There are a couple things you can try.

1) Look at the interaction graphs for each pair of factors. If you see something that  looks interesting, you can pick it out.

2) Use Bayesian Analysis.

3) if the aliasing structure is not complete, like it is in a 4-factor, 8-run FFD, you can pick out interactions to analyze for, if you have enough DF. 

The basic use of a Taguchi design is to determine main effects.  However, as you state, they can provide some information on interactions.  The basic approach to using a Taguchi design to examine an interaction is to eliminate columns (5 columns in your experiment).  Eliminating columns will allow estimation of certain 2-factor interactions.

Which columns to eliminate?  To determine which columns to eliminate you will need to consult a Taguchi interaction table for your design.  A Taguchi interaction table will show how each column is confounded.  To learn how the tables are constructed look up Taguchi linear graphs which show the logic of how the Taguchi orthogonal arrays are constructed.  There is lots of good information available on using interaction tables and linear graphs.

The interaction tables are used when designing the experiment so as too assign factors in such a way to minimize or eliminate the confounding effects relevant to a particular experiment.  Some software packages provide an option to specify potential interactions, then the software will generate a Taguchi and assign factors appropriately.

Hi Dominik,

There are two things I would like to share with you:

1) Selection of the Design. According to Schoen (2007), there exist 15 designs of 18 runs with one two-level and five three-level factors. Some designs are better than others and one way to classify them is according to the Generalized Minimum Aberration (GMA) criterion. GMA is based on the Generalized Word-Length Patter (Ap); for example, A3 is proportional to the sum of squared correlations between Main Effects contrast vectors and Two-Factor Interactions contrast vectors. A4 is proportional to the sum of squared correlations among Two-Factor interactions contrast vectors. A3 and A4 do not depend on the set of normalized contrast used.

A2 is zero since your design is an orthogonal array.

According to this criterion, the design you chose has (A3, A4) = (9.67, 8.5), but there is other design with the same characteristics but with (A3, A4) = (9.06, 10.33), see Schoen (2007). If you would like to have less ‘overall’ correlation between main effects and two-factor interactions, then the second design is better. How to choose your design depends on your specific problem.

By knowing these measures you can have a better idea of the design you choose. And such measures have a relation also with the aliasing the design produces.

2) Analyzing the Data. Yes, there is an advantage of having this complex aliasing between main effects and two-factor interactions. Such advantage is that you do not have fully confounded effects; as a result you can estimate all these effects. According to the characteristics of your design, you will have 6 degrees of freedom for estimating two factor interactions. Remember that a two-factor interaction between two three-level factor involves 4 degrees of freedom, while interactions between one two-level factor and one three-level factor involve only two. So you have, for example, space to estimate three interactions between the two-level factor and one three-level factor. The strategy of how to fit the model rely on the idea proposed by Hamada and Wu (1991). They take advantages of the effect heredity and effect sparsity principles and of the stepwise regression procedure. You can also consult all the details in their book (Chapter 9). They also mention a Bayesian Strategy that can cope with more complex models.

I hope this helps a bit.

http://www.amazon.com/Experiments-Planning-Analysis-Optimization-Jeff/dp/0471699462/ref=sr_1_1?ie=UTF8&qid=1409752641&sr=8-1&keywords=experiments+planning+analysis+and+parameter+design+optimization

Article Classification of all orthogonal arrays with run-size 18

Article Analysis of Designed Experiments With Complex Aliasing

Thank you, I will go through documents that you provided and eventually ask another question.

@Fausto

1) I do not have this book neither.

2) Yes I saw this problem with complicated interactions and I understood the fact that factors are entangled with many interactions. Unfortunately I collected results almost one year ago and I had not thought about it then.

3) Please provide me info about which of yours documents are you talking

Thank you. I will do this till the end of this week or am latest at the beginning of the next one. I will attach the document with calculated SS of main effects and interactions (calculated using contrasts). As it would be done for normal design (not entangled). It is obvious and I know that SS will not sum to SS total because of complex retaliation between main effects and interactions in this case. Hence, the sum of SS of main effects and interaction of second level will be higher than SS total what is a wrong result.

As I promised I am attaching L18 design with results and calculated SS (with separation for square and linear retaliations).

Hi All,

I think that I found a nice piece of paper and assuming that an author is right I tried to calculated aliasing matrix according to a paper:

John Lawson, "Regression analysis of experiments with complex confounding patterns guided by the alias matrix," Computational Statistics & Data Analysis 39 (2002) 227–241

He wrote that if X1 is a design matrix (main effects) , X2 interaction matrix I can calculate aliasing matrix as follow:

A = (X1'X1)-1 X1'X2

I obtained aliasing matrix as in the attachment. Please let me know if I am right.

It is a general matrix for the whole L18 design so in case of my particular case it will be simpler.

Sorry, I just had a chance to read it. I think that the aliasing matrix that I calculated indeed is valid only when linear contrasts of analysed main effects is considered. It is invalid if we want to verified square relations. But I have one idea and I will try to present today upgraded version of this matrix which I hope will be better.

What will you say if we will define:

X1 - design matrix which include contrasts (linear and square)

X2 - interaction matrix between contrasts linear and square

If we will assume that we can use given before formula which is once more given below:

A = (X1'X1)-1 X1'X2

We can see much more complex relations between main effects and interactions, please have a look at attached file.

When we will conduct such calculations for full factorial designs e.g. 32 then any aliasing is detected so I am assuming that the procedure can be correct.

Can you please provide me any feedback? Am I correct or I did a big mistake and I am wasting time and I shall go just with main effects and AB interaction with this design?

I am attaching reduced L18 aliasing matrix for 6 factors design. There are two aliasing matrices, first one divided into each contrast and second one without constants for interactions. The design is the same in this case as the one that I attached to the first post.

@Fausto

It will be easier if I will attach full calculations

1) aliasing matrix - there are calculation that you claimed as invalid for more than 2 factor

2) aliasing matrix L18 -upgraded  - there are calculation that you were asking for

Additional description

X1- design matrix

X2 - interaction matrix

@Fausto.

First of all thank you.

1) You do not have to analyse two next documents. This answer satisfies me.

2) The problem right now is how to use this aliasing matrix in my experiment? I wrote that I can pick several main effects for which the difference of results between -1 and 1 levels is the highest. Then I should sum all values for each column of the aliasing matrix (only for picked main effects) and pick several of interaction for which the sum is the highest. Then I should investigate in the experiment only these interactions.  Do you agree?

Please have a look at attached file. You will have there also used formulas.

1) At first row after design matrix I am summing multiplication (R1+R2+R3)*wij (where: R1,R2,R3 - results achieved in three measuring runs, wij - particular design matrix coefficient)

2) I am picking three contrasts of inputs which give the highest negative or positive result.

3) At aliasing matrix I sum values from each column (but only for 3 picked main effects contsrats)

4) I am choosing three interaction which absolute value of sums is the highest

I am obtaining following aliasing relations:

Al (not aliased)

Bl (not aliased)

BQ (not aliased)

Cl - 3/4 BQDQ

CQ - 1/4 BQDQ

Dl - 1/2 AlCl  - 1/4 BlCQ

DQ - 1/6 AlCl  - 1/4 BlCQ

El - 1/2 AlCl  - 1/2 BlCQ + 1 1/2 BQDQ

EQ - 1/6 AlCl + 1/2 BQDQ

Fl + 3/4 BlCQ

FQ - 1/4 BlCQ + 1/2 BQDQ

I have 11 equations, 11 variables so in theory I can compute a little bit dealiased Cl, CQ, Dl, El, EQ, Fl, FQ, and BQDQ, BlCQ, AlCl

It is just an estimation and each of the results is additionally aliased with other factors but I am assuming that the influence of this three interactions is the highest.

Alternatively to the previous post I can pick arbitrary 3 interaction which I predict influence the result the most.The choice is made based on the knowledge about a process which is investigated.

Unfortunately I made a mistake. I have only the number of independent equations equal to number of aliased factors. Hence, I have no idea how to compute it right now.

I do not know how to separate influences of main effects from interactions. Because all SS of interaction contrasts are as well aliased with main factors.

1) I thought that if diagonal coefficients of (X1'X1)-1 are bigger than 1/18 in this case it means that design is not orthogonal. Am I right? If yes then design with contrasts is not orthogonal because of Bl, Cl,Dl,El,Fl. They are 1/12.  I read also that if factors are each other confounded that means that there is no orthogonality. So in this case design is orthogonal only for main effects and if interactions other than AB is concerned there is no orthogonality.Am I right?

2) If not can you provide me a method for orthogonality verification?

3) How would you include in your analysis derived aliasing matrix?

Hi Fausto,

1) Yes I estimated confounded factors and I understand it.

2) I cannot compute using contrasts or any other known by me method separate influence of interactions beside of AB interaction - I understand it.

3) My first question was is there any method to approximate an influence of interaction e.g. EF in given L18 design??? Or I shall show entanglement of factors and try to explain how complicate are the relation between interactions and factors???

4) My second question was how to verify orthogonality of any design???

I will have very limited access to internet from tomorrow for a week.

@Fausto,

thanks.

3)  I do not who to include this entanglement. I will provide you an example at the beginning of next week and I will ask for the confirmation if it is the thing that you are talking about.

4) Do I have to do it factor by factor (each column of design with each column of design)? Or I can compare the whole matrix with some other matrix to get the orthogonality for the whole design matrix at once. I suppose that I can compare design matrix with 'I' matrix. Is it true?

I will come back to you for further clarification when I will be back at the beginning of next week.

@Fausto,

OK, the design is orthogonal when multiplication of the design matrix and its transposition is related as follows:

X*X'=X'*X=I

I got it, I hope.

I am attaching several examples can you confirm me if I am right.

Only in example 4 design is not orthogonal, right?

@Fausto

1) If I know that L18 is not orthogonal when second order interaction are used and that I cannot in such situation use contrasts it means that attached aliasing matrix (upgraded one) is useless from practical point of view, is it? I cannot use it.

2) I can use only calculations of interactions as AB, AC, AD etc. Which are computed as always. But is such case I have to find once more aliasing matrix.

Dear Fausto,

it was not my question and I am not  missing "ENTANGLEMENT is more IMPORTANT"

I asked if I still can use aliasing matrix that I computed if I cannot use contrasts. And if yes how?

And in the second question I asked for confirmation if I have to find an aliasing matrix which include only interaction without contrasts.

Hence, my sentence has a sense because I still want to get an answer how looks like the aliasing matrix in this case (L18 design).

And if I will compute a matrix as I did it at the beginning (see attached file). This matrix indicates the same entanglements as my second one. But I am not sure if coefficient are correct.

You are right. Indeed, they are not the same - unfortunately.