Analysis of similarities (ANOSIM) provides a way to test statistically
whether there is a significant difference between two or more groups
of sampling units. Function anosim operates directly on a
dissimilarity matrix. A suitable dissimilarity matrix is produced by
functions dist or vegdist. The
method is philosophically allied with NMDS ordination
(isoMDS), in that it uses only the rank order of
dissimilarity values. If two groups of sampling units are really different in their species
composition, then compositional dissimilarities between the groups
ought to be greater than those within the groups. The anosim
statistic $R$ is based on the difference of mean ranks between
groups ($r_B$) and within groups ($r_W$):
$$R = (r_B - r_W)/(N/(N-1)/4)$$
The divisor is chosen so that $R$ will be in the interval
$-1 \dots +1$, value $0$ indicating completely random
grouping.
The statistical significance of observed $R$ is assessed by
permuting the grouping vector to obtain the empiricial
distribution of $R$ under null-model.
The function has summary and plot methods. These both
show valuable information to assess the validity of the method: The
function assumes that all ranked dissimilarities within groups should
have about equal median and range. The plot method uses
boxplot with options notch=TRUE and
varwidth=TRUE.