There are three conditional independence models (AB, BC), (AC, BC) and (AB, AC). We’ll look at the first of these; the others are similar. The model for (AB, BC) consists of the saturated model with the λAC and λABC terms dropped. It therefore has (a – 1)(c – 1) + (a – 1)(b – 1)(c – 1) = (a – 1)b(c – 1) degrees of freedom.
The probability that an observation will occur in cell ijk is
Let’s look in particular at the conditional independent model (CG, CT) for Example 1 of Three-way Contingency Tables (see Figure 1).
Figure 1 – Expected frequencies for (CG, CT) model
All the marginal totals for the expectations are set equal to the corresponding totals for the observations. Each of the cells in the range J6:L9 are calculated as described above. For example the expected frequency of a positive result for males taking therapy 1 (cell J6) is calculated via the formula =M6*J12/M12.
The degrees of freedom for this model is (2 – 1)∙2∙(3 – 1) = 4. From these results, we can calculate the results in Figure 2.
Figure 2 – Residuals and chi-square for (CG, CT) model
This table is similar to Figure 2 of the Independence Model for Two-way Contingency Tables. We see that χ2 (maximum likelihood) is 5.59 with 4 degrees of freedom, which is not significant. This means that this model is a good fit for the data.