Suppose we have following three factors:
Factor A: 5 possible values
Factor B: 4 possible values
Factor C: 2 possible values
How can I construct an Orthogonal array for these?
Main thing which I don't understand is making the combinations. I remember we used to follow '11112222', '11221122', '12121212' this kinda combinations, but it seems everyone has different approach for filling the values in array. Is there any standard approach?
There isn't a single neat algorithm that generates orthogonal arrays to order. Instead there are a variety of constructions that have been discovered in a host of different areas of mathematics, and some techniques for modifying orthogonal arrays to change their parameters in some way or another. For instance see http://www.itl.nist.gov/div898/handbook/pri/section3/pri33a.htm and http://www.win.tue.nl/~aeb/preprints/oa3.pdf. Many statistics packages have an orthogonal array design utility which uses these rules and a list of known orthogonal arrays to try and find an orthogonal array that will satisfy the requirements it has been given.
In your case I can find nothing closer at the moment than the six five-level factors design at http://www.york.ac.uk/depts/maths/tables/l25.htm using 25 runs. You can certainly discard three columns. Where you have e.g. five levels in the design and only 4 (or 2) levels in the experiment I would be inclined to consistently relabel e.g. {1,2,3,4,5} -> {1,2,3,4,4} and {1,2,3,4,5} => {1,2,1,2,1} but I have no clear idea of what this does to the experimental properties.
The computing of orthogonal arrays can be computationally expensive, so designs are generally made available in the form of a library.
The R package DOE.base has a oa.design() function that retrieves a design with a given number of factors and factor levels. For example, to retrieve a design with 3 factors and levels of 3, 4 and 5, use these commands.
library(DOE.base)
oa.design(nlevels=c(3,4,5))
In this case, the returned design is a full factorial with 60 runs. This still is an orthogonal array, but a much more expensive experiment than the alternatives with equal factor levels.
To obtain an orthogonal array 3 factors with 5 levels each, use:
oa.design(nlevels=c(5,5,5))
A B C
1 1 5 4
2 2 1 5
3 3 4 5
4 3 5 2
5 5 2 4
6 3 3 3
7 5 5 5
8 5 4 3
9 2 5 3
10 5 1 2
11 4 1 3
12 5 3 1
13 4 4 4
14 1 1 1
15 1 2 3
16 3 2 1
17 2 3 4
18 4 3 2
19 4 5 1
20 3 1 4
21 1 3 5
22 1 4 2
23 4 2 5
24 2 2 2
25 2 4 1
The entering 3 factors with 4 levels each returns an orthogonal array of 16 runs and entering 3 factors of 3 levels returns an orthogonal array of 9 runs.
Alternatively, the Python package OApackage is available in PyPi (https://pypi.org/project/OApackage/).
For more information, see:
Complete Enumeration of Pure-Level and Mixed-Level Orthogonal Arrays, E.D. Schoen, P.T. Eendebak, M.V.M. Nguyen, Journal of Combinatorial Designs, Volume 18, Issue 2, pages 123-140, 2010.
Two-Level Designs to Estimate All Main Effects and Two-Factor Interactions, Pieter T. Eendebak, Eric D. Schoen, Technometrics Vol. 59 , Iss. 1, 2017
