5.3 - Models of Independence and Associations in 3-Way Tables

For multinomial sampling and a two-dimensional table, only independence of row and column is of interest. With three-dimensional tables, there are at least eight models of interest. In the next sections of this lesson, we are going to look at different models of independence and dependence that can capture relationships between variables in a three-way table. Please note that these concepts extend to any number of categorical variables (e.g., \(k\)-way table), and are not unique to categorical data only. But we will consider their mathematical and graphical representation and interpretations within the context of categorical data and links to odd-ratios, marginal, and partial tables. In the later lessons, we will see different ways of fitting these models (e.g., log-linear models, logistic regression, etc.).

Recall that independence means no association, but there are many types of association that might exist among variables. Here are the types of independence and associations relationships (i.e., models) that we will consider.

  1. Mutual independence -- all variables are independent from each other, denoted \((X,Y,Z)\) or \(X\perp\!\!\perp Y \perp\!\!\perp Z\).
  2. Joint independence -- two variables are jointly independent of the third, denoted \((XY,Z)\) or \(X Y \perp\!\!\perp Z\).
  3. Marginal independence -- two variables are independent while ignoring the third, e.g., \(\theta_{XY}=1\), denoted \((X,Y)\).
  4. Conditional independence -- two variables are independent given the third, e.g., \(\theta_{XY(Z=k)}=1\), for all \(k=1,2,\ldots,K\), denoted \((XY, YZ)\) or \(X \perp\!\!\perp Y | Z\).
  5. Homogeneous associations -- conditional (partial) odds-ratios don't depend on the value of the third variable, denoted \((XY, XZ, YZ)\).

Before we look at the details, here is a summary of the relationships among these models:

  • Mutual independence implies joint independence, i.e., all variables are independent of each other.
  • Joint independence implies marginal independence, i.e., one variable is independent of the other two.
  • Marginal independence does NOT imply joint independence.
  • Marginal independence does NOT imply conditional independence.
  • Conditional independence does NOT imply marginal independence.

It is worth noting that a minimum of three variables is required for all the above types of independence to be defined.