DOI: 10.5061/dryad.p2ngf1vn5
title: Data from: Extending null scenarios with Faddy distributions in a probabilistic randomization protocol for presence-absence data

description: 1. The analysis of species occurrences at discrete locations makes use of
 statistical methods intended to elucidate whether a random process can
 explain a particular observed pattern of presences-absences (1-0). Various
 statistical methods have contributed to the development of null model
 analysis of (1-0) data in community ecology using randomization tests.
 Frequentist techniques assuming probability distributions under the null
 scenarios have been proposed, as in the work by Navarro and Manly (2009)
 (NM), a protocol that has been applied in the analysis of plant and
 microbial communities, and chemical hazards. 2. The NM method assumes that
 presences-absences are governed by independent Bernoulli random variables,
 and that a non-observable non-negative random variable (“quasi-abundance”)
 is associated to each species on each location. The quasi-abundance is
 presumed to follow any of three possible distributions (Poisson, Binomial
 and Negative Binomial) and to be log-linearly related to the qualitative
 effects of species and location. By connecting the probability of
 occurrence of each species on each location, and the "best"
 quasi-abundance distribution (chosen by profile deviance), it is possible
 to estimate that probability by generalized linear modelling, which is
 used, in turn, to generate random matrices via parametric bootstrap. The
 question now is whether just three distributions are enough to support an
 “optimal” null model. 3. We provide the theoretical formulation of the
 original NM protocol for null model analysis, and then expand the
 quasi-abundance distributions, based on extended Poisson processes (Faddy
 1997), to allow general distributions of over-dispersed and
 under-dispersed discrete random variables. The method is illustrated using
 presence-absence data of island lizard communities. 4. For the binomial
 case and Faddy distributions, nonlinear constrained optimization
 algorithms are needed in order to get maximum likelihood estimates thus,
 the null-model selection process faces challenging numerical problems
 (non-convergence to the global optimum). In addition, the process may end
 up suggesting that the best fitted probabilities for the generation of
 null matrices are those obtained from links different to the canonical
 logistic link. This property of the NM protocol should not be ignored, as
 an improper choice of the null matrix universe may impact the outcome of
 randomization tests.
description: The file contains <em>coded presence-absence
 data</em> of 20 lizard species on 25 islands in the Gulf of
 California (Case 1983; Manly 1995). Species and island numbers follow the
 order shown in Manly (1995). Each space-delimited line corresponds to a
 single species (20 lines in the example); each line always ends with a
 –1.  A positive number correspond to an island number, i.e., the column in
 the presence-absence matrix where that species appeared. With the
 exception of the last number on each line, a negative number will always
 be preceded by a positive number indicating the range of islands where the
 species occurs, from the positive number to the absolute value of the
 negative number. Example. The third line in lizgal.txt
 (species <em>Scleoporus orcutti</em>) contains the
 numbers: 13_–16_22_23_–1    (_ indicates a
 whitespace) The line indicates that
 <em>Scleoporus orcutti</em> occurred on islands 13, 14, 15,
 16, 22 and 23.
 <strong>References</strong> Case,
 T. J. (1983) Niche overlap and the assembly of island lizard communities.
 <em>Oikos</em> 41:427-433. Manly, B. F. J.
 (1995) A note on the analysis of species co-occurrences.
 <em>Ecology</em> 76:1109-1115.  
description: This data file is read by R-scripts “FaddyP-NB.r”, “FaddyBinom.r”
 and “Faddyover-under.r” for null model selection. Each program, included
 in the Supporting Information section, detects the total number of rows
 and columns from the coded presence-absence data, and converts these into
 a binary matrix.