Why Does Categorical Variable Anticipated Coefficient Change Sign?

[#1]

I am looking for a more detailed explanation of why the Anticipated Coefficient changes sign between subsets of a multi-level categorical factor in the DOE platform. 

From JMP help: Evaluate Design Options (jmp.com)

I understand why factor subsets need to be created for a categorical variable with 3 or more levels, and would expect a table like the following to be going on in the background of the DOE platform. This table is from here: FAQ: How do I interpret the coefficients of an effect-coded variable involved in an interaction in a...

However when I make a design with a 3-level Cat variable, the following snip is the Power table I get. Why is the "X1 2" anticipated coeff set to -1, when I would expect "L3" of X1 to implicitly be given that designation for both "X1 1" and "X1 2" as it is the 'opposing' factor level in both of those subsets? Maybe I am misunderstanding the construction of "X1 1" and "X1 2"?

[#2 · Accepted solution]

Hello @V1N0V3R1T4S, 

The Anticipated Coefficients are generated by providing Delta (default=2), which is the desired difference you want to detect in the response due to changing from the low to high level (continuous factors) or between categories (categorical factor). JMP sets the coefficients for categorical factors as alternating +/- Delta/2 because this ensures, as we change between categories of the categorical factor, the difference in the mean response is Delta.
 
This is worked out in the attached PDF in case you want a more detailed answer.  Here, we also showed that -- if you were to use all positive ones as you might be suggesting -- the difference in the mean response would be greater than Delta.  See Power Considerations in Designed Experiments for further background. 

Cheers, 

@PatrickGiuliano (JMP Technical Support)