Lesson 5: Three-Way Tables: Different Types of Independence

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This lesson spells out analysis techniques for three-way tables which is a representative analysis of any K-way table. We begin with the structure of a three-way table, and its corresponding joint, marginal and conditional distributions. In particular, we distinguish between a marginal and a conditional odds ratio. We then consider how to extend the goodness-of-fit tests that we saw in the previous lessons. Now there will be a variety of models of independence and associations (see the list below).  It is natural to think about using the tests we developed for two-way tables on three-way tables.  Many of the models we discuss in this lesson correspond to a particular choice of how to turn a three-way table into a two-way table.

The concept of conditional independence is very important and it is the basis for many statistical models (e.g., latent class models, factor analysis, item response models, graphical models, etc.). With respect to conditional independence models we will see more on Cochran-Mantel-Haenszel (CMH) statistic, and describe the phenomenon known as Simpson's paradox. We will also learn about a model of homogeneous associations. In a later lesson on Log-Linear models, we will learn how to answer the same questions but via models instead of goodness-of-fit tests. 

Each of the models described in this lesson will be given a mathematical formulation, e.g., a formula for the expected counts when possible. If that's not your "cup of tea", you can skip those paragraphs and focus on conceptual understanding and how we do the relevant tests in SAS and/or R.

Key Concepts in Three-Way Tables

  1. Marginal and partial tables
  2. Marginal and conditional odds ratios
  3. Models (types of associations):
    1. Saturated
    2. Complete (mutual) independence
    3. Joint Independence (e.g., two variables independent of a third)
    4. Conditional Independence
    5. Homogeneous associations
  4. Inference
    1. Simpson’s paradox
    2. CMH tests for conditional independence
    3. Estimating common odds ratios
    4. Breslow-Day statistics for testing homogeneity


  • Understand the structure of three-way tables
  • Understand the concepts of independence and associations in three-way tables

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