Multi-way Frequency Tables

The reasoning presented for the analysis of the 2 by 2 table can be generalized to more complex tables.

For example, suppose we had a third variable in our study, namely whether the individuals in our sample experience stress at work. Because we are interested in the effect of stress on graying, we will consider Stress as another design variable.

Note: if our study were concerned with the effect of gray hair on subsequent stress, variable stress would be the response variable, and hair color would be the design variable.). The resultant table is a three-way frequency table.

Fitting models

We can apply our previous reasoning to analyze this table. Specifically, we could fit different models that reflect different hypotheses about the data. For example, we could begin with a model that hypothesizes independence between all factors. As before, the expected frequencies in that case would reflect the respective marginal frequencies. If any significant deviations occur, we would reject this model.

Interaction effects

Another conceivable model would be that age is related to hair color, and stress is related to hair color, but the two (age and stress) factors do not interact in their effect. In that case, we would need to simultaneously fit the marginal totals for the two-way table of age by hair color collapsed across levels of stress, and the two-way table of stress by hair color collapsed across the levels of age. If this model does not fit the data, we would have to conclude that age, stress, and hair color all are interrelated. Put another way, we would conclude that age and stress interact in their effect on graying.

The concept of interaction here is analogous to that used in analysis of variance (ANOVA/MANOVA). For example, the age by stress interaction could be interpreted such that the relationship of age to hair color is modified by stress. While age brings about only little graying in the absence of stress, age is highly related when stress is present. Put another way, the effects of age and stress on graying are not additive, but interactive.

If you are not familiar with the concept of interaction, we recommend that you read the Introductory Overview to ANOVA/MANOVA. Many aspects of the interpretation of results from a log-linear analysis of a multi-way frequency table are very similar to ANOVA.

Iterative proportional fitting

The computation of expected frequencies becomes increasingly complex when there are more than two factors in the table. However, they can be computed, and, therefore, we can easily apply the reasoning developed for the 2 by 2 table to complex tables. The commonly used method for computing the expected frequencies is the so-called iterative proportional fitting procedure.