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 pseudo-F ratios.

Usage

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adonis2(formula, data, permutations = 999, method = "bray",
    sqrt.dist = FALSE, add = FALSE, by = "terms",
    parallel = getOption("mc.cores"), ...)
adonis(formula, data, permutations = 999, method = "bray",
    strata = NULL, contr.unordered = "contr.sum",
    contr.ordered = "contr.poly", parallel = getOption("mc.cores"), ...)

Arguments

formula

Model formula. The LHS must be either a community data matrix or a dissimilarity matrix, e.g., from vegdist or dist. If the LHS is a data matrix, function vegdist will be used to find the dissimilarities. The RHS defines the independent variables. These can be continuous variables or factors, they can be transformed within the formula, and they can have interactions as in a typical formula. If a dissimilarity object is supplied, no species coefficients can be calculated in adonis (see Value below).

data

the data frame for the independent variables.

permutations

a list of control values for the permutations as returned by the function how, or the number of permutations required, or a permutation matrix where each row gives the permuted indices.

method

the name of any method used in vegdist to calculate pairwise distances if the left hand side of the formula was a data frame or a matrix.

sqrt.dist

Take square root of dissimilarities. This often euclidifies dissimilarities.

add

Add a constant to the non-diagonal dissimilarities such that all eigenvalues are non-negative in the underlying Principal Co-ordinates Analysis (see wcmdscale for details). Choice "lingoes" (or TRUE) use the recommended method of Legendre & Anderson (1999: “method 1”) and "cailliez" uses their “method 2”.

by

by = "terms" will assess significance for each term (sequentially from first to last), setting by = "margin" will assess the marginal effects of the terms (each marginal term analysed in a model with all other variables), and by = NULL will assess the overall significance of all terms together. The arguments is passed on to anova.cca.

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 parallel = 1 uses ordinary, non-parallel processing. The parallel processing is done with parallel package.

...

Other arguments passed to vegdist.

Details

adonis2 and adonis are functions for the analysis and partitioning sums of squares using dissimilarities. Function adonis is directly based on the algorithm of Anderson (2001) and performs a sequential test of terms. Function adonis2 is based on the principles of McArdle & Anderson (2001) and can perform sequential, marginal and overall tests. Function adonis2 also allows using additive constants or squareroot of dissimilarities to avoid negative eigenvalues. but both functions can handle semimetric indices (such as Bray-Curtis) that produce negative eigenvalues. Function adonis2 can be much slower than adonis, in particular with several terms. With the same random permutation, tests are identical in both functions, and the results are also identical to anova.cca of dbrda and capscale. With Euclidean distances, the tests are also identical to anova.cca of rda.

The functions partition sums of squares of a multivariate data set, and they are directly analogous to MANOVA (multivariate analysis of variance). McArdle and Anderson (2001) and Anderson (2001) refer to the method as “permutational manova” (formerly “nonparametric manova”). Further, as the inputs are linear predictors, and a response matrix of an arbitrary number of columns, they are a robust alternative to both parametric MANOVA and to ordination methods for describing how variation is attributed to different experimental treatments or uncontrolled covariates. Functions are also analogous to distance-based redundancy analysis in functions dbrda and capscale (Legendre and Anderson 1999). Functions provide 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.

Value

Function adonis2 returns an anova.cca result object. Function adonis returns an object of class "adonis" with following components:

aov.tab

Typical AOV table showing sources of variation, degrees of freedom, sequential sums of squares, mean squares, F statistics, partial R-squared 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 permustats and its support functions.

model.matrix

The model.matrix for the right hand side of the formula.

terms

The terms component of the model.

Note

Anderson (2001, Fig. 4) warns that the method may confound location and dispersion effects: significant differences may be caused by different within-group 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 (adonis) and Jari Oksanen (adonis2).

References

Anderson, M.J. 2001. A new method for non-parametric multivariate analysis of variance. Austral Ecology, 26: 32–46.

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. Distance-based 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 distance-based redundancy analysis. Ecology, 82: 290–297.

Warton, D.I., Wright, T.W., Wang, Y. 2012. Distance-based multivariate analyses confound location and dispersion effects. Methods in Ecology and Evolution, 3, 89–101.

See Also

mrpp, anosim, mantel, varpart.

Examples

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data(dune)
data(dune.env)
## default test by terms
adonis2(dune ~ Management*A1, data = dune.env)
## overall tests
adonis2(dune ~ Management*A1, data = dune.env, by = NULL)

### 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)
### Ellipsoid hulls show treatment
with(dat, ordiellipse(mod, field, kind = "ehull", label = TRUE))
### Spider shows fields
with(dat, ordispider(mod, field, lty=3, col="red"))

### Incorrect (no strata)
perm <- how(nperm = 199)
adonis2 (Y ~ NO3, data = dat, permutations = perm)

## Correct with strata
setBlocks(perm) <- with(dat, field)
adonis2(Y ~ NO3, data = dat, permutations = perm)

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