# dbrda: Principal Coordinates Analysis and [Partial] Distance-based... In vegan: Community Ecology Package

 dbrda R Documentation

## Principal Coordinates Analysis and [Partial] Distance-based Redundancy Analysis

### Description

Distance-based redundancy analysis (dbRDA) is an ordination method similar to Redundancy Analysis (`rda`), but it allows non-Euclidean dissimilarity indices, such as Manhattan or Bray-Curtis distance. Despite this non-Euclidean feature, the analysis is strictly linear and metric. If called with Euclidean distance, the results are identical to `rda`, but dbRDA will be less efficient. Functions `dbrda` is constrained versions of metric scaling, a.k.a. principal coordinates analysis, which are based on the Euclidean distance but can be used, and are more useful, with other dissimilarity measures. Function `capscale` is a simplified version based on Euclidean approximation of dissimilarities. The functions can also perform unconstrained principal coordinates analysis (PCO), optionally using extended dissimilarities. `pco()` is a wrapper to `dbrda()`, which performs PCO.

### Usage

``````dbrda(formula, data, distance = "euclidean", sqrt.dist = FALSE,
add = FALSE, dfun = vegdist, metaMDSdist = FALSE,
na.action = na.fail, subset = NULL, ...)
capscale(formula, data, distance = "euclidean", sqrt.dist = FALSE,
comm = NULL, add = FALSE,  dfun = vegdist, metaMDSdist = FALSE,
na.action = na.fail, subset = NULL, ...)
pco(X, ...)
``````

### 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` or function given in `dfun` will be used to find the dissimilarities. The RHS defines the constraints. The constraints 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 `rda` or `cca`. This allows the use of partial dbRDA. `X` Community data matrix. `data` Data frame containing the variables on the right hand side of the model formula. `distance` The name of the dissimilarity (or distance) index if the LHS of the `formula` is a data frame instead of dissimilarity matrix. `sqrt.dist` Take square roots of dissimilarities. See section `Details` below. `comm` Community data frame which will be used for finding species scores when the LHS of the `formula` was a dissimilarity matrix. This is not used if the LHS is a data frame. If this is not supplied, the “species scores” are unavailable when dissimilarities were supplied. N.B., this is only available in `capscale`: `dbrda` does not return species scores. Function `sppscores` can be used to add species scores if they are missing. `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). `"lingoes"` (or `TRUE`) uses the recommended method of Legendre & Anderson (1999: “method 1”) and `"cailliez"` uses their “method 2”. The latter is the only one in `cmdscale`. `dfun` Distance or dissimilarity function used. Any function returning standard `"dist"` and taking the index name as the first argument can be used. `metaMDSdist` Use `metaMDSdist` similarly as in `metaMDS`. This means automatic data transformation and using extended flexible shortest path dissimilarities (function `stepacross`) when there are many dissimilarities based on no shared species. `na.action` Handling of missing values in constraints or conditions. The default (`na.fail`) is to stop with missing values. Choices `na.omit` and `na.exclude` delete rows with missing values, but differ in representation of results. With `na.omit` only non-missing site scores are shown, but `na.exclude` gives `NA` for scores of missing observations. Unlike in `rda`, no WA scores are available for missing constraints or conditions. `subset` Subset of data rows. This can be a logical vector which is `TRUE` for kept observations, or a logical expression which can contain variables in the working environment, `data` or species names of the community data (if given in the formula or as `comm` argument). `...` Other parameters passed to underlying functions (e.g., `metaMDSdist`). For `pco()` argument are passed to `dbrda()`.

### Details

Functions `dbrda` and `capscale` provide two alternative implementations of dbRDA. Function `dbrda` is based on McArdle & Anderson (2001) and directly decomposes dissimilarities. With Euclidean distances results are identical to `rda`. Non-Euclidean dissimilarities may give negative eigenvalues associated with imaginary axes. Function `capscale` is based on Legendre & Anderson (1999): the dissimilarity data are first ordinated using metric scaling, and the ordination results are analysed as `rda`. `capscale` ignores the imaginary component and will not give negative eigenvalues (but will report the magnitude on imaginary component).

If the user supplied a community data frame instead of dissimilarities, the functions will find dissimilarities using `vegdist` or distance function given in `dfun` with specified `distance`. The functions will accept distance objects from `vegdist`, `dist`, or any other method producing compatible objects. The constraining variables can be continuous or factors or both, they can have interaction terms, or they can be transformed in the call. Moreover, there can be a special term `Condition` just like in `rda` and `cca` so that “partial” analysis can be performed.

Function `dbrda` does not return species scores, and they can also be missing in `capscale`, but they can be added after the analysis using function `sppscores`.

Non-Euclidean dissimilarities can produce negative eigenvalues (Legendre & Anderson 1999, McArdle & Anderson 2001). If there are negative eigenvalues, the printed output of `capscale` will add a column with sums of positive eigenvalues and an item of sum of negative eigenvalues, and `dbrda` will add a column giving the number of real dimensions with positive eigenvalues. If negative eigenvalues are disturbing, functions let you distort the dissimilarities so that only non-negative eigenvalues will be produced with argument `add = TRUE`. Alternatively, with `sqrt.dist = TRUE`, square roots of dissimilarities can be used which may help in avoiding negative eigenvalues (Legendre & Anderson 1999).

The functions can be also used to perform ordinary metric scaling a.k.a. principal coordinates analysis by using a formula with only a constant on the right hand side, or `comm ~ 1`. The new function `pco()` implements principal coordinates analysis via `dbrda()` directly, using this formula. With `metaMDSdist = TRUE`, the function can do automatic data standardization and use extended dissimilarities using function `stepacross` similarly as in non-metric multidimensional scaling with `metaMDS`.

### Value

The functions return an object of class `dbrda` or `capscale` which inherit from `rda`. See `cca.object` for description of the result object. Function `pco()` returns an object of class `"vegan_pco"` (which inherits from class `"dbrda"`) to avoid clashes with other packages.

### Note

Function `dbrda` implements real distance-based RDA and is preferred over `capscale`. `capscale` was originally developed as a variant of constrained analysis of proximities (Anderson & Willis 2003), but these developments made it more similar to dbRDA. However, it discards the imaginary dimensions with negative eigenvalues and ordination and significance tests area only based on real dimensions and positive eigenvalues. `capscale` may be removed from vegan in the future. It has been in `vegan` since 2003 (CRAN release 1.6-0) while `dbrda` was first released in 2016 (version 2.4-0), and removal of `capscale` may be disruptive to historical examples and scripts, but in modern times `dbrda` should be used.

The inertia is named after the dissimilarity index as defined in the dissimilarity data, or as `unknown distance` if such information is missing. If the largest original dissimilarity was larger than 4, `capscale` handles input similarly as `rda` and bases its analysis on variance instead of sum of squares. Keyword `mean` is added to the inertia in these cases, e.g. with Euclidean and Manhattan distances. Inertia is based on squared index, and keyword `squared` is added to the name of distance, unless data were square root transformed (argument `sqrt.dist=TRUE`). If an additive constant was used with argument `add`, `Lingoes` or `Cailliez adjusted` is added to the the name of inertia, and the value of the constant is printed.

Jari Oksanen

### References

Anderson, M.J. & Willis, T.J. (2003). Canonical analysis of principal coordinates: a useful method of constrained ordination for ecology. Ecology 84, 511–525.

Gower, J.C. (1985). Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra and its Applications 67, 81–97.

Legendre, P. & Anderson, M. J. (1999). Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecological Monographs 69, 1–24.

Legendre, P. & Legendre, L. (2012). Numerical Ecology. 3rd English Edition. Elsevier.

McArdle, B.H. & Anderson, M.J. (2001). Fitting multivariate models to community data: a comment on distance-based redundancy analysis. Ecology 82, 290–297.

`rda`, `cca`, `plot.cca`, `anova.cca`, `vegdist`, `dist`, `cmdscale`, `wcmdscale` for underlying and related functions. Function `sppscores` can add species scores or replace existing species scores.

The function returns similar result object as `rda` (see `cca.object`). This section for `rda` gives a more complete list of functions that can be used to access and analyse dbRDA results.

### Examples

``````data(varespec, varechem)
## dbrda
dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="bray")
## avoid negative eigenvalues with sqrt distances
dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="bray",
sqrt.dist = TRUE)
## avoid negative eigenvalues also with Jaccard distances
(m <- dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="jaccard"))
## add species scores
sppscores(m) <- wisconsin(varespec)
## pco
pco(varespec, dist = "bray", sqrt.dist = TRUE)
``````

vegan documentation built on Sept. 11, 2024, 7:57 p.m.