Standard and Non-Standard Log-Linear Models For 2 × 2 Contingency Tables

Log-linear models can be used to model the joint relationship of two or more categorical variables in a multiway contingency table. In a log-linear model, the logarithm of the expected joint counts (or the logarithm of the joint probabilities) in a contingency table can be written as a linear model. Most log-linear models used in practice are standard. Standard log-linear models include the traditional parameter terms we see in ANOVA models: an overall effect, main effects, and various kinds of interaction terms. Standard log-linear models are divided into hierarchical and non-hierarchical. Hierarchical models satisfy the hierarchy principle: if a higher-order term is included in the log-linear model, then so are all the lower-order terms. Most of the standard log-linear models used in practice are hierarchical models. Nonhierarchical standard models appear rarely in the literature because they are difficult to interpret. In this thesis, we examine standard and non-standard log-linear models for 2 × 2 contingency tables. To show their application, we use a thromboembolism data set that first appeared in Vessey and Doll (1968) and was later analyzed by Worcester (1971) using a multiplicative model, which can be equivalently written as a non-standard log-linear model. Although the above data were collected in a one-to-two matching design, Worcester (1971) analyzed them using multinomial sampling where only the total was fixed. In this thesis, however, we also examine a product multinomial sampling design for these data, which is a more correct probability model for a matched design. We use the free statistical software R to estimate the above log-linear models. We compare the estimated log-linear models using the Pearson chi-square test, the G-square test, and the AIC, and we discuss the results.