Cascade multivariate regression tree: a novel approach for modelling nested explanatory sets

1. Ecological data analysis frequently calls for the assessment of the relationship between species composition and a set of explanatory variables of interest. The assessment may have to be pursued while taking into account the influence of another set of explanatory variables. The hypothetical nature and structure of the influence of an explanatory set on the effect of a distinct explanatory set guides the proper choice of modelling methodology for a combined explanatory assessment. 2. Here, we describe a framework where the relationship between the response data and a main set of explanatory variables is not linear. It may, for example, take the form of abrupt changes in the response following thresholds of the explanatory variables, or any other nonlinearizable relationship. The influence of a second set of explanatory variables is determined a posteriori, after the influence of the main explanatory set has been taken into account. This is useful when one of the sets is thought to have an effect that varies as a function of the other. 3. To achieve this type of assessment, we propose a cascade of multivariate regression trees ( CMRT). We decompose the total dispersion of a response matrix between two explanatory data sets in a nested manner. By handling each leaf ( group) resulting from the first-level multivariate regression tree ( MRT) analysis as separate independent data sets in following analyses, we can separate the explanatory power of the first partition from those of the subordinate partitions computed using a second explanatory set. A preliminary biological hypothesis will guide the choice of which set of explanatory variables should be used to compute the main partition. The method could be extended to more than two explanatory data sets whose effects on the response data are hierarchical. 4. Cascade of multivariate regression trees allows the users to impose a nested structure to their causal hypotheses in MRT analysis. To illustrate this new procedure, we use the well-known and readily available Doubs fish and oribatid mite data sets and provide the necessary R functions in a package available onCRAN( http:// cran. r-project. org).