
24 1 0 -1 1 1 0 -1 -1 1 0 0 -1 1
25 1 1 0 -1 -1 1 0 1 0 -1 0 -1 1
26 1 1 0 -1 0 -1 1 -1 1 0 1 0 -1
27 1 1 0 -1 1 0 -1 0 -1 1 -1 1 0

L36

formula => 11 Factors on 2 levels and 12 Factors on 3 levels
Tests ↓ Factors
A B C D E F G H I J K L M N O P Q R S T U V W
1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0
3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1
4 -1 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 1 1 1
5 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 -1 -1 -1 -1
6 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 1 1 1 -1 -1 -1 -1 0 0 0 0
7 -1 -1 0 0 0 -1 -1 -1 0 0 0 -1 -1 0 1 -1 0 1 1 -1 0 0 1
8 -1 -1 0 0 0 -1 -1 -1 0 0 0 0 0 1 -1 0 1 -1 -1 0 1 1 -1
9 -1 -1 0 0 0 -1 -1 -1 0 0 0 1 1 -1 0 1 -1 0 0 1 -1 -1 0
10 -1 0 -1 0 0 -1 0 0 -1 -1 0 -1 -1 1 0 -1 1 0 1 0 -1 1 0
11 -1 0 -1 0 0 -1 0 0 -1 -1 0 0 0 -1 1 0 -1 1 -1 1 0 -1 1
12 -1 0 -1 0 0 -1 0 0 -1 -1 0 1 1 0 -1 1 0 -1 0 -1 1 0 -1
13 -1 0 0 -1 0 0 -1 0 -1 0 -1 -1 0 1 -1 1 0 -1 1 1 0 -1 0
14 -1 0 0 -1 0 0 -1 0 -1 0 -1 0 1 -1 0 -1 1 0 -1 -1 1 0 1
15 -1 0 0 -1 0 0 -1 0 -1 0 -1 1 -1 0 1 0 -1 1 0 0 -1 1 -1
16 -1 0 0 0 -1 0 0 -1 0 -1 -1 -1 0 1 0 -1 -1 1 0 1 1 0 -1
17 -1 0 0 0 -1 0 0 -1 0 -1 -1 0 1 -1 1 0 0 -1 1 -1 -1 1 0
18 -1 0 0 0 -1 0 0 -1 0 -1 -1 1 -1 0 -1 1 1 0 -1 0 0 -1 1
19 0 -1 0 0 -1 -1 0 0 -1 0 -1 -1 0 -1 1 1 1 -1 0 0 -1 0 1
20 0 -1 0 0 -1 -1 0 0 -1 0 -1 0 1 0 -1 -1 -1 0 1 1 0 1 -1
21 0 -1 0 0 -1 -1 0 0 -1 0 -1 1 -1 1 0 0 0 1 -1 -1 1 -1 0
22 0 -1 0 -1 0 0 0 -1 -1 -1 0 -1 0 0 1 1 -1 0 -1 -1 1 1 0
23 0 -1 0 -1 0 0 0 -1 -1 -1 0 0 1 1 -1 -1 0 1 0 0 -1 -1 1
24 0 -1 0 -1 0 0 0 -1 -1 -1 0 1 -1 -1 0 0 1 -1 1 1 0 0 -1
25 0 -1 -1 0 0 0 -1 0 0 -1 -1 -1 1 0 -1 0 1 1 -1 1 -1 0 0
26 0 -1 -1 0 0 0 -1 0 0 -1 -1 0 -1 1 0 1 -1 -1 0 -1 0 1 1
27 0 -1 -1 0 0 0 -1 0 0 -1 -1 1 0 -1 1 -1 0 0 1 0 1 -1 -1
28 0 0 0 -1 -1 -1 -1 0 0 -1 0 -1 1 0 0 0 -1 -1 1 0 1 -1 1
29 0 0 0 -1 -1 -1 -1 0 0 -1 0 0 -1 1 1 1 0 0 -1 1 -1 0 -1
30 0 0 0 -1 -1 -1 -1 0 0 -1 0 1 0 -1 -1 -1 1 1 0 -1 0 1 0
31 0 0 -1 0 -1 0 -1 -1 -1 0 0 -1 1 1 1 0 1 0 0 -1 0 -1 -1
32 0 0 -1 0 -1 0 -1 -1 -1 0 0 0 -1 -1 -1 1 -1 1 1 0 1 0 0
33 0 0 -1 0 -1 0 -1 -1 -1 0 0 1 0 0 0 -1 0 -1 -1 1 -1 1 1
34 0 0 -1 -1 0 -1 0 -1 0 0 -1 -1 1 -1 0 1 0 1 -1 0 0 1 -1
35 0 0 -1 -1 0 -1 0 -1 0 0 -1 0 -1 0 1 -1 1 -1 0 1 1 -1 0
36 0 0 -1 -1 0 -1 0 -1 0 0 -1 1 0 1 -1 0 -1 0 1 -1 -1 0 1

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