Coefficients of association between nominal and fully ranked ordinal variables with applications to ecological network analysis

Abstract

A central issue of ecological data analysis is the pairwise comparison of variables describing biological entities and the environment. Difficulties arise with calculations if the measurement scales of the variables differ. In particular, no method is available for measuring the association between a nominal and a fully ranked ordinal variable. Here two coefficients are suggested by reducing this problem to the evaluation of pattern in string representations. The first one is a topological measure that counts the number of other types of elements occurring between pairs of elements of a given state along the entire length of the string, thus providing a global coefficient of aggregation/segregation. The second coefficient is based on counting the number of different elements within substrings generated from the complete string with the moving window technique. Thus, it is a local measure. There is no compact and general formula for calculating these measures, and heuristics are involved for finding the possible minimum and maximum values by algorithmic approximation and Markov Chain Monte Carlo simulation. An R function is provided for computations. The methods are applied to the comparison of nominal variables (biological traits) categorizing marine food web nodes with fully ranked variables describing major graph theory properties of the same nodes in the network. The most descriptive traits (mobility, major functional group) significantly associated with network metrics (weighted indices) were identified from a variety of combinations across three marine ecosystems. These coefficients thus provide an objective, statistically-sound method for identifying ecologically meaningful traits.