The structure of 3-D tables

• In 2-way tables, independence means the cell counts are given by the marginals.
• In 3-way tables, we have three pairs of variables (A*B, A*C and B*C), but a 3-way tabulation (A*B*C): more complex:
  • We could have overall 3-way independence: knowing the 1-D A, B and C margins (frequencies) is enough to predict the cell counts in the 3-way table.
  • We could have any one, two or all three of the 2-way interactions - pairwise associations.
  • We could need the A*B*C interaction, if the association between e.g., A and B differs across the categories of C (if this is true then in general the association between A and C differs across B, etc. ).

• The equation for a saturated model of a three-way table is
  
  $$F_{ijk} = \tau_0 \tau_i^A \tau_j^B \tau_k^C\tau_{ij}^{AB}\tau_{ik}^{AC}\tau_{jk}^{BC}\tau_{ijk}^{ABC}$$
  
  
  • A grand mean;
  • Three `main' or first-order effects;
  • Three two-way interactions;
  • One three-way interaction.