Centre for Computational Statistics and Machine Learning Model Selection with Penalised Likelihood In a Multi-Dimensional Contingency Table

In multi-dimensional contingency tables sparse data occur frequently. For example, with bi-nary comorbidity data [1]; or genetic data, such as exons of spliced genes [2] or thresholded microarray data. We assume a log-linear model to analyse such tables, and use conditional Poisson likelihood (Birch, 1963), whence interactions between the variables measure depen-dence. In particular, we consider the class of hierarchical log-linear models. With sparse data, the maximum likelihood estimates of the parameters may not exist (Agresti, 2002). Then, the natural focus is on low order interactions and we explore a multiple Lasso-based penalised like-lihood approach, ` (θ), in this context. Like [2], we use penalised likelihood to analyse simulated contingency tables [1] in order to produce simpler hierarchical models. However, we extend the approach by introducing separate penalties for interactions of different orders, viz: ` (θ) = `Pois (θ) − λ2 |θ2 | − λ3 |θ3|, (1) where λ2 and λ3 are tuning constants (> 0) and θi i = 2, 3, are the vectors of unknown parame-ters associated with the 2-way and 3-way interactions, respectively. Below, we present contour plots (Y v λ2 and λ3) where Y = the percentage of significant two-way interactions (left), and the Pearson chi squared statistic (right) for a particular 5-dimensional table: Percentage of sign. two ways λ2 λ