Log-mean linear models for binary data

Details Log-mean linear models for binary data Log-mean linear models for binary data

2011-09-28

arxiv.org/abs/1109.6239v2

Statistics

Methodology

Scientific paper

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.

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