Analysis of relationship among nominal variables

Loglinear models

Loglinear models (LLM) studies the relationships among two or more discrete variables. Often referred to as multiway frequency analysis, it is an extension of the familiar chi-square test for independence in two-way contingency tables.

LLM may be used to analyze surveys and questionnaires which have complex interrelationships among the questions. Although questionnaires are often analyzed by considering only two questions at a time, this ignores important three-way (and multi-way) relationships among the questions. The use of LLM on this type of data is analogous to the use of multiple regression rather that simple correlations on continuous data.

Since the use of LLM requires few assumptions about population distributions, it is remarkably free of limitations. It may be applied to almost any circumstance in which the variables are (or can be made) discrete. It can even be used to analyze continuous variables which fail to meet distributional assumptions (by collapsing the continuous variables into a few categories).

Three basic assumptions should be considered when using LLM:

Observations are independent from each other. In practice, this means that each observation comes from a different subject, that the subjects were randomly selected from the population of interest, and that no specific group of subjects is purposefully omitted.
All observations are identically distributed. This means that they are obtained in the same way. For example, you could not mix the results of a telephone survey with those of a door-to-door survey.
The number of observations is large. Since LLM makes use of large sample approximations, it requires large samples. The LLM algorithm begins by taking the natural logarithm of each of the cell frequencies, so empty cells (those with frequencies of zero) are not allowed. LLM appears to be less restrictive than traditional chi-square contingency tests, so rules that are used for those tests may be used for LLM analysis as well.

Fundamental Approach

Selecting an appropriate model.
The first step is to find an appropriate model of the data. Several techniques may be used to find an appropriate LLM. One of the most popular is the step-down technique in which complex terms are removed until all terms remaining are significant. This search for an appropriate model is restricted to those models which are hierarchical. Hierarchical models are those in which the inclusion of a term forces the inclusion of all components of that term. For example, the inclusion of the two-way interaction, AB, forces terms A and B to also be included. Before the model is accepted, you should study the residuals to determine if the model fits the data reasonably well.
Interpreting the selected model.
Once a model is selected, it must be interpreted. This is the step in which you determine what your data are telling you.