Overview
While alpha diversity examines differences within treatment groups, beta diversity measures the similarity (or dissimilarity) between samples and groups.
Each variable is plotted on principal coordinates analysis (PCoA) plots, to examine the variance between samples based on a dissimilarity matrix. A detailed explanation of PCoA and other ordination methods can be seen here: http://albertsenlab.org/ampvis2-ordination/
Sample distance can be measured using 3 distance-based ordination methods. These methods are:
Bray–Curtis dissimilarity measures the fraction of overabundant counts.
Sorenson, T. (1948) “A method of establishing groups of equal amplitude in plant sociology based on similarity of species content.” Kongelige Danske Videnskabernes Selskab 5.1-34: 4-7.
Cao index is a minimally biased index for high beta diversity and variable sampling intensity. Chao index tries to take into account the number of unseen species pairs.
Cao, Y., Bark, A. W., & Williams, W. P. (1997). Analysing benthic macroinvertebrate community changes along a pollution gradient: a framework for the development of biotic indices. Water Research, 31(4), 884-892.
Jaccard similarity index measures the fraction of unique features, regardless of abundance..
Jaccard, P. (1908). “Nouvellesrecherches sur la distribution florale.” Bull. Soc. V and. Sci. Nat., (44):223-270.
Significance tests
For each beta diversity method, both overall significance and pairwise significance can be calculated using a Permutational Multivariate Analysis of Variance (PERMANOVA), a non-parametric multivariate statistical test. The R-squared value represents the percentage of variance explained by the examined groups. E.g. if R = 0.23 then 23% of the total diversity is explained by groupwise differences. PERMANOVA is based on groupwise differences, thus cannot be applied to continuous data.