Fractional Factorial Design - Alias terms

[#1]

Hello,

I have created a fractional factoral design with 4 continuous factors.

I get the following design evaluation (see attachment).

What explains that X2*X3, X2*X4 and X3*X4 are not in the model?

Thank you for your support!

Claire

[#2]

Adding to my previous questions:

- Do I understand correctly, that the terms that can be investigated are the following:

X1, X2, X3, X4, X1*X2, X1*X3, X1*X4?

- In the alias matrix, we see that X1*X2 is aliased with X3*X4. Does that mean that if either X1*X2 or X3*X4 are significant in the model, that we won't be able to tell which of the two terms is impacting the response? Or since only X1*X2 is in the model, we will know that the impact on the response is due to X1*X2?

[#3]

You have 7 degrees of freedom and you can't put 2 of the same degrees of freedom in the model, so yes to the first question.  Although you can use the alias of each of the 2nd order effects as well. Yes to your second question, no to the third question. You have a resolution IV design.  This means the design was generated by aliasing a third order effect with a first order effect (e.g., D=ABC).  Identity for this experiment: I=ABCD.  In this case, there are pairs of 2nd order effects that are aliased (e.g., AB=CD, AC=BD and AD=BC).  Aliasing is equivalent to confounded. The effects of both AB AND CD are "captured" together and are not separable in this experiment. Just because you call it something in the model does not mean that is the interaction that is the only effect being estimated.

Fractional factorials are based on the following principles:

1. Sparsity: Only a few of the many factors have a causal effect on the response

2. Hierarchy:  1st>2nd>>3rd>>>4th...

2. Heredity: For a higher order term to be "active" at least one parent should be active

[#4 · Accepted solution]

Hi  @cbaril,

Some elements to answer your questions :

Since you're using a fractional factorial design with 4 factors and 8 experiments, that means you can estimate 8 terms : the intercept, the 4 main effects (of each factors), and 3 two-factors interactions. 

You have at least two choices here to keep the same number of experiments (but you'll still not be able to estimate all 2FIs with the number of experiments you fixed (8)):

• Either you build a fractional factorial design the "traditional way" or the domain-expertise oriented way (like you did), so you can choose which 3 two-factors interactions can be estimated, but the rest of the interactions can't be estimated, because of the aliases (correlations) between these interactions. You won't have enough experiments/degree of freedom to test and estimate the remaining 2FIs. So you end up with a model able to estimate main effects X1, X2, X3, X4, and 2FIs X1*X2, X1*X3 and X1*X4. If two 2FIs are aliased (like X1*X3 and X2*X4), that means these two 2FIs can't be estimated separately, "isolated", so as you mention, you won't be able to differentiate if a change in the response comes from X1*X3 or from X2*X4. The analysis will by default imply that the changes comes from X1*X3 as this term was entered in your model.
• Or you build a fractional factorial design by setting the estimability of main effect as "Necessary" and all the 2FIs as "If Possible" or you directly enter all 2FIs in the model terms from "Fit Model" and use Stepwise regression or similar methods, so that you can test among all possible 2FIs which are the "most significant ones" to enter the model.

Last option (if possible), you can also augment the number of experiments to be able to estimate each main effects and 2FIs separately (3 more runs). For this, the minimum required number of runs would be 11 to estimate the terms you need : 1 intercept, 4 main effects, and 6 two-factors interactions.

Hope this answer will help you,