What is the correct way to specify a DOE factor that is a discrete numerical variable?

[#1]

I was trying to duplicate an analysis in a paper by Christopher Nachtsheim and @bradleyjones called Split-Plot Designs: What, Why, and How, Journal of Quality Technology, Vol. 41, No. 4, October 2009. They use the Box et al data for studying the corrosion resistance of steel bars (attached image). In a split-plot design, Temp is treated as a hard-to-change (whole plot) factor with three levels: 360, 370, 380 degrees. Each whole-plot treatment is replicated twice, so we have 6 whole plots with the 4 treatments of Coating within each. 

I was trying to duplicate this using the JMP DOE Custom Design dialog:

Since I designated Temp as hard to change, and I asked for 6 whole plots, JMP creates a design as expected with 6 whole plots (2 whole plots for each of the 3 Temp treatment levels), with each whole plot holding all 4 treatments of Coating, or 24 runs.

But here's the confusing thing: If I treat Temp as a categorical, I get 2 reps of each of the 3 Temp levels. But if I treat Temp as a discrete numerical, I get whole plots where Temp is either 360 or 380 (3 whole plots of each) and no whole plots with 370.

It's not surprising that I get a different design when treating Temp as categorical or numeric. But I would have thought Temp should be treated as a discrete numerical. Are there other parts of the Custom Design dialog that I'm using incorrectly?  

[#2]

I cannot reproduce your results. When I specify Temp as discrete numeric with 3 levels and hard to change with 6 whole plots, I get the expected design. I have attached my results. What did we do differently? Did you perhaps remove the Temp*Temp "if possible" term from the model? If so, you can't do that as that is the term that forces the middle level of Temp to be used.

[#3]

Indeed I did remove the quadratic term, and I now see why that dropped the middle (370) treatment value. I assume that by designating the quadratic term as ‘if possible’ that it is not estimable.

[#4]

When I add Temp as a 3-level discrete numeric hard-change factor in the Custom Design dialog, it defaults to a continuous whereas yours appears to be nominal. I included the Temp*Coating interaction term as 'Necessary'. Leaving the Temp*Temp in the model, with estimability 'if possible', my resulting JMP data table indicates that Temp is continuous whereas yours shows Temp as nominal. Is that perhaps the reason yours is including all 3 treatment levels across the whole plots? In my resulting design, JMP still assigns treatments of only 360 and 380 to the whole plots.  I am asking for 6 whole plots, and 24 runs. Here is the my input to the Custom Design dialog:

[#5]

Well, this is interesting.

I cannot reproduce my original results now! In fact, if you look at the design script that was attached to my original table, Temp is clearly marked as continuous. But the resulting table from the script has Temp as nominal. I don't know how that happened as every time I try to recreate the design from scratch, I get a continuous Temp and only two levels for Temp are used.

I have submitted this to JMP Technical Support as I think there may be a bug lurking in there somewhere. I will let you know if they shed any more light on this situation.

[#6 · Accepted solution]

The case study by Box treated Temp as categorical. In that case there is no order - the levels are discrete but unordered. That is 370 is not greater than 360 and less than 380. Treating Temp as categorical gives the replicated design that Box wanted.

However, I think that Temp should be treated as Discrete Numeric or even Continuous. In that case, you might be interested in the quadratic effect of Temp. You might also be interested in the complex interaction effect of Temp^2*Coating.

There is nothing wrong with what you did. In fact, I believe it is a more principled approach.