Probabilistic model of species co-occurrence

I’m continuing further work to develop the probabilistic model of species co-occurrence.  This paper (Veech_2006_J_Biogeography) was an intial step in developing the probabilistic model, although it still relied on data randomization (i.e., null models) to test for non-random co-occurrence between species.  In its most recent form the model is presented in this paper (Veech_2013_GEB).

The next phase in improving the model involves extending its applicability.  I would like to make the model more amenable to testing for multispecies (>2) co-occurrence patterns. As it stands now the model only tests for non-random co-occurrence between two species at a time.  It could actually use the same basic probabilistic framework to test for simultaneous co-occurrence of three species although the equations (to find the probabilities) are very long.  For more than three species, the model is just not mathematically tractable.  A mathematician colleague convinced me of this.  Nonetheless, there may be a work-around way to extend the model to multispecies.  Another colleague with good math and stats skills who is also an ecologist (Yoni Gavish) has corresponded with me on this issue – so we may eventually be able to figure something out.  I would also like to extend the model so that it can take into account that survey sites (or samples) may differ in some property or characteristic (that should be statistically controlled) or that may be of interest as an explanatory variable (e.g., percent cover of habitat).  I’ve got a good idea how to accomplish this – more details to come ….