radfit {vegan}R Documentation

Rank – Abundance or Dominance / Diversity Models


Functions construct rank – abundance or dominance / diversity or Whittaker plots and fit brokenstick, pre-emption, log-Normal, Zipf and Zipf-Mandelbrot models of species abundance.


## S3 method for class 'data.frame':
radfit(df, ...)
## S3 method for class 'radfit.frame':
plot(x, order.by, BIC = FALSE, model, legend = TRUE,
     as.table = TRUE, ...)
## Default S3 method:
radfit(x, ...)
## S3 method for class 'radfit':
plot(x, BIC = FALSE, legend = TRUE, ...)  
radlattice(x, BIC = FALSE, ...)
rad.null(x, family=poisson, ...)
rad.preempt(x, family = poisson, ...)
rad.lognormal(x, family = poisson, ...)
rad.zipf(x, family = poisson, ...)
rad.zipfbrot(x, family = poisson, ...)
## S3 method for class 'radline':
plot(x, xlab = "Rank", ylab = "Abundance", type = "b", ...)
## S3 method for class 'radline':
lines(x, ...)
## S3 method for class 'radline':
points(x, ...)
## S3 method for class 'rad':
plot(x, xlab = "Rank", ylab = "Abundance", log = "y", ...)


df Data frame where sites are rows and species are columns.
x A vector giving species abundances in a site, or an object to be plotted.
order.by A vector used for ordering sites in plots.
BIC Use Bayesian Information Criterion, BIC, instead of Akaike's AIC. The penalty for a parameter is k = log(S) where S is the number of species, whereas AIC uses k = 2.
model Show only the specified model. If missing, AIC is used to select the model. The model names (which can be abbreviated) are Preemption, Lognormal, Veiled.LN, Zipf, Mandelbrot.
legend Add legend of line colours.
as.table Arrange panels starting from upper left corner (passed to xyplot).
family Error distribution (passed to glm). All alternatives accepting link = "log" in family can be used, although not all make sense.
xlab,ylab Labels for x and y axes.
type Type of the plot, "b" for plotting both observed points and fitted lines, "p" for only points, "l" for only fitted lines, and "n" for only setting the frame.
log Use logarithmic scale for given axis. The default log =" y" gives the traditional plot in community ecology where the pre-emption model is a straight line, and with log = "xy" Zipf model is a straight line. With log = "" both axes are in the original arithmetic scale.
... Other parameters to functions.


Rank – Abundance Dominance (RAD) or Dominance/Diversity plots (Whittaker 1965) display logarithmic species abundances against species rank order. These plots are supposed to be effective in analysing types of abundance distributions in communities. These functions fit some of the most popular models mainly following Wilson (1991). Function as.rad constructs observed RAD data. Functions rad.XXXX (where XXXX is a name) fit the individual models, and function radfit fits all models. The argument of the function radfit can be either a vector for a single community or a data frame where each row represents a distinct community. All these functions have their own plot functions. When the argument is a data frame, plot uses Lattice graphics, and other plot functions use ordinary graphics. The ordinary graphics functions return invisibly an ordiplot object for observed points, and function identify.ordiplot can be used to label selected species. The most complete control of graphics can be achieved with rad.XXXX methods which have points and lines functions to add observed values and fitted models into existing graphs. Alternatively, radlattice uses Lattice graphics to display each radfit model in a separate panel together with their AIC or BIC values.

Function rad.null fits a brokenstick model where the expected abundance of species at rank r is a[r] = J/S sum(from x=r to S) 1/x (Pielou 1975), where J is the total number of individuals (site total) and S is the total number of species in the community. This gives a Null model where the individuals are randomly distributed among observed species, and there are no fitted parameters. Function rad.preempt fits the niche preemption model, a.k.a. geometric series or Motomura model, where the expected abundance a of species at rank r is a[r] = J*alpha*(1-alpha)^(r-1). The only estimated parameter is the preemption coefficient α which gives the decay rate of abundance per rank. The niche preemption model is a straight line in a RAD plot. Function rad.lognormal fits a log-Normal model which assumes that the logarithmic abundances are distributed Normally, or a[r] = exp(log(mu) + log(sigma) * N), where N is a Normal deviate. Function rad.zipf fits the Zipf model a[r] = J*p1*r^gamma where p1 is the fitted proportion of the most abundant species, and gamma is a decay coefficient. The Zipf – Mandelbrot model (rad.zipfbrot) adds one parameter: a[r] = J*c*(r+beta)^gamma after which p1 of the Zipf model changes into a meaningless scaling constant c. There are grand narratives about ecological mechanisms behind each model (Wilson 1991), but several alternative and contrasting mechanisms can produce similar models and a good fit does not imply a specific mechanism.

Log-Normal and Zipf models are generalized linear models (glm) with logarithmic link function. Zipf-Mandelbrot adds one nonlinear parameter to the Zipf model, and is fitted using nlm for the nonlinear parameter and estimating other parameters and log-Likelihood with glm. Pre-emption model is fitted as purely nonlinear model. There are no estimated parameters in the Null model. The default family is poisson which is appropriate only for genuine counts (integers), but other families that accept link = "log" can be used. Family Gamma may be appropriate for abundance data, such as cover. The ``best'' model is selected by AIC. Therefore ``quasi'' families such as quasipoisson cannot be used: they do not have AIC nor log-Likelihood needed in non-linear models.


Function rad.XXXX will return an object of class radline, which is constructed to resemble results of glm and has many (but not all) of its components, even when only nlm was used in fitting. At least the following glm methods can be applied to the result: fitted, residuals.glm with alternatives "deviance" (default), "pearson", "response", function coef, AIC, extractAIC, and deviance. Function radfit applied to a vector will return an object of class radfit with item y for the constructed RAD, item family for the error distribution, and item models containing each radline object as an item. In addition, there are special AIC, coef and fitted implementations for radfit results. When applied to a data frame radfit will return an object of class radfit.frame which is a list of radfit objects; function summary can be used to display the results for individual radfit objects. The functions are still preliminary, and the items in the radline objects may change.


The RAD models are usually fitted for proportions instead of original abundances. However, nothing in these models seems to require division of abundances by site totals, and original observations are used in these functions. If you wish to use proportions, you must standardize your data by site totals, e.g. with decostand and use appropriate family such as Gamma.

The lognormal model is fitted in a standard way, but I do think this is not quite correct – at least it is not equivalent to fitting Normal density to log abundances like originally suggested (Preston 1948).

Some models may fail. In particular, estimation of the Zipf-Mandelbrot model is difficult. If the fitting fails, NA is returned.

Wilson (1991) defined preemption model as a[r] = J*p1*(1 - alpha)^(r-1), where p1 is the fitted proportion of the first species. However, parameter p1 is completely defined by α since the fitted proportions must add to one, and therefore I handle preemption as a one-parameter model.

Veiled log-Normal model was included in earlier releases of this function, but it was removed because it was flawed: an implicit veil line also appears in the ordinary log-Normal. The latest release version with rad.veil was 1.6-10.


Jari Oksanen


Pielou, E.C. (1975) Ecological Diversity. Wiley & Sons.

Preston, F.W. (1948) The commonness and rarity of species. Ecology 29, 254–283.

Whittaker, R. H. (1965) Dominance and diversity in plant communities. Science 147, 250–260.

Wilson, J. B. (1991) Methods for fitting dominance/diversity curves. Journal of Vegetation Science 2, 35–46.

See Also

fisherfit and prestonfit. An alternative approach is to use qqnorm or qqplot with any distribution. For controlling graphics: Lattice, xyplot, lset.


mod <- rad.lognormal(BCI[1,])
mod <- radfit(BCI[1,])
## Standard plot overlaid for all models
## Pre-emption model is a line
## log for both axes: Zipf model is a line
plot(mod, log = "xy")
## Lattice graphics separately for each model
# Take a subset of BCI to save time and nerves
mod <- radfit(BCI[2:5,])
plot(mod, pch=".")

[Package vegan version 1.16-32 Index]