Other models for two-way contingency tables

Basic Concepts

The saturated and independence models are the only log-linear models for two-way contingency tables that use both variables. There are, however, a number of other models. The complete list of models is given in Figure 1.

Figure 1 – Model types for two-way contingency tables

Here A and B are the names for the two independent variables. As such, they correspond to the terminology for one-factor ANOVA. We only consider what are called hierarchical (or nested) models in this book. This means that if AB is included in the model, then both A and B are also included. Thus, for example, we won’t consider models of form

ln y = b₀ + b₁ t₁ + b₂ t₁t₂

It is common to abbreviate the models using superscripted lambdas. Thus, Figure 1 can be re-expressed as in Figure 2.

Figure 2 – Revised list of model types

There are two conditional equiprobability models: one corresponding to variable A and the other to variable B. In the first conditional equiprobability model, cell differences are only attributable to A and cell frequencies are equally likely between any of the levels for B.  Thus only the t₁ term contributes to the model (and so the t₂ term is not present). In the second conditional equiprobability model, the roles of A and B are reversed. The equiprobability model assumes that observations as equally likely to appear in any of the cells (i.e. they are randomly distributed between cells).

Example

Example 1: Create the conditional equiprobability and equiprobability models for the data in Example 2 of Independence Testing

Figure 3 shows the expected frequencies for each of the three new models and the value of the maximum likelihood (chi-square) statistic for each.

Figure 3 – Other models for Example 1

The degrees of freedom values are equal to the number of cells in the contingency table that are not free to take on any value (i.e. have lost their freedom). In the first conditional equiprobability model, the two cells in the first row can take on any value, but the cells in the second row are then bound. Thus df = 2. The situation is similar for the second conditional equiprobability model. For the equiprobability model, one cell is free to take on any value, but then the other three cells are bound, and so df = 3.

Since p-value < α = .05 for all of these models, none is a significant fit for the data.

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