There are three partial independence models (*A, BC*), (*B, AC*) and (*C, AB*). We’ll look at the first of these; the others are similar. The model for (*A, BC*) consists of the saturated model with the* λ ^{AB}, λ^{AC} *and

*λ*terms dropped. It therefore has (

^{ABC}*a*– 1)(

*b*– 1) + (

*a*– 1)(

*c*– 1) + (

*a*– 1)(

*b*– 1)(

*c*– 1) = (

*a*– 1)(

*bc*– 1) degrees of freedom.

In this model *A* and *B* are independent, *A* and *C* are independent, but *B* and *C* are not independent. The probability that an observation will occur in cell *ijk* is

Let’s look in particular at the conditional independent model (*C, GT*) for Example 1 of Three-way Contingency Tables (see Figure 1).

**Figure 1 – Expected frequencies for ( C, GT) model**

Once again, all the marginal totals for the expectations are set equal to the corresponding totals for the observations. Each of the cells in the range J23:L26 are calculated as described above. For example the expected frequency of a positive result for males taking therapy 1 (cell J23) is calculated via the formula =J32*M29/M27.

The degrees of freedom for this model is (2 – 1)(2∙3 – 1) = 5. From these results, we can create a table similar to that found in Figure 2 of Conditional Independence Model. From this table we would find that χ^{2} (maximum likelihood) is 60.30 with 5 degrees of freedom, which is significant. This means that this model is not a good fit for the data.