Comprehension of correlation coefficiant |r| (what is the cut-off)

[#1]

Hello all, 

I am currently working on a L9 Taguchi matrix design of experiments. I have three parameters that have three levels each and I observe their effects on continuous outputs like pH. I want to see how the pH evolves depending on my three inital factors. 

By construction, the L9 matrix considers only principal effects and neglects the interaction between terms. More specifically, it aliases two-factor interractions with main effects. This allows us to limit the number of experiments. In this case, only 9 experiments are necessary.

Since I am doing this work for my PhD, I wish to also justify leaving interraction terms behind by giving the correlation between terms (in Evaluate window). How do I understand the limit of what to consider correlated? I have values of |r| going up to 0,58. In my old statistics classes, I remember the cutoff being at |r| = 0,7 or 0,8 typically. 

How can I justify not using two-factor interractions in my final model ?

Thanks for your help!

Anna

[#2]

Feel free to ask for any extra details !

[#3]

I would graph the relationships with scatter plots and consider the practical significance of the relationships rather than relying specifically on a statistic which may be influenced by unusual data points.

[#4]

Hi @FrancesBudgie97,

Welcome in the Community !

There are no strict rules regarding the correlation between terms in the design, and which correlation value is ok to do the analysis. As I have already answered in your other post, I wouldn't remove by default any terms that may be correlated to others in the model. Correlation between terms are often found in designs, and that doesn't prevent them to be included in the model. The consequence will be some restrictions in the terms inclusion (you won't be able to include them all, or you'll end up with a singularity in your model), some lack of precision to estimate them (with broader confidence intervals around the estimate values) and some level of collinearity (you can check Variance Inflation Factors (VIF) values for the terms included in your model to assess the degree of collinearity in your model).

Interaction effects are very important and their absence may be the cause of a lack-of-fit. It's also rare (particularly in chemical systems studied with DoE) to have no interaction effects. For pH for example, the value can change based on acid type, concentration, ratio, etc... but the evolution of pH may be different based on acid types and ratio, so you may find interaction effects in your model.

As a rule of thumb, and from the article “Regularities in Data from Factorial Experiments”, by Xiang Li, Nandan Sudarsanam, and Daniel D. Frey, March 2006 and the meta-analysis they have done on 113 data sets, you can expect (at least) for three factors to have 40% of main effects that are active,  28% of active intercation effects, and 14% of potential three-factor interactions. In you case with three factors (maximum 3 main effects, 3 interaction effects, 3 quadratic effects and 1 three-factor intercation), I wouldn't be surprised to have 2 to 3 active main effects, with at least one active interaction effect.

As @statman mentioned it, always plot your data before building models. Try to see patterns, links and correlations between factors and responses, before trying to build your model. There are many checks you can do graphically before modeling.

Hope this answer will help you,