Log-Mean Linear Parameterizations for Smooth Independence Models

This contribution aims to show how the log-mean linear parameterization for a set of categorical data can be used to specify models of marginal independence including also independence constraints in partial tables. The collection of these independencies can be jointly specified in a single log-mean linear model preserving the smoothness property. In the graphical modelling area, this approach allows us to define a class of parsimonious bi-directed graph models where all the constraints have a clear interpretation in terms of independencies. Moreover, whenever the data at hand seem to satisfy both marginal and conditional independencies that jointly result in a non-smooth model, the log-mean linear parameterization provides a partial solution which allows us to preserve smoothness by specifying the conditional independence statements only in partial tables. Finally, we suggest a criterion for variable coding such that these independencies are tested in partial tables with many observations. A simulation study illustrates how this method increases the efficiency of inferential procedures especially for sparse tables.