Saturated model for three-way contingency tables

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

Saturated model three way

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

image2301

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

tC = 1 if cured and = 0 otherwise
tG = 1 if male and = 0 otherwise
tT1 = 1 if therapy 1 and = 0 otherwise
tT2 = 1 if therapy 2 and = 0 otherwise

In the general case where

image2285

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.

Number coefficients saturated model

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, expijk = obsijk, or alternatively the probability that an observation will occur in cell ijk is pijk = obsijk / n.

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