# 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

#### 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

• 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: