Combining conditional log-linear structures

Graphical models offer simple and intuitive interpretations in terms of conditional independence relationships, and these are especially valuable when large numbers of variables are involved. In some settings, restrictions on experiments and other forms of data collection may result in our being able to estimate only parts of a large graphical model; for example, when the data in a large contingency table are extremely sparse. In other settings, we might use a model building strategy that constructs component pieces first, and then tries to combine those pieces into a larger model. In this article we address this problem of combining component models in the context of cross-classified categorical data, and we show how to derive partial information about an underlying log-linear structure from its conditional log-linear structures and then how to use this information to choose a log-linear structure under the assumption that it is graphical. We illustrate the results using a simulated dataset based on a problem arising in cognitive psychology applied to learning.