envfit {vegan}R Documentation

Fits an Environmental Vector or Factor onto an Ordination


The function fits environmental vectors or factors onto an ordination. The projections of points onto vectors have maximum correlation with corresponding environmental variables, and the factors show the averages of factor levels.


## Default S3 method:
envfit(ord, env, permutations = 0, strata, choices=c(1,2), 
   display = "sites", w  = weights(ord), na.rm = FALSE, ...)
## S3 method for class 'formula':
envfit(formula, data, ...)
## S3 method for class 'envfit':
plot(x, choices = c(1,2), arrow.mul, at = c(0,0), axis = FALSE, 
    p.max = NULL, col = "blue", add = TRUE, ...)
## S3 method for class 'envfit':
scores(x, display, choices, ...)
vectorfit(X, P, permutations = 0, strata, w, ...)
factorfit(X, P, permutations = 0, strata, w, ...)


ord An ordination object or other structure from which the ordination scores can be extracted (including a data frame or matrix of scores)
env Data frame, matrix or vector of environmental variables. The variables can be of mixed type (factors, continuous variables) in data frames.
X Matrix or data frame of ordination scores.
P Data frame, matrix or vector of environmental variable(s). These must be continuous for vectorfit and factors or characters for factorfit.
permutations Number of permutations for assessing significance of vectors or factors.
formula, data Model formula and data.
na.rm Remove points with missing values in ordination scores or environmental variables
x A result object from envfit.
choices Axes to plotted.
arrow.mul Multiplier for vector lengths. The arrows are automatically scaled similarly as in plot.cca if this is not given and add = TRUE.
at The origin of fitted arrows in the plot. If you plot arrows in other places then origin, you probably have to specify arrrow.mul.
axis Plot axis showing the scaling of fitted arrows.
p.max Maximum estimated P value for displayed variables. You must calculate P values with setting permutations to use this option.
col Colour in plotting.
add Results added to an existing ordination plot.
strata An integer vector or factor specifying the strata for permutation. If supplied, observations are permuted only within the specified strata.
display In fitting functions these are ordinary site scores or linear combination scores ("lc") in constrained ordination (cca, rda, capscale). In scores function they are either "vectors" or "factors" (with synonyms "bp" or "cn", resp.).
w Weights used in fitting (concerns mainly cca and decorana results which have nonconstant weights).
... Parameters passed to scores.


Function envfit finds vectors or factor averages of environmental variables. Function plot.envfit adds these in an ordination diagram. If X is a data.frame, envfit uses factorfit for factor variables and vectorfit for other variables. If X is a matrix or a vector, envfit uses only vectorfit. Alternatively, the model can be defined a simplified model formula, where the left hand side must be an ordination result object or a matrix of ordination scores, and right hand side lists the environmental variables. The formula interface can be used for easier selection and/or transformation of environmental variables. Only the main effects will be analysed even if interaction terms were defined in the formula.

Functions vectorfit and factorfit can be called directly. Function vectorfit finds directions in the ordination space towards which the environmental vectors change most rapidly and to which they have maximal correlations with the ordination configuration. Function factorfit finds averages of ordination scores for factor levels. Function factorfit treats ordered and unordered factors similarly.

If permutations > 0, the `significance' of fitted vectors or factors is assessed using permutation of environmental variables. The goodness of fit statistic is squared correlation coefficient (r^2). For factors this is defined as r^2 = 1 - ss_w/ss_t, where ss_w and ss_t are within-group and total sums of squares.

User can supply a vector of prior weights w. If the ordination object has weights, these will be used. In practise this means that the row totals are used as weights with cca or decorana results. If you do not like this, but want to give equal weights to all sites, you should set w = NULL. The weighted fitting gives similar results to biplot arrows and class centroids in cca. For complete similarity between fitted vectors and biplot arrows, you should set display = "lc" (and possibly scaling = 2).

The lengths of arrows for fitted vectors are automatically adjusted for the physical size of the plot, and the arrow lengths cannot be compared across plots. For similar scaling of arrows, you must explicitly set the arrow.mul argument in the plot command.

The results can be accessed with scores.envfit function which returns either the fitted vectors scaled by correlation coefficient or the centroids of the fitted environmental variables.


Functions vectorfit and factorfit return lists of classes vectorfit and factorfit which have a print method. The result object have the following items:

arrows Arrow endpoints from vectorfit. The arrows are scaled to unit length.
centroids Class centroids from factorfit.
r Goodness of fit statistic: Squared correlation coefficient
permutations Number of permutations.
pvals Empirical P-values for each variable.

Function envfit returns a list of class envfit with results of vectorfit and envfit as items.
Function plot.envfit scales the vectors by correlation.


Fitted vectors have become the method of choice in displaying environmental variables in ordination. Indeed, they are the optimal way of presenting environmental variables in Constrained Correspondence Analysis cca, since there they are the linear constraints. In unconstrained ordination the relation between external variables and ordination configuration may be less linear, and therefore other methods than arrows may be more useful. The simplest is to adjust the plotting symbol sizes (cex, symbols) by environmental variables. Fancier methods involve smoothing and regression methods that abound in R, and ordisurf provides a wrapper for some.


Jari Oksanen. The permutation test derives from the code suggested by Michael Scroggie.

See Also

A better alternative to vectors may be ordisurf.


ord <- metaMDS(varespec)
(fit <- envfit(ord, varechem, perm = 999))
scores(fit, "vectors")
plot(fit, p.max = 0.05, col = "red")
## Adding fitted arrows to CCA. We use "lc" scores, and hope
## that arrows are scaled similarly in cca and envfit plots
ord <- cca(varespec ~ Al + P + K, varechem)
plot(ord, type="p")
fit <- envfit(ord, varechem, perm = 999, display = "lc")
plot(fit, p.max = 0.05, col = "red")
## Class variables, formula interface, and displaying the
## inter-class variability with `ordispider'
ord <- cca(dune)
fit <- envfit(ord ~ Moisture + A1, dune.env)
plot(ord, type = "n")
ordispider(ord, Moisture, col="skyblue")
points(ord, display = "sites", col = as.numeric(Moisture), pch=16)
plot(fit, cex=1.2, axis=TRUE)

[Package vegan version 1.16-32 Index]