The analysis of fixed and random effects in a log-linear modeling framework

In statistical analysis involving categorical explanatory variables, the traditional distinction defines effects as fixed when all categories are observed and random when only a subset of categories is observed. This distinction is well established in the context of Analysis of Variance. This paper extends the theory to situations where all variables in the analysis – not just the explanatory ones – are categorical. In such cases, log-linear models are the most common analytical approach. The main theoretical result demonstrates that if a log-linear model holds when considering all categories of the involved variables (fixed effects), it also holds when some variables are restricted to a subset of their categories (random effects). Consequently, model fit tests that are valid in the fixed effects case remain applicable in the random effects case, although this may lead to a reduction of power. These findings are illustrated through simulations that explore various scenarios in which only a subset of a variable's categories is observed.