  Large contingency tables summarizing categorical variables arise in many
areas. For example in biology when a large number of biomarkers are
cross-tabulated according to their discrete expression level. Interactions of
the variables are generally studied with log-linear models and the structure of
a log-linear model can be visually represented by a graph from which the
conditional independence structure can then be read off. However, since the
number of parameters in a saturated model grows exponentially in the number of
variables, this generally comes with a heavy burden as far as computational
power is concerned. If we restrict ourselves to models of lower order
interactions or other sparse structures we face similar problems as the number
of cells remains unchanged. We therefore present a divide-and-conquer approach,
where we first divide the problem into several lower-dimensional problems and
then combine these to form a global solution. Our methodology is
computationally feasible for log-linear interaction modeling with many
categorical variables each or some of them having many categories. We
demonstrate the proposed method on simulated data and apply it to a bio-medical
problem in cancer research.
