How do you design an experiment when one factor depends on another?

[#1]

I want to use design of experiment principles to select factor levels for an test. For the system of interest, one easy to change factor depends on a more difficult to change factor. How do you define the levels in a statistically valid manner? To illustrate, say I want to study the performance of a machine. One of the factors of interest is operating speed, a factor with 7 levels than are is mostly simple to change between. They are only "mostly simple" because the machine requires two different configurations to reach those 7 speed levels, configuration 1 for speeds 1-5, and configuration 2 for speeds 5-7 (yes, speed 5 can be achieved using either configuration). Configuration is more difficult to change than speed; it requires stopping the machine, then starting it up again. I recognize this would require a split-plot design. How do I define this type of dependency in JMP?

[#2]

Do I simply need to add constraints on the factor levels?

[#3]

In JMP, there is an option to specify disallowed combinations, but it is only available when using the "Custom Design" tool which does not visit every level of a discrete numeric factor, regardless of how many points I let it use. Using the "Screening Design" tool, the design includes all levels for discrete numeric variables but does not accept disallowed combinations. I get the sense that I need to implement this separate to the software and I want to do it in a statistically robust manner.

[#4 · Accepted solution]

You are correct. Defining factor constraints for a design using JMP is unique to custom design. Other design methods, except for extreme vertices simplex designs, do not provide a way to introduce factor constraints.

Custom design does not use factorial combinations to build a design. It uses the coordinate exchange algorithm. Custom design does not attempt to satisfy any goal based on combinations. It satisfies an optimality criterion, D-optimality by default. The estimation of the effects of the continuous factor Speed are better served with the custom design than by the factorial design. If you require all combinations, for some reason other than estimating the model parameters, you could enter Speed as a categorical factor with 7 levels and then adding runs should eventually achieve at least one run for every combination.

Yes, disallowed combinations are required because the linear constraint only works with continuous factors.

The design is really just about data collection to fit the linear model using regression. You can always modify the custom design, or the design from using any method, to suit your purpose or situation, and it won't break. I recommend that you simulate the response. The simulation can be nonsense - it is just so that you can carry your experiment through the analysis step to be sure that there are no singularities. If so, then your design really did break.