  Multivariate sample spaces may be incomplete Cartesian products, when certain
combinations of the categories of the variables are not possible. Traditional
log-linear models, which generalize independence and conditional independence,
do not apply in such cases, as they may associate positive probabilities with
the non-existing cells. To describe the association structure in incomplete
sample spaces, this paper develops a class of hierarchical multiplicative
models which are defined by setting certain non-homogeneous generalized odds
ratios equal to one and are named after Aitchison and Silvey who were among the
first to consider such ratios. These models are curved exponential families
that do not contain an overall effect and, from an algebraic perspective, are
non-homogeneous toric ideals. The relationship of this model class with
log-linear models and quasi log-linear models is studied in detail in terms of
both statistics and algebraic geometry. The existence of maximum likelihood
estimates and their properties, as well as the relevant algorithms are also
discussed.
