This paper is devoted to the theory and application of a novel class of
models for binary data, which we call log-mean linear (LML) models. The
characterizing feature of these models is that they are specified by linear
constraints on the LML parameter, defined as a log-linear expansion of the mean
parameter of the multivariate Bernoulli distribution. We show that marginal
independence relationships between variables can be specified by setting
certain LML interactions to zero and, more specifically, that graphical models
of marginal independence are LML models. LML models are code dependent, in the
sense that they are not invariant with respect to relabelling of variable
values. As a consequence, they allow us to specify sub-models defined by
code-specific independencies, which are independencies in sub-populations of
interest. This special feature of LML models has useful applications. Firstly,
it provides an alternative, flexible, way to specify parsimonious sub-models of
marginal independence models. The main advantage of this approach concerns the
interpretation of the sub-model, which is fully characterized by independence
relationships, either marginal or code-specific. Secondly, the code-specific
nature of these models can be exploited to focus on a fixed, arbitrary, cell of
the probability table and on the corresponding sub-population. This leads to an
innovative family of models, which we call pivotal code-specific LML models,
that is especially useful when the interest of researchers is focused on a
small sub-population obtained by stratifying individuals according to some
features. The application of LML models is illustrated on three datasets, one
of which concerns the use of pivotal code-specific LML models in the field of
personalized medicine.