Fitting a response surface model with four factors (one factor is a nuisance factor)

[#1]

I've designed a DoE to study the influence of factors on a filtration process. First, I designed a fractional factorial DoE to investigate 4 factors at two levels (low and high) and also includes two center points. In the evaluation, I find that one of the four factors has no influence on the responses and is not considered in further DoE designs. The design is then augmented to include star points with the remaining three factors. It is then noticed that there is a nuisance factor which was not controlled in the fractional factorial DoE and was also not controlled in the augmented design. It is expected to have an impact on one or more responses. This nuisance factor is however dependent on one of the three factors (from a calculation perspective). Values of the nuisance factor were calculated for both experimental designs and included in the JMP data table for evaluation. Now, the design contains four factors.

As there are no `levels`for the nuisance factor per se, is it valid to fit a response surface model to the data? Or would a different model/approach be appropriate in this case?

[#2]

Every situation requires thoughtful understanding of the mechanisms at work.  Unfortunately you don't provide enough to provide specific guidance, but I'll share my thoughts.  How did you discover the noise factor and it's effect on the responses?  Is the noise variable measurable?  If so you might consider treating it as a covariate. How close are you to "optimum" or desired results?  What is the noise variable?  Is it something to do with raw materials?  Can you work with the supplier of the raw materials to reduce variability? Have you studied the process for consistency/stability via directed sampling?  I'm not sure what you mean by "This nuisance factor is however dependent on one of the three factors (from a calculation perspective)."  If your design factors interact with the noise variable, that is a problem.  It is an indication your process is not robust.  You'll have to seek out how to become robust to the noise variable.  This might include finding levels for the design factors or finding another design factor that minimizes the noise factor effect.  It might also require re-design of the process.  Of course the good news is you have identified the noise factor and therefore can replicate your experiments over that changing noise factor to estimate robustness.  Was there any evidence of curvature from the center points?  If not, why are you adding star points to estimate non-linear effects?

[#3]

Here I'll provide more details on the experimental design.

One of the factors in the DoE was the protein concentration (in g/L) in the input solution. It was intended to filter the same volume of input solution for each experiment (we measure this). Therefore, the protein content filtered in each experiment would be different. The protein content is reported as g/m2 filter and is considered as the nuisance factor (perhaps a better term is required). In some experiments the filters blocked and not all of the solution could be filtered (most likely due to high protein concentrations). Therefore, we proposed to determine the protein content filtered in each experiment and to see if this had an impact on the responses (most interesting would be the effect on flux decay).  

To answer your question about curvature, we in fact detected curvature from the center points and therefore added star points to the design. Also, we are not trying to optimise our process but establish ranges that ensure process robustness.

[#4]

@Michael_Mart from your second reply it sure looks like so far you may want to follow @statman 's advice to treat the 'nuisance factor' as you call it, as what JMP calls a 'covariate', from an experimental design and analysis point of view. Do you have the ability to backtrack and analyze the fractional factorial design with the covariate included? That might also be informative. Here's a link to the JMP documentation section related to covariates: Experiments with Covariates in JMP 

You may also want to get a hold of Goos and Jones book, "Optimal Design of Experiments" for more details and examples of using covariates within a DOE based investigation.

[#5]

@P_Bartell yes, I am able to backtrack and analyze the design.

Thanks both for your inputs!