10.2 - Log-linear Models for Three-way Tables

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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 varous models of independence discussed in Lesson 5, e.g., conditional independence, joint independence and homogenous associations model. We will also learn additional statistics, besides the usual X2 and G2, to assess the model fit, and to choose the "best" model.

Key concepts:

  • Three-way Log-linear models
  • Parameters Constraints, Estimation and Interpretation
  • Model selection and Inference for log-linear models
  • Test about partial associations

Objectives:

  • Understand the structure of the log-linear models in three-way tables
  • Understand the concepts of independence and associations described via log-linear models in three-way tables

Readings

  • Agresti (2007) Ch. 7, 8
  • Agresti (2013) Ch. 8, 9

Expanding the log-linear model notation to 3-way tables:

\(\text{log}(\mu_{ijk})=\lambda+\lambda_i^A+\lambda_j^B+\lambda_k^C+\lambda_{ij}^{AB}+\lambda_{ik}^{AC}+\lambda_{jk}^{BC}+\lambda_{ijk}^{ABC}\)

The main questions for this lesson are:

  • What do the λ terms mean in this model?
  • What hypothesis 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

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

Dept.
Men rejected
Men accepted
Women rejected
Women accepted
A
313
512
19
89
B
207
353
8
17
C
205
120
391
202
D
278
139
244
131
E
138
53
299
94
F
351
22
317
24

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. You will work with this example on the next homework as well. Let y be the frequency or count in a particular cell of the three-way table. See berkelyLoglin.sas (and berkeley.sas) or berkeleyLoglin.R (and berkeley.R).