gms | German Medical Science

Background: Tools for modelling multi-species distributions are of great interest in ecology. The reasons for this are twofold: first, they allow for both a marginal interpretation which provides information about the distribution one species at a time. Secondly, they allow for a joint interpretation, providing information about the correlations between the species abundances and their possible dependency on other covariates .

Recently, Kinshofer et al. [1] investigated interspecific competition between three species of fish-eating birds through the analysis of counts collected at lake Seehammer See (Upper Bavaria) between 2001 and 2016. By comparing standardized regression coefficients for the trends of the three species, their analysis showed that the Winter populations were not limited by the local food resources. Instead, positive trends were observed.

Methods: Our approach is to apply multivariate conditional transformation models (MCTMs) first introduced by Klein et al. [2] to the problem of multi-species distribution modelling.

Klein et al. transform an absolutely continuous response vector Y via an unknown, bijective, strictly monotonically increasing transformation function. Then, they factorize the multivariate distribution of the transformed response vector as a product of conditional distributions. The parameters in the model amount to parameters of a Gaussian copula if one imposes a triangular structure on the transformation function and makes specific assumptions on the distribution of the transformed response. This is convenient because the parameters are directly interpretable as parameters for the marginal distributions or as correlation coefficients between the species.

Results: Although MCTMs are initially formulated for an absolutely continuous random vector, Klein et al. also derived an extension for the estimation of distribution functions of discrete random vectors. However, in the Gaussian copula framework, estimation of the parameters in this extension is numerically more demanding because it requires evaluation of a high dimensional integral. We derived analytic-form expressions for the log-likelihood and scores in the discrete case and implemented these in R to estimate the parameters of interest while still remaining in the Gaussian copula framework.

Conclusion: Moreover, MCTMs allow for multivariate regression by letting the estimated coefficients depend on the covariates. This enables estimation of time-dependent correlation coefficients, similar to what was done by Kinshofer et al. On the other hand, as opposed to Kinshofer et al., we are able to estimate all parameters simultaneously and do not have to fit separate models for the three bird species.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.