Permutational Multivariate Analysis of Variance Using Distance Matrices
Description
Analysis of variance using distance matrices — for partitioning distance matrices among sources of variation and fitting linear models (e.g., factors, polynomial regression) to distance matrices; uses a permutation test with pseudoF ratios.
Usage
1 2 3 
Arguments
formula 
a typical model formula such as 
data 
the data frame from which 
permutations 
a list of control values for the permutations
as returned by the function 
method 
the name of any method used in 
strata 
groups (strata) within which to constrain permutations. 
contr.unordered, contr.ordered 
contrasts used for the design matrix (default in R is dummy or treatment contrasts for unordered factors). 
parallel 
Number of parallel processes or a predefined socket
cluster. With 
... 
Other arguments passed to 
Details
adonis
is a function for the analysis and partitioning
sums of squares using semimetric and metric distance matrices. Insofar
as it partitions sums of squares of a multivariate data set, it is
directly analogous to MANOVA (multivariate analysis of
variance). M.J. Anderson (McArdle and Anderson 2001, Anderson 2001) refers to the
method as “permutational manova” (formerly “nonparametric manova”). Further, as its inputs are
linear predictors, and a response matrix of an arbitrary number of
columns (2 to millions), it is a robust alternative to both parametric
MANOVA and to ordination methods for describing how variation is
attributed to different experimental treatments or uncontrolled
covariates. It is also analogous to redundancy analysis (Legendre and
Anderson 1999).
Typical uses of adonis
include analysis of ecological community
data (samples X species matrices) or genetic data where we might have a
limited number of samples of individuals and thousands or millions of
columns of gene expression data (e.g. Zapala and Schork 2006).
adonis
is an alternative to AMOVA (nested analysis of molecular
variance, Excoffier, Smouse, and Quattro, 1992;
amova
in the ade4 package) for both crossed
and nested factors.
If the experimental design has nestedness, then use strata
to
test hypotheses. For instance, imagine we are testing whether a
plant community is influenced by nitrate amendments, and we have two
replicate plots at each of two levels of nitrate (0, 10 ppm). We have
replicated the experiment in three fields with (perhaps) different
average productivity. In this design, we would need to specify
strata = field
so that randomizations occur only within
each field and not across all fields . See example below.
Like AMOVA (Excoffier et al. 1992), adonis
relies on a
longunderstood phenomenon that allows one to partition sums of squared
deviations from a centroid in two different ways (McArdle and Anderson
2001). The most widely recognized method, used, e.g., for ANOVA and
MANOVA, is to first identify the relevant centroids and then to
calculated the squared deviations from these points. For a centered
n x p response matrix Y, this method uses the
p x p inner product matrix Y'Y. The less
appreciated method is to use the n x n outer product
matrix YY'. Both AMOVA and adonis
use this latter
method. This allows the use of any semimetric (e.g. BrayCurtis, aka
Steinhaus, Czekanowski, and Sørensen) or metric
(e.g. Euclidean) distance matrix (McArdle and Anderson 2001). Using
Euclidean distances with the second method results in the same analysis
as the first method.
Significance tests are done using Ftests based on sequential sums
of squares from permutations of the raw data, and not permutations of
residuals. Permutations of the raw data may have better small sample
characteristics. Further, the precise meaning of hypothesis tests will
depend upon precisely what is permuted. The strata argument keeps groups
intact for a particular hypothesis test where one does not want to
permute the data among particular groups. For instance, strata = B
causes permutations among levels of A
but retains data within
levels of B
(no permutation among levels of B
). See
permutations
for additional details on permutation tests
in Vegan.
The default contrasts
are different than in R in
general. Specifically, they use “sum” contrasts, sometimes known
as “ANOVA” contrasts. See a useful text (e.g. Crawley,
2002) for a transparent introduction to linear model contrasts. This
choice of contrasts is simply a personal
pedagogical preference. The particular contrasts can be set to any
contrasts
specified in R, including Helmert and treatment
contrasts.
Rules associated with formulae apply. See "An Introduction to R" for an overview of rules.
print.adonis
shows the aov.tab
component of the output.
Value
This function returns typical, but limited, output for analysis of variance (general linear models).
aov.tab 
Typical AOV table showing sources of variation, degrees of freedom, sequential sums of squares, mean squares, F statistics, partial Rsquared and P values, based on N permutations. 
coefficients 
matrix of coefficients of the linear model, with rows representing sources of variation and columns representing species; each column represents a fit of a species abundance to the linear model. These are what you get when you fit one species to your predictors. These are NOT available if you supply the distance matrix in the formula, rather than the site x species matrix 
coef.sites 
matrix of coefficients of the linear model, with rows representing sources of variation and columns representing sites; each column represents a fit of a sites distances (from all other sites) to the linear model. These are what you get when you fit distances of one site to your predictors. 
f.perms 
an N by m matrix of the null F
statistics for each source of variation based on N
permutations of the data. The permutations can be inspected with

model.matrix 
The 
terms 
The 
Note
Anderson (2001, Fig. 4) warns that the method may confound
location and dispersion effects: significant differences may be caused
by different withingroup variation (dispersion) instead of different
mean values of the groups (see Warton et al. 2012 for a general
analysis). However, it seems that adonis
is less sensitive to
dispersion effects than some of its alternatives (anosim
,
mrpp
). Function betadisper
is a sister
function to adonis
to study the differences in dispersion
within the same geometric framework.
Author(s)
Martin Henry H. Stevens HStevens@muohio.edu, adapted to vegan by Jari Oksanen.
References
Anderson, M.J. 2001. A new method for nonparametric multivariate analysis of variance. Austral Ecology, 26: 32–46.
Crawley, M.J. 2002. Statistical Computing: An Introduction to Data Analysis Using SPLUS
Excoffier, L., P.E. Smouse, and J.M. Quattro. 1992. Analysis of molecular variance inferred from metric distances among DNA haplotypes: Application to human mitochondrial DNA restriction data. Genetics, 131:479–491.
Legendre, P. and M.J. Anderson. 1999. Distancebased redundancy analysis: Testing multispecies responses in multifactorial ecological experiments. Ecological Monographs, 69:1–24.
McArdle, B.H. and M.J. Anderson. 2001. Fitting multivariate models to community data: A comment on distancebased redundancy analysis. Ecology, 82: 290–297.
Warton, D.I., Wright, T.W., Wang, Y. 2012. Distancebased multivariate analyses confound location and dispersion effects. Methods in Ecology and Evolution, 3, 89–101.
Zapala, M.A. and N.J. Schork. 2006. Multivariate regression analysis of distance matrices for testing associations between gene expression patterns and related variables. Proceedings of the National Academy of Sciences, USA, 103:19430–19435.
See Also
mrpp
, anosim
,
mantel
, varpart
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  data(dune)
data(dune.env)
adonis(dune ~ Management*A1, data=dune.env, permutations=99)
### Example of use with strata, for nested (e.g., block) designs.
dat < expand.grid(rep=gl(2,1), NO3=factor(c(0,10)),field=gl(3,1) )
dat
Agropyron < with(dat, as.numeric(field) + as.numeric(NO3)+2) +rnorm(12)/2
Schizachyrium < with(dat, as.numeric(field)  as.numeric(NO3)+2) +rnorm(12)/2
total < Agropyron + Schizachyrium
dotplot(total ~ NO3, dat, jitter.x=TRUE, groups=field,
type=c('p','a'), xlab="NO3", auto.key=list(columns=3, lines=TRUE) )
Y < data.frame(Agropyron, Schizachyrium)
mod < metaMDS(Y)
plot(mod)
### Hulls show treatment
with(dat, ordihull(mod, group=NO3, show="0"))
with(dat, ordihull(mod, group=NO3, show="10", col=3))
### Spider shows fields
with(dat, ordispider(mod, group=field, lty=3, col="red"))
### Correct hypothesis test (with strata)
adonis(Y ~ NO3, data=dat, strata=dat$field, perm=999)
### Incorrect (no strata)
adonis(Y ~ NO3, data=dat, perm=999)

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