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, ..., permustat = c("pseudo-F", "modified-F"),
    groups = NULL)

Arguments

formula: Model formula. The function can be called only with the formula interface. Most usual features of formula hold, especially as defined in cca and rda. 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. The RHS can have a special term Condition that defines variables to be “partialled out” before constraints, just like in dbrda or cca. This allows the use of partial `adonis2`.
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.
permustat: The permutation statistic. The default "pseudo-F" retains the ordinary pseudo-\(F\) test. "modified-F" compares groups while allowing their within-group variation (dispersion) to differ. The argument name must be supplied in full.
groups: A factor identifying the groups to compare when permustat = "modified-F". For a formula containing a single factor, that factor is used automatically. For a model with several factors and all their interactions, supply a factor identifying each combination of factor levels, for example interaction(A, B) for the model Y ~ A * B. See Details for the supported models.
...: 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.

With permustat = "modified-F", the modified pseudo-\(F\) statistic of Anderson et al. (2017) compares all groups in a one-way analysis, allowing their within-group variation to differ. A model containing several factors and all their interactions can also be used for an overall test, treating each combination of factor levels as a group. For example, Y ~ A * B with groups = interaction(A, B) gives the same overall test as a one-way analysis using the combined grouping factor. Supply this factor in groups; it does not need to be added to the model formula. All factors are treated as fixed. Separate tests of individual factors or interactions are not currently supported.

There must be at least two groups and at least two observations in every group. The modified test does not support continuous predictors, models that omit interactions between factors, Condition terms, by tests, or model = "full" permutations. Tests for mixed or random-factor models would require further development. The modified test is not available for cca or capscale. Chi-square dissimilarities can instead be supplied explicitly to adonis2 or dbrda when those dissimilarities are appropriate for the analysis.

The modified test uses the same between-group and within-group sums of squares as the ordinary pseudo-\(F\) test. Instead of pooling within-group variation across all observations, it estimates variation separately for each group and combines these estimates using weights based on group sizes. The test statistic is recalculated for every permutation.

The residual degrees of freedom reported for the modified test use the Brown–Forsythe/Satterthwaite approximation and may be non-integer. The residual and total sums of squares, and hence \(R^2\), retain their usual definitions.

For a one-way model with a single response variable (and Euclidean distances for distance-based analyses), permustat = "modified-F" reduces to the Brown–Forsythe statistic for testing equality of group means under unequal variances. This differs from Welch's test used by oneway.test with var.equal = FALSE. With two groups, the modified statistic equals the square of Welch's \(t\) statistic from t.test, and its effective denominator degrees of freedom agree with those of Welch's test. With three or more groups, the statistics and denominator degrees of freedom generally differ because the Brown–Forsythe and Welch methods use different variance weighting and degrees-of-freedom approximations. The univariate Brown–Forsythe statistic and denominator degrees of freedom can be compared with onewaytests::bf.test(), described by Dag et al. (2018).

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.

Anderson, M.J., Walsh, D.C.I., Clarke, K.R., Gorley, R.N. and Guerra- Castro, E. 2017. Some solutions to the multivariate Behrens–Fisher problem for dissimilarity-based analyses. Australian & New Zealand Journal of Statistics, 59: 57–79.

Dag, O., Dolgun, A. and Konar, N.M. (2018). onewaytests: An R Package for One-Way Tests in Independent Groups Designs. The R Journal, 10(1): 175–199. https://journal.r-project.org/articles/RJ-2018-022/.

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  0.004 **
#> 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  0.002 **
#> A1             1   0.4409 0.10256 2.9387  0.019 * 
#> Management:A1  3   0.5892 0.13705 1.3090  0.225   
#> Residual      12   1.8004 0.41878                 
#> Total         19   4.2990 1.00000                 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## Published omnibus test allowing heterogeneous dispersions
data("norwegianShelf")
take23 <- norwegianShelf$area %in% c("2", "3")
shelf23 <- 1L * (norwegianShelf$comm[take23, , drop = FALSE] > 0L)
area23 <- droplevels(norwegianShelf$area[take23])
shelf.perms <- rbind(rev(seq_len(nrow(shelf23))),
                     c(2:nrow(shelf23), 1))
adonis2(shelf23 ~ area23, method = "jaccard", binary = TRUE,
        permutations = shelf.perms, permustat = "modified-F")
#> Permutation test for adonis under reduced model
#> Modified pseudo-F allowing heterogeneous dispersions
#> Permutation: supplied matrix
#> Number of permutations: 2
#> 
#> adonis2(formula = shelf23 ~ area23, permutations = shelf.perms, method = "jaccard", binary = TRUE, permustat = "modified-F")
#>              Df SumOfSqs      R2      F Pr(>F)
#> Model     1.000   1.7613 0.12919 6.8388 0.3333
#> Residual 43.659  11.8717 0.87081              
#> Total    45.000  13.6330 1.00000              

### 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
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  0.02565 0.13671 1.5835  0.235
#> Residual 10  0.16198 0.86329              
#> Total    11  0.18763 1.00000              
## 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  0.02565 0.13671 1.5835  0.015 *
#> Residual 10  0.16198 0.86329                
#> Total    11  0.18763 1.00000                
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1