How do I perform a multi-state decomposition with interaction effects? I am trying to perform a decomposition with interaction effects.  This paper provides a solution for n-factors where each factor has a binary state (see section 2).  I have a problem with 2 factors, one of which has binary state and the other has three states.
For example:
Factor 1 is sky condition which can be sunny or cloudy
Factor 2 is type of ice cream which can be vanilla, chocolate, or strawberry
The "start" case is vanilla on a sunny day.  The "end" case is strawberry on a cloudy day.  I can measure the money an ice cream shop makes for each combination of Factor1 and Factor2.  So for example, selling chocolate on a sunny day results in $5.50.
I'd like to perform a full decomposition with interaction effects.  What would be the formula?  Ideally the solution is expressed in terms of the example I have provided for maximum clarity.
 A: I dont think your example fits into the framework used in the paper. 
I think what you have in mind is how revenue of the shop is determined by weather and ice cream type (btw it is odd that the shop only sells one type of icecream). 
What you then need is the Regression:
$$revenue = b_0 + b_1\times cloudy + a_1\times chocolate + a_2\times strawberry + c_1\times cloudy\times chocolate + c_2\times cloudy \times strawberry$$
the interaction effects are given by $c_1$, $c_2$
Hope this helps
Martin
A: Yes, with the regression you measure the ceteris paribus effect of cloudy (=b1), of chocolate (=a1), and of strawberry (=a2) as well as the interaction effects (=c1, c2)
I don't really see a decomposition problem in your example where a 100 percent effect is split up into different parts. I also don't really see the purpose of your analysis.
You could split up the difference between f00 and f11 into the cp effect of strawberry, the cp effect of cloudy, and the interaction of the both. But its unclear how the third possibility enters.
