3.17 The loglinear structure model for joint attribute distribution
A log-linear model takes the form of a function whose logarithm equals a linear combination of the parameters of the model. X. Xu & Davier (2008) proposed to use a loglinear model to smooth the joint attribute distribution, which can be written as
\[\begin{equation} \label{loglinear} \log[n({\alpha}_c)]=\lambda_0+\sum_{k=1}^K\lambda_k\alpha_k+\sum_{k=1}^{K-1}\sum_{k'=k+1}^K\lambda_{kk'}\alpha_k\alpha_{k'}, \end{equation}\]where \(n({\alpha}_c)=N\pi_c\) is the number of individuals with attribute profile \({\alpha}_c\). The above loglinear structural model considers main effects and first-order interactions. It has \(1+K(K+1)/2\) parameters, namely, \({\lambda}=[\lambda_0,\lambda_1,\ldots,\lambda_{K-1,K}]^\top\). It is very flexible in that it can be simplified by removing all interactions between attributes, or extended by including higher-order attribute interactions.
Tan et al. (2022) use the log-linear model to examine the co-presence of attributes. They found that the conditional relationship between any pair of attributes is the same across different combinations of the remaining attributes (i.e., no three-way interactions). Substantially, they found that the alcohol-related problem attribute was not significantly associated with other attributes, whereas anxiety, hostility, and depression attributes exhibited statistically significant homogeneous associations. For example, the attributes depression and hostility were more likely to be co-present.