Permutational Multivariate Analysis of Variance Using Distance Matrices

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

adonis2(formula, data, permutations = 999, method = "bray",
    sqrt.dist = FALSE, add = FALSE, by = NULL,
    parallel = getOption("mc.cores"), na.action = na.fail,
    strata = NULL, ...)

Arguments

formula: Model formula. The left-hand side (LHS) of the formula 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 right-hand side (RHS) of the formula 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.
data: the data frame for the independent variables, with rows in the same order as the community data matrix or dissimilarity matrix named on the LHS of formula.
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 = NULL will assess the overall significance of all terms together, 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), by = "onedf" will analyse one-degree-of-freedom contrasts sequentially. The argument is passed on to anova.cca.
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.
na.action: Handling of missing values on the right-hand-side of the formula (see na.fail for explanation and alternatives). Missing values are not allowed on the left-hand-side. NB, argument subset is not implemented.
strata: Groups within which to constrain permutations. The traditional non-movable strata are set as Blocks in the permute package, but some more flexible alternatives may be more appropriate.
...: Other arguments passed to vegdist.

Details

adonis2 is a function for the analysis and partitioning sums of squares using dissimilarities. The function is based on the principles of McArdle & Anderson (2001) and can perform sequential, marginal and overall tests. The function also allows using additive constants or squareroot of dissimilarities to avoid negative eigenvalues, but can also handle semimetric indices (such as Bray-Curtis) that produce negative eigenvalues. The adonis2 tests are identical to anova.cca of dbrda. With Euclidean distances, the tests are also identical to anova.cca of rda.

The function partitions 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. The method is also analogous to distance-based redundancy analysis and algorithmically similar to dbrda (Legendre and Anderson 1999), and provides 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

The function returns an anova.cca result object with a new column for partial \(R^2\): This is the proportion of sum of squares from the total, and in marginal models (by = "margin") the \(R^2\) terms do not add up to 1.

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 adonis2 is less sensitive to dispersion effects than some of its alternatives (anosim, mrpp). Function betadisper is a sister function to adonis2 to study the differences in dispersion within the same geometric framework.

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.

Author

Martin Henry H. Stevens and Jari Oksanen.

Examples

data(dune)
data(dune.env)
## default is overall (omnibus) test
adonis2(dune ~ Management*A1, data = dune.env)
#> Permutation test for adonis under reduced model
#> Permutation: free
#> Number of permutations: 999
#> 
#> adonis2(formula = dune ~ Management * A1, data = dune.env)
#>          Df SumOfSqs      R2      F Pr(>F)   
#> Model     7   2.4987 0.58122 2.3792  0.002 **
#> Residual 12   1.8004 0.41878                 
#> Total    19   4.2990 1.00000                 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## sequential tests
adonis2(dune ~ Management*A1, data = dune.env, by = "terms")
#> Permutation test for adonis under reduced model
#> Terms added sequentially (first to last)
#> Permutation: free
#> Number of permutations: 999
#> 
#> adonis2(formula = dune ~ Management * A1, data = dune.env, by = "terms")
#>               Df SumOfSqs      R2      F Pr(>F)    
#> Management     3   1.4686 0.34161 3.2629  0.001 ***
#> A1             1   0.4409 0.10256 2.9387  0.020 *  
#> Management:A1  3   0.5892 0.13705 1.3090  0.214    
#> Residual      12   1.8004 0.41878                  
#> Total         19   4.2990 1.00000                  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

### 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
#>    rep NO3 field
#> 1    1   0     1
#> 2    2   0     1
#> 3    1  10     1
#> 4    2  10     1
#> 5    1   0     2
#> 6    2   0     2
#> 7    1  10     2
#> 8    2  10     2
#> 9    1   0     3
#> 10   2   0     3
#> 11   1  10     3
#> 12   2  10     3
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, trace = FALSE)
plot(mod)
### Ellipsoid hulls show treatment
with(dat, ordiellipse(mod, NO3, kind = "ehull", label = TRUE))
### Spider shows fields
with(dat, ordispider(mod, field, lty=3, col="red", label = TRUE))


### Incorrect (no strata)
adonis2(Y ~ NO3, data = dat, permutations = 199)
#> Permutation test for adonis under reduced model
#> Permutation: free
#> Number of permutations: 199
#> 
#> adonis2(formula = Y ~ NO3, data = dat, permutations = 199)
#>          Df SumOfSqs      R2      F Pr(>F)  
#> Model     1 0.036688 0.24632 3.2682   0.06 .
#> Residual 10 0.112256 0.75368                
#> Total    11 0.148944 1.00000                
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## Correct with strata
with(dat, adonis2(Y ~ NO3, data = dat, permutations = 199, strata = field))
#> Permutation test for adonis under reduced model
#> Blocks:  strata 
#> Permutation: free
#> Number of permutations: 199
#> 
#> adonis2(formula = Y ~ NO3, data = dat, permutations = 199, strata = field)
#>          Df SumOfSqs      R2      F Pr(>F)   
#> Model     1 0.036688 0.24632 3.2682  0.005 **
#> Residual 10 0.112256 0.75368                 
#> Total    11 0.148944 1.00000                 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1