All data were either simulated, adopted from published sources, or retrieved from data repositories. References to all original sources are consistently provided. The data were used to explore various analytical scenarios in which all variables are categorical and, for at least some of them, only a subset of categories is observed. When all variables are categorical, one of the most common analytical approaches is log-linear modeling. 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).