vegan (version 1.11-0)

betadisper: Multivariate homogeneity of groups dispersions (variances)

Description

Implements Marti Anderson's PERMDISP2 procedure for the analysis of multivariate homogeneity of group dispersions (variances). 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.

Usage

betadisper(d, group, type = c("centroid", "median"))
## S3 method for class 'betadisper':
anova(object, \dots)
permDisper(object, control = permControl(nperm = 999))
## 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", ...)

Arguments

d

a distance structure such as that returned by dist or vegdist.

group

vector describing the group structure, usually a factor or an object that can be coerced to a factor using as.factor.

type

the type of analysis to perform. Only type = "centroid" is currently supported.

object, x

an object of class "betadisper", the result of a call to betadisper.

control

a list of control values for the permutations to replace the default values returned by the function permControl.

display

character; partial match to access scores for "sites" or "species".

choices, axes

the principal coordinate axes wanted.

hull

logical; should the convex hull for each group be plotted?

cex, ylab, xlab, main, sub

graphical parameters. For details, see plot.default.

...

arguments, including graphical parameters, passed to other methods.

Value

• The anova method returns an object of class "anova" inheriting from class "data.frame".

  permDisper returns an object of class "permDisper", a list with components tab, the ANOVA table which is an object inheriting from class "data.frame", and control, the result of a call to permControl..

  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.

  betadisper returns a list of class "betadisper" with the following components:

• eig
  numeric; the eigenvalues of the principal coordinates analysis.
• vectors
  matrix; the eigenvectors of the principal coordinates analysis.
• distances
  numeric; the Euclidean distances in principal coordinate space between the samples and their respective group centroid.
• group
  factor; vector describing the group structure
• centroids
  matrix; the locations of the group centroids on the principal coordinates.
• call
  the matched function call.

Details

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 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.

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. permDisper permutes model residuals to generate a permutation distribution of F under the Null hypothesis of no difference in dispersion between groups.

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).

References

Anderson, M.J. (2006) Distance-based tests for homogeneity of multivariate dispersions. Biometrics 62(1), 245--253.

Anderson, M.J., Ellingsen, K.E. & McArdle, B.H. (2006) Multivariate dispersion as a measure of beta diversity. Ecology Letters 9(6), 683--693.

See Also

anova.lm, scores, boxplot

Examples

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
permDisper(mod)

## 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)