Saturated model for three-way contingency tables

Basic Concepts

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. Thus 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

Summary

Thus the saturated model has abc coefficients and no degrees of freedom. Other models will have a number of degrees of freedom which corresponds to the number of coefficients that 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.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Howell, D. C. (2010) Statistical methods for psychology (7^th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf