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New in JMP 10 DOE: Simultaneous addition of multiple covariate factors

Among other kinds of factors, the Custom Designer in JMP has a mechanism for adding factors with values that are not controllable but are known in advance of experimentation. I call these factors covariate factors, although in regression analysis covariates have a slightly different meaning.

Can you provide a realistic example?

Imagine that you have several lots of material and measurements of some important variables on each lot. You want to use items from these lots as the units for an experiment. You plan to process different units in a controlled way using an experiment design. The measured variables on each lot are not something you can change, so they are covariate factors. You want to design the experiment in such a way that you obtain the most information about the effects of the factors that you can control.

Peter Goos, a professor at the University of Antwerp, and I recently described an application just like the one above in our book, Optimal Design of Experiments – A Case Study Approach. In the case study, an auto manufacturing supplier was using plasma etching on lots of polypropylene to improve its adhesive characteristics. There were three measured characteristics on each lot. The investigators wanted to use these covariate factors along with the plasma etch control factors in an experiment.

How do you set up an experiment involving multiple covariate factors in JMP?

In JMP 9, this process was a bit tedious, especially if there were many covariate factors in a JMP data table. The interface required you to choose each covariate factor one at a time.

Now you can enter all the covariate factors at once using the dialog below. The yellow background color shows that I have selected all three factors.

The data table was called Polypropylene Covariates and had three columns: EPDM, PP and Color. EPDM and PP are two continuous measurements, and Color was categorical with two levels. The table has 40 rows – see Table 1.

What do you do after defining the covariate factors?

You add the factors you can control. In this case, they are Flow Rate, Power and Time. Table 2 shows the factors after this step.

Since there are 40 lots of material, the next step is to decide how many of these lots to use in the experiment. The minimum number would depend on the number of terms in the model. Of course, the maximum is 40 because that is all the material you have on hand.

Suppose I want the main effects of the factors, and I am willing to do 12 runs. What next?

You type 12 in the Number of Runs edit box. Then, to generate the design, the Custom Designer has to do two things. First, it has to pick the best 12 lots out of the 40 lots you have. Then, it has to choose the Flow Rate, Power and Time settings that maximize the information about all six factor effects. The table below shows the resulting design.

A table is necessary for providing a recipe for what to do for each run, but a graph tells more. I used the Graph Builder to get a feeling for how the Custom Designer chose the 12 lots for the covariate factors. The black points in Figure 1 show the 12 chosen lot values of EPDM and PP split by Color.

How do I know that this is a good design?

You evaluate designs using covariate factors using the same diagnostics as with any other design. Figure 2 shows the column correlation color map. The correlation that is largest in magnitude is 0.12 – not a matter for concern.

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Gerd Kopp wrote:

Dear Brad, thank you for making the process of including covariates easier in JMP 10!

Now let's assume I don't just want to assess the effect that factors and covariates have on the response, but I want to find robust factor settings, in the sense that settings of covariates affect the response as little as possible. I usually include interaction effects of factors and covariates in the model and use the factors that have significant interactions with covariates to try to minimise covariate effects. To find these settings I use the profiler, which is great, but in cases where there are several covariates which may interact with each other, this manual process is difficult and one can't be sure if one found the optimum setting of factors in the sense of achieving robustness. Is there a way in JMP to find these settings other than graphically, maybe similar to using maximise desirability - maximise robustness? Thanks in advance, Gerd

P.S. in your description it seemed that the order of including covariates and controllable factors mattered. Is that the case or could I just as well include them the other way round?

Staff

Gerd,

It sounds like I need to post on how to make a process robust to noise factors - in this case the covariates. Until then, here is the short answer. You are on the right track including the interaction effects of the control factors with the covariates in the model. Once you have a fitted model with these terms, save the prediction formula using the red triangle menu in the Fit Model platform. Then, open the Profiler platform in the Graph menu. Add your covariate factors to the noise factor list. This creates new pseudo responses that are the derivative of the response with respect to each noise factor. When you optimize using Maximize Desirability, you get a multiple response optimization that attempts to drive the previously mentioned derivatives to zero while simultaneously putting your response on target. Voila - robust process settings...

To answer your postscript, the order that you enter the factors is not important. You can enter the covariates first, last or in whatever order you want.

Happy experimenting,