Analysis of a Mixture DOE with stepwise regression

[#1]

In Mixture case studies (webcast & JMP documentation library) I notice that backward regression is used to analyse the results. Why is stepwise regression not used with forward regression or all possible models in regualr (non Pro) JMP? Is stepwise not possble because of the special nature of mixture models?

[#2]

Hi Victor, I definitely agree on your heridity principles and that's why in this case backward regression based on p-values does not work here.  So I used All Possible Models with forward AICC selection with very nice results. I am now working on another mixture case with a process variable and agian all possible models method generates a - maybe wrong - but very useful model! For non-pro JMP users I recommend using stepwise all possible model method for mixture DOE analysis. 

[#3]

Hmmm, you have one or two mixture experiments and conclude "I recommend using stepwise all possible model method for mixture DOE analysis".  I suggest you use caution with this recommendation or, as Danial puts it, you will have the failure of "premature generalization".

[#4]

 At least the models I get satisfy Cox pragmatic statement "Wrong but useful to meet my goal" What is a better alternative in regular, non JMP pro? Will these models be better? 

[#5]

For pro users I definitely recommend machine learning methods with validation

[#6]

My notion might have been expressed previously in this discussion, but I want to state it explicitly. First of all, we would not test the main effect parameter estimates. Why not? The constraint that the component proportions sum to one has ramifications beyond the mathematics of the linear model. An effect is a change in the response when a factor changes its level. The changes in the factor levels are independent in a non-mixture experiment, so the effect of changing a factor is uniquely determined in a properly designed experiment. It is impossible to change one mixture component independently from all the other components. So we cannot uniquely assign the effect to one component in a mixture experiment.

Second, we cannot test the main effect parameter estimates. JMP incorrectly tests the parameter estimate for the main effects against a null hypothesis that the parameter is zero. The main effect parameter is the mean response plus the mean change in the response of a pure blend so the null hypothesis is the mean response, not zero. Furthermore, the joint tests are impossible because the mean response is part of all the main effect parameters.

The true mean response in this simulated data set is 20. The true main effect parameters, which include the mean response, are 10, 20, and 30 for X1, X2, and X3, respectively. The parameters are significantly different from 0, but that is the wrong null hypothesis. Each of the main effect estimates is flagged with the note that the tests are confounded.

We don't ask about the main effects, but we can and should ask about the higher-order terms because they tell us about the shape of the surface and are testable.

[#7]

Very interesting Marc, what mixture DOE result analysis method do you recommend for non JMP pro users? In my example above backward regression with the logworth pareto did not work; AICc forward with stewise all possible models worked fine.

[#8]

I do not recommend a particular method for selecting the linear model for a mixture experiment. I reiterate @statman's advice to consider all a priori knowledge while designing the experiment and again while analyzing the experiment.

[#9]

I fully agree and that's what I did in my example above, did I use the right approach?

[#10]

This means that we should focus on interior points in the ternary plot trying to estimate the higher order terms? So a space filling DOE is a better option than trying to estimate a two way interaction model?

[#11]

Just to clarify how the points are generated when using a mixture optimal design (with the case of 3 mixture factors) :
- Points at the vertices help estimate main effects parameters,
- Points at the middle of edges help estimate 2-factors interactions.
- Point in the centre is used for 3rd degree interaction.

Space-filling design create randomly and homogeneously distributed points in the experimental space, in the absence of any a-priori model.
If you already have an idea about the type of model you will use (like a two-way interaction model), an optimal design may be more useful and effective, as the generation of points will be optimized for the parameters estimation of the effect terms you have specified in your assumed model (unlike the "randomness" of space-filling design points generation).

Space-filling designs are useful in the absence of any a-priori knowledge about a possible model, with a probability of non-linear response surface, and have more flexibility regarding the modelling possibilities in case of points that are non-measurable (stability problems, very high or low values, ...).

Hope this clarify the difference between the use of these two types of methodologies.