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* λ ^{AC}* and

*λ*terms dropped. It therefore has (

^{ABC}*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 χ^{2} (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.