We have repeatedly made reference to the "significance" of deviations of the observed frequencies from the expected frequencies. We can evaluate the statistical significance of the goodness-of-fit of a particular model via a Chi-square test. Log-Linear Analysis computes two types of Chi-squares: the traditional Pearson Chi-square statistic, and the maximum likelihood ratio Chi-square statistic (the term likelihood ratio was first introduced by Neyman and Pearson, 1931; the term maximum likelihood was first used by Fisher, 1922a). In practice, the interpretation and magnitude of those two Chi-square statistics are essentially identical. Both tests evaluate whether the expected cell frequencies under the respective model are significantly different from the observed cell frequencies. If so, the respective model for the table is rejected.

Reviewing and plotting residual frequencies. After we have chosen a model for the observed table, it is always a good idea to inspect the residual frequencies, that is, the observed minus the expected frequencies. If the model is appropriate for the table, all residual frequencies should be "random noise," that is, consist of positive and negative values of approximately equal magnitudes that are distributed evenly across the cells of the table. Log-Linear Analysis also is used to produce various plots of residual frequencies and related statistics.

Statistical significance of effects. The Chi-squares of models that are hierarchically related to each other can be directly compared. For example, if we first fit a model with the age by hair color interaction and the stress by hair color interaction, and then fit a model with the age by stress by hair color (three-way) interaction, the second model is a superset of the previous model. We could evaluate the difference in the Chi-square statistics, based on the difference in the degrees of freedom; if the differential Chi-square statistic is significant, we would conclude that the three-way interaction model provides a significantly better fit to the observed table than the model without this interaction. Therefore, the three-way interaction is statistically significant.

In general, two models are hierarchically related to each other if one can be produced from the other by either adding terms (variables or interactions) or deleting terms (but not both at the same time).