The saturated model for Example 1 of Three-way Contingency Tables takes the form:

Where terms involving *C* and *G* require 2 – 1 = 1 coefficient and terms involving *T* require 3 – 1 = 2 coefficients, and so terms involving *CG, CT, GT* and *CGT* require 1 ∙ 1 = 1, 1 ∙ 2 = 2, 1 ∙ 2 = 2, and 1 ∙ 1 ∙ 2 = 2, respectively. This can be seen from the following expanded form

using a suitable coding of the categorical variables, such as:

*t _{C} *= 1 if cured and = 0 otherwise

*t*

_{G }= 1 if male and = 0 otherwise

*t*

_{T1 }= 1 if therapy 1 and = 0 otherwise

*t*

_{T2 }= 1 if therapy 2 and = 0 otherwise

In the general case where

Figure 1 shows how many coefficients are required for each term, assuming *A* takes on *a* values, *B* takes on *b* values and *C* takes on *c* values.

**Figure 1 – Number of coefficients**

Thus the saturated model has terms and no degrees of freedom. Other models will have a number of degrees of freedom which corresponds to number of coefficients which are missing from the model (based on Figure 1).

As in the two-way case, the expected value of any cell in the contingency table for the saturated model is the same as the observed value of that cell. Thus, *exp _{ijk} = obs_{ijk}*, or alternatively the probability that an observation will occur in cell

*ijk*is

*p*/

_{ijk}= obs_{ijk}*n*.