In the case of categorical data one is commonly looking for association between two variables. The χ2 test is one example. Usually the χ2 test is performed for a 2 × 2 contingency table. Even though the test is still valid for larger tables, one can run into difficulties with interpretation. All that a significant χ2 test tells us is that the pattern of data as depicted in the table could not arise by chance. In a 2 x 2 contingency table the presence or absence of association between the two variables is often clear from inspection alone. The formal statistical test merely confirms (or refutes) it. In the case of complicated contingency tables involving several variables a more robust form of analysis is the log-linear analysis.
Recall that the χ2 test involves entering the frequency counts for the two categorical variables in rows and columns together with the marginal totals (i.e. totals for each row and each column), as well as the full overall total. From these totals the expected frequency for each cell is calculated. Then χ2 =Σ(Observed frequency − Expected)2 ÷ Expected. (Recall also that the probability of the joint occurrence of two independent events is the product of their separate probabilities). A log-linear model is best thought of as a model for the expected frequencies in a contingency table. But it is more than just an alternative form of the χ2 test. Its strength lies in that it can be extended to quite complicated contingency tables involving several variables.
In a 2 × 2 contingency table the probability of an individual occupying a given... [continues]
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