One measure of multivariate dispersion (variance) for a group of
samples is to calculate the average distance of group members to the
group centroid or spatial median (both referred to as 'centroid' from
now on unless stated otherwise) in multivariate space. To test if the
dispersions (variances) of one or more groups are different, the
distances of group members to the group centroid are subject to
ANOVA. This is a multivariate analogue of Levene's test for
homogeneity of variances if the distances between group members and
group centroids is the Euclidean distance. However, better measures of distance than the Euclidean distance are
available for ecological data. These can be accommodated by reducing
the distances produced using any dissimilarity coefficient to
principal coordinates, which embeds them within a Euclidean space. The
analysis then proceeds by calculating the Euclidean distances between
group members and the group centroid on the basis of the principal
coordinate axes rather than the original distances.
Non-metric dissimilarity coefficients can produce principal coordinate
axes that have negative Eigenvalues. These correspond to the
imaginary, non-metric part of the distance between objects. If
negative Eigenvalues are produced, we must correct for these imaginary
distances.
The distance to its centroid of a point is $$z_{ij}^c =
\sqrt{\Delta^2(u_{ij}^+, c_i^+) - \Delta^2(u_{ij}^-, c_i^-)},$$ where
$\Delta^2$ is the squared Euclidean distance between
$u_{ij}$, the principal coordinate for the $j^{th}$
point in the $i^{th}$ group, and $c_i$, the
coordinate of the centroid for the $i^{th}$ group. The
super-scripted $+$ and $-$ indicate the real and imaginary
parts respectively. This is equation (3) in Anderson (2006). If the
imaginary part is greater in magnitude than the real part, then we
would be taking the square root of a negative value, resulting in
NaN. From vegan 1.12-12 betadisper takes the absolute
value of the real distance minus the imaginary distance, before
computing the square root. This is in line with the behaviour of Marti
Anderson's PERMDISP2 programme.
To test if one or more groups is more variable than the others, ANOVA
of the distances to group centroids can be performed and parametric
theory used to interpret the significance of F. An alternative is to
use a permutation test. permutest.betadisper permutes model
residuals to generate a permutation distribution of F under the Null
hypothesis of no difference in dispersion between groups.
Pairwise comparisons of group mean dispersions can also be performed
using permutest.betadisper. An alternative to the classical
comparison of group dispersions, is to calculate Tukey's Honest
Significant Differences between groups, via
TukeyHSD.betadisper. This is a simple wrapper to
TukeyHSD.aov. The user is directed to read the help file
for TukeyHSD before using this function. In particular,
note the statement about using the function with
unbalanced designs.
The results of the analysis can be visualised using the plot
and boxplot methods.
One additional use of these functions is in assessing beta diversity
(Anderson et al 2006). Function betadiver
provides some popular dissimilarity measures for this purpose.
As noted in passing by Anderson (2001) and in a related
context by O'Neill (2000), estimates of dispersion around a
central location (median or centroid) that is calculated from the same data
will be biased downward. This bias matters most when comparing diversity
among treatments with small, unequal numbers of samples. Setting
bias.adjust=TRUE when using betadisper imposes a
$\sqrt{n/(n-1)}$ correction (Stier et al. 2012).