The saturated model

The saturated model is the model where all $i \times j$ parameters must be estimated. This leaves no degrees of freedom for testing the goodness of fit of the model. In this model, the $G^2$ statistic yields 0, with 0 degrees of freedom.

This model always perfectly reproduces the cell frequencies. This is similar to the case of regression analysis using $n$ parameters to fit an equation of $n$ individuals. Perfect prediction is achieved. This model is not interesting per se , because the goal of any analysis to build a model as parsimonious as possible (using a minimum number of parameters) that satisfactorily fit a set of data.

For the stress example, one perfectly predict any cell frequency using the parameters. If we were to predict, say $m_{01}$, the transition from BSTRE=0 to BSTRE=1, we would simple compute: $\log m_{01} = 3.79 + 0.64 - 0.65 = 3.80055$ (when not rounding the answer), yielding $m_{01} = \exp{3.80055} = 44.72$, the exact expected frequency.

Needless to say that this model is of little interest, as a computed statistical result. It is nonetheless useful as a reference model; since it completely accounts for the "variance" of the matrix, subsequent models are to compared with it. The smaller the difference between a specified model and this one, the better the fit (and conversely).