Cox mixture estimates and Profiler

[#1]

Martin here:

Recently I did one cementitous adhesives with 4 factors (CSA cement, OPC cement, retarder, accelerator), one reponse (adhesion).

See the attached JMP design.

I am prettty happy with the results. Lou advsed me before to focus on Cox mixture and profiler. 

Estimate in Cox mixture model represnt the slope of the curves in profiler,  it appears to me CSA and OPC all show maximum value

at certain point, why the estimate of CSA and OPC is signed differently? one is 2.02 (CSA), the other -3.31 (OPC). 

Since CSA, OPC, as factors are all significant, what more can the significance of CSA^2, OPC^2, CSA*OPC tell me?  Do I simply say, OPC,CSA interact significantly?

I am very interested in the interaction of OPC, CSA and retarder, but Cox mixture can not run for three factor interaction. Is there any way I can test the significance

of three factors interaction?  I am told the conventional paramemter estimate is not effective in interperating mixture design results.

Many thanks.

Martin

[#2 · Accepted solution]

There are two types of mixture models that JMP can fit: the Scheffe mixture model (the default) or the Cox mixture model. These two models are equivalent, but their interpretation of the models are quite different.

The parameters of the Cox mixture model are interpreted like typical regression coefficients except that it is all relative to your reference mixture. Picture yourself in the middle of your design space, at your reference mixture spot. As you look towards the CSA vertex (where it would be all CSA), you will be looking uphill because the coefficient is positive. From that same location, if you look towards the OPC vertex you will be looking downhill (a negative coefficient). I think the coefficients will be more obvious if you set the profiler at your reference mixture. 

The squared terms are telling you that there is curvature as you move towards the vertex for each factor. From your reference blend, looking towards CSA is uphill, but the squared term indicates that the slope of the hill changes. You can see this on your profiler. As CSA is increased, your response increases to a point and then it starts decreasing.

Interactions indicate that the slopes of the lines will change depending on the settings of other variables. You can see this from the profiler, too. Drag the red line for OPC, but watch the curve for CSA. You should see the CSA curve change shape. That is the interaction. 

As for the three factor interaction, they are rare and often difficult to interpret. This is one of the difficulties with a Cox mixture model. However, you can estimate the curvature of the 3-factor interaction by looking at a Scheffe model rather than Cox. The Scheffe model is interpreted VERY differently from the Cox mixture model, but does not have any limitations of the Cox model (and is the default mixture model in JMP). A full explanation of the Scheffe model is probably going to be too long for this discussion board, but realize that for a Scheffe model a 2-way interaction is actually a quadratic term and a 3-way interaction is actually a cubic term. You may want to read about it from a text such as Experiments with Mixture by John Cornell, the JMP help system, or by taking a class. JMP Education does offer a full class on the design and analysis of mixture experiments that goes through all of these details. 

[#3]

Thanks for the detailed explanation.  I have a much better understanding about Cox mixture model now.

Martin