BetaDispersion

This is a small package aimed at providing equivalent functionality to betadisper in the R package vegan, namely multivariate test for homogeneity of variance (dispersion). This is often used in ecology to compare beta diversity between metacommunities. This analysis comes from work by Marti Anderson (2006). However, a number of tweaks have been made to the preferred way of performing the analysis since then. We aim to keep this package aligned with decisions made by vegan devs. However, the current implementation only offers permutation tests for P-values, and only spatial medians (not centroids) are used. Use of permutation tests and spatial medians are considered more robust than alternatives, but please open an issue/PR if you feel that these options are not appropriate for your use case.

The basic steps are to calculate spatial medians for groups of data points, calculate F-statistics (including those for each pairwise combination of groups), then permute the residuals (distances from medians) and re-calculate F for each permutation. The P-value is the proportion of permuted F values that are greater than or or equal to the original F value.

Two functions are exported, dispersion takes either a data array (where each row is an observation), a vector containing group identities and a distance function type from Distances.jl, or a distance matrix and the grouping vector. Alternatively, if working with Euclidean distances, pass in the original data (where each row is an observation) instead of a distance matrix and set the key work argument metric = true. This function returns a Disp struct containing:

• F = Global F-Statistic
• pairwise_F = pairwise F-statistics
• medians = spatial (aka geometric) median of each group in the transformed coordinates
• residuals = vector containing each observations distance from its group median
• means = vector containing the means distance to median for each group
• group = the original grouping vector
• levels = unique items from 'group'
• group_residuals = vector containing each groups distance from the overall median

permutest takes the named tuple returned by dispersion and returns a named tuple containing:

• P = Global P-value
• pairwise_P = pairwise P-values
• F = Global F-Statistic
• pairwise_F = pairwise F-statistics

Additionally, If Turing.jl is loaded before BetaDispersion you can call bayesdisp on the Disp returned by dispersion to perform Bayesian inference to estimate mean and standard deviation of the residuals of each group. By default 4 chains are run simultaneously using the No U-turns (NUTS) sampler for 1000 iterations each. The function returns the Markov chains along with diagnostics, more details can be found here. Check the source code to see the function arguments. If fine control over model implementation is required, the user is urged to construct their model directly in Turing.jl or the users preferred PPL.

Example usage

using Turing, StatsPlots
using Distances, BetaDispersion
x = rand(30,5)
g = rand(1:2,30)
d = dispersion(x,g,BrayCurtis)
p = permutest(d)
chns = bayesdisp(d)
plot(chns)