Fienberg and Kim [1999. J. Amer. Statist. Assoc. 94 (445), 229-239] investigated the relationship between a log-linear model (LLM) and its conditional log-linear model (CLLM). While they considered a single conditional variable, we explored the relationship further considering multiple conditional variables. It is shown that we can find the exact model structure of an LLM from the sets of CLLMs of the LLM conditional on some sets of variables under a certain condition for the conditional sets. A graphical interpretation of the relation is presented when the LLM is graphical. The results are applied effectively for analyzing categorical real data from a lecture evaluation. (c) 2005 Elsevier B.V. All rights reserved.