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
\(\log(\mu_{ijk})=\lambda+\lambda_i^A+\lambda_j^B+\lambda_k^C+\lambda_{ij}^{AB}+\lambda_{ik}^{AC}+\lambda_{jk}^{BC}+\lambda_{ijk}^{ABC}\)
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
- Saturated
- Complete Independence
- Joint Independence
- Conditional Independence
- 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.