betadisper is a multivariate analogue of Levene's test for
homogeneity of variances. Non-euclidean distances between objects and
group centroids are handled by reducing the original distances to
principal coordinates. This procedure has latterly been used as a
means of assessing beta diversity. There are anova,
scores, plot and boxplot methods. TukeyHSD.betadisper creates a set of confidence intervals on
the differences between the mean distance-to-centroid of the levels of
the grouping factor with the specified family-wise probability of
coverage. The intervals are based on the Studentized range statistic,
Tukey's 'Honest Significant Difference' method.
betadisper(d, group, type = c("centroid", "median"))## S3 method for class 'betadisper':
anova(object, \dots)
## S3 method for class 'betadisper':
scores(x, display = c("sites", "centroids"),
choices = c(1,2), ...)
## S3 method for class 'betadisper':
plot(x, axes = c(1,2), cex = 0.7, hull = TRUE,
ylab, xlab, main, sub, ...)
## S3 method for class 'betadisper':
boxplot(x, ylab = "Distance to centroid", ...)
## S3 method for class 'betadisper':
TukeyHSD(x, which = "group", ordered = FALSE,
conf.level = 0.95, \ldots)
as.factor.type =
"centroid" is currently supported."sites" or "species"."betadisper", the result of a
call to betadisper.plot.default.TukeyHSD.plot.betadisper and boxplot.betadisper), passed to
other methods.anova method returns an object of class "anova"
inheriting from class "data.frame". The scores method returns a list with one or both of the
components "sites" and "centroids".
The plot function invisibly returns an object of class
"ordiplot", a plotting structure which can be used by
identify.ordiplot (to identify the points) or other
functions in the ordiplot family.
The boxplot function invisibly returns a list whose components
are documented in boxplot.
TukeyHSD.betadisper returns a list. See TukeyHSD
for further details.
betadisper returns a list of class "betadisper" with the
following components:
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 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 comprisons 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.
Anderson, M.J., Ellingsen, K.E. & McArdle, B.H. (2006) Multivariate dispersion as a measure of beta diversity. Ecology Letters 9(6), 683--693.
permutest.betadisper, anova.lm,
scores, boxplot,
TukeyHSD. Further measure of beta diversity
can be found in betadiver.data(varespec)
## Bray-Curtis distances between samples
dis <- vegdist(varespec)
## First 16 sites grazed, remaining 8 sites ungrazed
groups <- factor(c(rep(1,16), rep(2,8)), labels = c("grazed","ungrazed"))
## Calculate multivariate dispersions
mod <- betadisper(dis, groups)
mod
## Perform test
anova(mod)
## Permutation test for F
permutest(mod, pairwise = TRUE)
## Tukey's Honest Significant Differences
(mod.HSD <- TukeyHSD(mod))
plot(mod.HSD)
## Plot the groups and distances to centroids on the
## first two PCoA axes
plot(mod)
## Draw a boxplot of the distances to centroid for each group
boxplot(mod)Run the code above in your browser using DataLab