How do I make a DoE for a mix of factors I have a physical model for (custom nonlinear model) and factors I dont (traditional factorial model)

[#1]

Hello - I know you can make non-linear custom DoEs. However, I can't seem to find a source, even a theoretical discussion in a journal article, that shows how to handle a sort of hybrid model, wherein there is a physical model that is known that relates a response with one of the factors being studied, but the model does not incorporate other factors that are under study and known to affect the system.

In my case, we have the model Y1 = q*K*X1/(1+K*X1), where

Y1 is a continuous response,

X1 is a continuous factor,

q and K are parameters.

q and K are themselves unknown continuous functions of a second factor, X2, making Y1 also a function of X2. Finally, Y1 is also an unknown function of continuous factor X3 and of categorical factor X4.

Right now we simply doing a RSM for Y1 = f(X1,X2,X3,X4), and then fitting the nonlinear model described above to the partial derivative of the RSM model with respect to X1. This seems laborious, seriously error prone, and very inefficient. However, I can't find literature on what combining physical models with factorial/quadratic models would look like, especially with regard to interactions and such. I'm but a humble chemical engineer, so I likely just don't know where to look, as I can't imagine this problem hasn't come up before.

Does anyone know a solution to this, either implemented in JMP in a sub-menu where I haven't seen it, or a literature source that I could work off of?

Thanks in advance.

(PS. It says to include if I want scripting or not. I don't know JMP scripting, so I would prefer an interactive solution. However, if there is a solution, and it would only work with scripting, I know people that could help me with that. So scripting is not preferred, but ok if its the best option. Thanks!)

[#2]

Sorry, I'm a bit confused by your request.  Are you trying to validate the theoretical model?  What questions are you trying to answer?  Typically experimentation is done to create an empirical model or understand causal structure.  I'm not sure what you are trying to do?

[#3]

Thanks for the reply! Sorry it was confusing. Ill try to clear up what I was saying.

We have a good physical, nonlinear model (derived from reaction kinetics and thermodynamics, if that's more clear) for one of the four factors under study. There currently is no model for the effect of the other three factors under study. That means we either 1) design the experiment factorially/quadratically as if we didn't know what the underlying physical model involving the first factor was, 2) only design for the first factor and block our experiments by the other three factors, or 3) figure out how to blend or hybridize a factorial design and a nonlinear design.

1) This option is ok (and what we are doing for the time being), but there may be a lot of physical insight lost with this approach.

2) We tried this last year for just one second factor. If you know anything about sliding levels, this is it. It was a nightmare with just one extra blocking factor, and I can't imagine what three would be like.

3) This seems is the approach I am suggesting.

We know the shape of Y's variation with X1. We don't know how it varies with X2-X4. We would like to capture that unknown variation with a statistical model on top of the physical model that we already have.

Does that help or did I make it more confusing.

PS. If this helps, for full context, I have attached the spreadsheet with the data. X1-X4 are the factors. Y1-Y3 are the responses. Y3 is the response that we have a physical model for in terms of X1 (holding X2-X4 constant), and I've put that model in the last column just so you can see it. Y1 and Y2 are fine modeled by a RSM or Full Factorial. Y3 is the response that I am posting about.

We have a fourth response, Y4, that is not part of the design because the analysis to obtain data for takes too long to make including it in the design worth it from a time-management point of view. At the end of the day, the point is to have a series of Y4 vs X3. The reason we need to do a DoE is because we need Y2 and Y3 to be constant for all members of the final series Y4 vs X3.

I worry that that may have made things more confusing. If so, what else can I clear up? I'm trying to stay away from scienc-y terms, but that may be making things more confusing.

[#4]

And the reason we care about including the physical model if possible is because it could give some concrete physical insight into the system. Its not necessary, but it would be very useful if possible.

[#5]

Simplifying your use case a bit, I think you are saying that you believe a response r = f(x,y). You already know f(x), so you really only need to identify f(y). Is that right? If so, a few questions:

• Can you just avoid changing x during your experiment?
• If you have to adjust x, does it change in increments such that you could you block for it?
• Do you expect interactions between x and y?

If for some reason x cannot be held constant (maybe it is a disturbance), then perhaps you could let your target variable be the difference between the response and the predicted response after accounting for x. So, instead of r = f(x,y), look for r' = r - f(x) = f(y).  You could of course extend that to y1, y2, y3, etc.

[#6]

It is very challenging to give advice when I don't completely understand the situation, so please disregard comments that are out of context Here are some things to think about and consider:

1. The model you posted earlier what you are calling a "good physical model". is theoretical.  Not sure what you mean by good?  Does it do a good job predicting?  What do residuals look like?

2. There are 3 additional factors of interest that you don't have a model for.  Realize that ALL models are conditional!  The coefficients (both magnitude and direction) can change dependent on what other terms are in the model.  Since I can't tell by your description what those other variables were doing when you derived the model you have, I would suggest including all variables in the experiment.  

3. The strategy of blocking is intended for noise variables, not factors you expect to be included in your model.  If you want to restrict randomization for design factors, split-plot designs would be appropriate.

4. Do you have a predicted model and explanations as to why that predicted model would be appropriate (hypotheses)?  Rank order your model effects to 2nd order (linear and non-linear).  This will help decide appropriate resolution and polynomial degree you need for your experiment.

5. I'm confused with your statement: "We know the shape of Y's variation with X1. We don't know how it varies with X2-X4."  Do you suspect interactions?  I don't see how you know the shape of Y with X1 and yet you don't know the effect of the other X's?  See point 1.

6. Doing Multivariate analysis on your Y1-Y3 (and Y3 restricted), Y1 and Y2 look to be strongly correlated, Y3 is not.  Y3 is marginally correlated with Y3 restricted.  Does that make sense form an engineering perspective?

7. Is the data set you attached real?  It does not appear to be data from a designed experiment? X2 varies from 4-2325 (some numbers are whole numbers and others decimals)? and X3 from 0-194?  You are missing Y3 data?  

8. It appears you have done Multiple Regression (Fit Model) on this data.  Do those models make sense from an engineering perspective?

9. I don't understand this: "The reason we need to do a DoE is because we need Y2 and Y3 to be constant for all members of the final series Y4 vs X3."  You hope to be able to manage X1-X4 so that Y2 and Y3 are constant so you can work with your Y4=f(X3) model?

[#7]

So you have a non-linear model of the response. See the JMP documentation about the Nonlinear Design platform and process.