Bi-dimensional null model analysis of presence-absence binary matrices.

Giovanni Strona,Nicholas J Gotelli,Werner Ulrich

doi:10.1002/ecy.2043

Giovanni Strona, Nicholas J Gotelli + Show 1 more

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https://doi.org/10.1002/ecy.2043

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Abstract

Comparing the structure of presence/absence (i.e., binary) matrices with those of randomized counterparts is a common practice in ecology. However, differences in the randomization procedures (null models) can affect the results of the comparisons, leading matrix structural patterns to appear either "random" or not. Subjectivity in the choice of one particular null model over another makes it often advisable to compare the results obtained using several different approaches. Yet, available algorithms to randomize binary matrices differ substantially in respect to the constraints they impose on the discrepancy between observed and randomized row and column marginal totals, which complicates the interpretation of contrasting patterns. This calls for new strategies both to explore intermediate scenarios of restrictiveness in-between extreme constraint assumptions, and to properly synthesize the resulting information. Here we introduce a new modeling framework based on a flexible matrix randomization algorithm (named the "Tuning Peg" algorithm) that addresses both issues. The algorithm consists of a modified swap procedure in which the discrepancy between the row and column marginal totals of the target matrix and those of its randomized counterpart can be "tuned" in a continuous way by two parameters (controlling, respectively, row and column discrepancy). We show how combining the Tuning Peg with a wise random walk procedure makes it possible to explore the complete null space embraced by existing algorithms. This exploration allows researchers to visualize matrix structural patterns in an innovative bi-dimensional landscape of significance/effect size. We demonstrate the rational and potential of our approach with a set of simulated and real matrices, showing how the simultaneous investigation of a comprehensive and continuous portion of the null space can be extremely informative, and possibly key to resolving longstanding debates in the analysis of ecological matrices.

Highlights

The problem of identifying non-random structural patterns of species distribution compared to a background expectation of ecological noise has received much attention, leading to the development of statistical standards for pattern identification in binary presence-absence matrices
The nearly totality of null matrices in all replicates and across the whole landscape had NODF and C-score values not different from those observed in the random artificial matrices with uniform marginal totals (U), while they resulted more nested (NODF) and with a smaller C-score than those observed in the artificial nested matrices (N)
We provide a new perspective on the issue, by showing how different null models can be reconciled on a bi-dimensional continuum of restrictiveness, whose dimensions are defined by the marginal mismatch in row and column totals between the target matrix and its randomized counterparts

Summary

Introduction

The problem of identifying non-random structural patterns of species distribution compared to a background expectation of ecological noise has received much attention, leading to the development of statistical standards for pattern identification in binary presence-absence matrices (rows = species, columns = sites; Gotelli and Graves 1996, Gimenez et al 2014). The resulting effect size quantifies the strength of the observed pattern, and the simulation generates a frequentist estimate of statistical significance. This procedure, raises several issues at each step of Manuscript received 27 June 2017; revised 8 September 2017; accepted 4 October 2017. Null models for randomizing presence-absence matrices retain, at minimum, the observed number of rows (usually species) and columns (usually sample sites). The key difference among existing null models is how they treat marginal sums of the matrix, which represent the occurrence frequencies of the different species (= row sums) and the number of species per site (= column sums)

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