10.2 - Log-linear Models for Three-way Tables

In this section we will extend the concepts we learned about log-linear models for two-way tables to three-way tables. We will learn how to fit various models of independence discussed in Lesson 5 (e.g., conditional independence, joint independence, etc.) and will see additional statistics, besides the usual \(X^2\) and \(G^2\), to assess the model fit and to choose the "best" model.

The notation for log-linear models extends to three-way tables as follows. If \(\mu_{ijk}\) represents the mean for the \(i\)th, \(j\)th, and \(k\)th levels of variables \(A\), \(B\), and \(C\), respectively, then we write


The main questions of interest are:

  • What do the \(\lambda\) terms mean in this model?
  • What hypotheses about them correspond to the models of independence we already know?
  • What are some efficient ways to specify and interpret these models and tables?
  • What are some efficient ways to fit and select among many possible models in three and higher dimensions?

As before for three-way tables, there are multiple models we can test and now fit. The log-linear models we will fit and evaluate are

  1. Saturated
  2. Complete Independence
  3. Joint Independence
  4. Conditional Independence
  5. Homogeneous Association

Example: Graduate Admissions Section

Let us go back to our familiar dataset on graduate admissions at Berkeley:

Dept. Males admitted Males rejected Females admitted Females rejected
A 512 313 89 19
B 353 207 17 8
C 120 205 202 391
D 139 279 131 244
E 53 138 94 299
F 22 351 24 317

Let D = department, S = sex, and A = admission status (rejected or accepted). We analyzed this as a three-way table before, and specifically, we looked at partial and marginal tables. Now we will look at it from a log-linear model point of view. Let \(Y\) be the observed frequency or count in a particular cell of the three-way table.