continuous factor needs to be treated like a mixture factor

[#1]

I have a factor I'd like to treat as continuous in order to assess curvature of response, but I need to restrain "total" to 100% a kin to a mixture setting.

linear constraints on mixture dont allow to limit to equal as far as I can tell, rather only less then or greater then or equal too.

Thoughts on how to address this problem?

[#2]

I think a bit more detail is needed. What are all of the factors? There must be more than just this one. You say this is "a kin" to a mixture, but why is it NOT a mixture situation?

[#3]

Thats a good question.

Its not a mixture in as much as a curved response is expected, and as the mixture function doesn't allow for 2nd power interactions it doesnt suit my needs unfortunately.

As it happens the problem is determining the optimal plasmid dna ratio for producing a recombinant protein product.  The total amount of DNA is set and the amount of plasmid needs to add to a total of 100%.  3 plasmids in question.

[#4]

Mixtures do indeed allow quadratic effects. They LOOK different, but they are there.

Any binary blending term that looks like an interaction is actually a quadratic effect.

Here is a simple example:

Notice that X1*X2 and X1*X3 are significant "interactions". Those are actually quadratic terms as evidenced by the curves in the Prediction Profiler.

To understand why you need to understand how the Scheffe mixture model is reparameterized due to the mixture constraint. X1*X1 is rewritten as X1*(1-X2-X3) for a 3 component blend. That results in "additional terms" for X1, X1*X2, and X1*X3.

So, why not treat a mixture as a mixture?

[#5]

Indeed.  Thank you for that, I think ill just proceed with a mixture, thank you!

[#6]

I will ask one further question actually.  With using this method, although the mixture functions fits my parameters well. The predicted statistical power is no more then 0.2 even upon lowering my RMSE to .2 (I expect 0.3 to 0.4 for reference)

truthfully seeing this id rather use the mixture function but i am nervous in light of the low power.  Do you have any thoughts on this?

[#7]

Honestly, I would not trust those power numbers, except possibly for the cross-product terms (which are interactions and quadratics combined as discussed earlier). There are a few reasons for this. First, mixtures are naturally correlated which will always lower power. There is nothing you can do about it. Second, I don't believe the power calculations are correct because the parameter estimates for a mixture main effect also includes the overall mean of the data. So comparing that parameter estimate to zero (the traditional test that is performed) makes no sense. Third, what if a main effect is not significant? What does that really mean? It does not mean that factor has no effect. After all, changing all of the other factors will force you to change the "non-significant" factor. All factors must be considered together.

For these reasons (and probably some more), mixture models are all about prediction. You cannot really derive cause-and-effect conclusions like a traditional regression model. The correlation structure forces you to look at your model in a different way.

[#8]

thank you for that!