BiogTest.individual and BiogTest.sample present randomization tests for the statistical comparison of two or more individual-based, sample-based, or coverage-based rarefaction curves. The biogeographical null hypothesis H0 is that is that two (or more) reference samples, represented by either abundance or incidence data, were drawn randomly from different assemblages that, nonetheless, share similar species richness and species abundance distributions.
To calculate the BiogTest statistic, Zobs, we summed the area between all unique pairs of the K sample rarefaction curves. If H0 is true, then Zobs should be relatively small because all of the rarefaction curves should have similar profiles, regardless of their species composition. In the limit, if all of the rarefaction curves had an identical profile, Zobs would equal 0. To construct the null distribution, the algorithm creates random assemblages by sampling from an underlying species abundance distribution. Of the many possible distributions which distribution should be used? The algorithm offers the possibility to choose among different underlying relative abundance distributions, including the log-normal, geometric and broken-stick distributions, although the former has some advantages, and thus its use is recommended.
The statistical parameters of the log-normal rank abundance distribution are the number of species in the assemblage and the variance of the distribution; the latter controls the differences in abundance between common and rare species. If these underlying parameters are known, then sample size effects can be estimated by random sampling of individuals from the specified distribution. However, it is very difficult to directly estimate these parameters from a sample or set of samples (O'Hara 2005). Instead, a suite of log-normal distributions is generated, that might act as a reasonable sampling universe for comparison with a set of reference samples to test the biogeographic null hypothesis. Our strategy was to specify a distribution for each of the two parameters in the log normal: species number and variance. As in a random effects model, each replicate of the null distribution reflects a single sample from a log-normal distribution in which the two model parameters were first determined by random assignment.
For the lower boundary of species richness, the minimum possible value cannot be smaller than the maximum number of species observed in the richest single sample among a set of samples. For the upper boundary of species richness, we calculated the upper bound of the Chao1 95
For the standard deviation of the log-normal, we sampled a random uniform value between 1.1 and 33 (0.1 and 3.5 on a log scale). For empirical assemblages, standard deviations typically fall within this range (Limpert et al. 2001). Once the null assemblage was specified by selection of parameters for species richness and the standard deviation, we sampled (with replacement) the specified number of individuals for each sample in an individual-based data set. For incidence data, we sampled the observed number of species in each sampling unit, sampling species without replacement, with sample probabilities set proportional to relative abundances in the log-normal distribution. We then used the analytic formulas in Chao et al. (2014) to construct the rarefaction curves for each of the pseudosamples, simulated a distribution of Zsim, and compared it to Zobs to estimate \(p(Z_{obs}|H_{0})\).
For the broken stick, the number of species in the meta-community must be known, and then a random partition is made to define the relative abundance of each species. For the geometric series, two parameters are needed: the number of species in the meta-community and a constant ratio D (D<1), which determines the abundance of the next species in the sequence. In the geometric series, D was obtained by sampling a random uniform value between 0.1 and 1. In all cases, the number of species (S), as in the log-normal distribution, was obtained by randomly drawing from a uniform distribution that was bounded at the low end by the maximum observed S and at the high end by the maximum upper bound of the 95
To better mimic the sampling process, a negative binomial random error was added to the abundance counts every time a sample was randomly drawn from the simulated meta-community. The negative binomial distribution was used to generate realistic heterogeneity that often results from spatial clustering of individuals and other small-scale processes. The expectation \(mu\) of the negative binomial was represented by the abundance count of each species in the meta-community.
Analytic estimators for individual-based, sample-based, and coverage-based rarefaction on Hill numbers were derived by Chao et al. (2014), and are currently implemented in this package (see link{rarefaction.individual} and link{rarefaction.sample} functions).