In order to interpret our results we intend to restrict our approach to a hierarchical family of models. The rule for hierarchical models is the same one we used for building ANOVA models:
Hierarchical Terms Rule: If we want to fit an interaction term we must also fit the corresponding lower order interactions and main effects.
Now consider a 3-way table.
We will consider the following models, abbreviations, and corresponding pictures.
¯
¯
1 indep 2 indep 3
Model Sufficient Stats Picture Meaning [0.5em]
-0.25in
1 indep 2 indep 3 [0.5em]
-0.25in
(1 and 2) indep 3 [0.5em]
-0.25in
2 indep 3 given 1
[0.5em]
none I don't know
The 
are the indicate the margins of the table that make up
the sufficient statistics---thus 
is the margin formed by summing
over the 2 and 3 indices of the table and 
is the margin formed
by summing over the 3 index.