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("median","centroid"), bias.adjust = FALSE)## 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':
eigenvals(x, \dots)
## 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, \dots)
as.factor. Can consist of a factor with a single
level (i.e., one group)."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.
eigenvals.betadisper returns a named vector of eigenvalues.
TukeyHSD.betadisper returns a list. See TukeyHSD
for further details.
betadisper returns a list of class "betadisper" with the
following components:
type="centroid" had the wrong type I error and was
anti-conservative. As such, the default for type has been
changed to "median", which uses the spatial median as the group
centroid. Tests suggests that the permutation test for this type of
analysis gives the correct error rates.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 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. 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 (2006) 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. 2013).
Anderson, M.J., Ellingsen, K.E. & McArdle, B.H. (2006) Multivariate dispersion as a measure of beta diversity. Ecology Letters 9, 683--693.
O'Neill, M.E. (2000) A Weighted Least Squares Approach to Levene's Test of Homogeneity of Variance. Australian & New Zealand Journal of Statistics 42, 81-–100.
Stier, A.C., Geange, S.W., Hanson, K.M., & Bolker, B.M. (2013) Predator density and timing of arrival affect reef fish community assembly. Ecology 94, 1057--1068.
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, permutations = 99)
## 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)
## can also specify which axes to plot, ordering respected
plot(mod, axes = c(3,1))
## Draw a boxplot of the distances to centroid for each group
boxplot(mod)
## `scores` and `eigenvals` also work
scrs <- scores(mod)
str(scrs)
head(scores(mod, 1:4, display = "sites"))
# group centroids/medians
scores(mod, 1:4, display = "centroids")
# eigenvalues from the underlying principal coordinates analysis
eigenvals(mod)
## try out bias correction; compare with mod3
(mod3B <- betadisper(dis, groups, type = "median", bias.adjust=TRUE))
## should always work for a single group
group <- factor(rep("grazed", NROW(varespec)))
(tmp <- betadisper(dis, group, type = "median"))
(tmp <- betadisper(dis, group, type = "centroid"))
## simulate missing values in 'd' and 'group'
## using spatial medians
groups[c(2,20)] <- NA
dis[c(2, 20)] <- NA
mod2 <- betadisper(dis, groups) ## warnings
mod2
permutest(mod2, permutations = 99)
anova(mod2)
plot(mod2)
boxplot(mod2)
plot(TukeyHSD(mod2))
## Using group centroids
mod3 <- betadisper(dis, groups, type = "centroid")
mod3
permutest(mod3, permutations = 99)
anova(mod3)
plot(mod3)
boxplot(mod3)
plot(TukeyHSD(mod3))Run the code above in your browser using DataLab