Terms in parameter estimates for two-way interactions of coded factors are being given as e.g. (EtOH + 0.1)*(n-PrOH-0.1)

[#1]

I'm trying to work through an example out of Goos and Jones (Ch 3), where they've set up a DoE screen and model of select two-way ineractions. All factors except "Block" are continuous, and the factors are coded as -1 and +1 for the two levels tested. When fitting the model, I'm confused that the Parameter Estimates pulldown gives each interaction term not as e.g. EtOH*n-PrOH, but rather as (EtOH + 0.1)*(n-PrOH-0.1) or (EtOH + 0.1)*Time (see IMG1.jpg). I'm not sure how to interpret this. Can anyone explain when and why JMP adds/subtracts a constant from each factor? It doesn't seem to always do this - I have another model where it does what I expect and gives the term as e.g. n-PrOH*EtOH (IMG2.jpg). Thanks!

[#2]

It looks as though the midpoint of the range is being subtracted from those particular model terms. I do not see this issue when I ran through the same example. Some more information is likely needed to know exactly why this is happening.

What version of JMP are you using? The design in chapter 3 is not originally in terms of -1 and +1. Did you type all of the results in? Have you verified that there were no typographical errors? Do you have the coding property turned on? Is the "center polynomials" on the Fit Model dialog checked? Perhaps you could share your JMP data table so we can see EXACTLY what you see.

[#3]

Hi Dan,

Thank you for your reply! I'm using JMP 15.0.0.

You wrote "The design in chapter 3 is not originally in terms of -1 and +1." That's true, but it seems that when reporting the factor-effect estimates i.e. in Tables 3.6 and 3.7, the estimates are reported for coded +1/-1 factor values. So I'd added columns of +1's and -1's in an attempt to replicate the analysis.

Attached is the JMP table file - "Ch03 - Adding Runs to Screening Expt." If the "Model - Table 3.7" script is run, you can see that in Parameter Estimates the EtOH*n-PrOH interaction term is given as "(n-PrOH-0.1)*(EtOH+0.1)" with an estimated value of -1.872. 

You mentioned coding - I hadn't realized this was a column option. It turns out that when I let JMP know that those are coded columns (as in file "Ch03 - Adding Runs to Screening Expt - Coded"), it reports the interaction term as I expected, as "EtOH*n-PrOH,"  and also with estimated value of -1.872. So the parameter value is the same regardless.

But I'd still like to know what JMP sees differently that the +/- 0.1 is in there when the columns aren't coded, and what it means? In neither case does the Effects Summary table show the +/- 0.1, it's just in the Parameter Estimates.

Thanks again!

Jason

[#4 · Accepted solution]

Ah, now I see what is going on. I did not know that you created your own coded columns, because you don't need to do that. But since you did, here is what is happening.

By default whenever you use Fit Model, JMP will automatically center higher order terms of a polynomial. So, before multiplying two terms together to make an interaction, it will center those factors by their mean. The mean of your EtOH column is -0.1. The mean of the n-PrOH column is 0.1. Since you are crossing them, JMP centers them first by subtracting those values. The primary reason for doing this is to reduce multicollinearity between model terms that is caused by scale differences. This is a good thing and why the coding property is usually turned on automatically for designed experiments. This will only occur in the parameter estimates table because that is the only table that is affected by this coding.

To remove the centering option (which I do NOT recommend, but for learning purposes only), when you choose Fit Model, specify the model and then go to the red triangle at the top and uncheck "Center Polynomials". This will turn the feature off and you will get a report that looks like you expect.

I should point out that for this situation there is no difference because you are already analyzing factors that are on a -1 to +1 scale. The means being slightly different than 0 does not impact anything, especially since they are in opposite directions (one negative, the other positive). To see the typical impact, use the data in the original units (without a coding property) and turn off the center polynomials. That will be a much larger difference.

Now for more fun, with the centering turned off on the originally scaled variables, right-click the parameter estimates table and choose Columns > VIF. This will add the variance inflation factors to the table. You would like to see values near 1. Anything larger is indicating how much the variance on the parameter estimates has been increased due to multicollinearity. Now use your coded table and do the same analysis on the original units. Now add the VIFs to that parameter estimates table. You should see values that are all closer to 1. This is what the coding (or even just polynomial centering) is giving you -- smaller variances on your parameter estimates.