radfit {vegan} | R Documentation

## Rank – Abundance or Dominance / Diversity Models

### Description

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.

### Usage

    ## 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, ...)
    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, ...)
    as.rad(x)
    ## S3 method for class 'rad':
    plot(x, xlab = "Rank", ylab = "Abundance", ...)

### Arguments

`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.
`...` | Other parameters to functions.

### Details

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 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.

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.

### Value

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.

### Note

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`.

### Author(s)

Jari Oksanen

### References

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`.

### Examples

    data(BCI)
    mod <- rad.lognormal(BCI[1,])
    mod
    plot(mod)
    mod <- radfit(BCI[1,])
    plot(mod)
    # Take a subset of BCI to save time and nerves
    mod <- radfit(BCI[2:5,])
    mod
    plot(mod, pch=".")

[Package vegan version 1.8-8 Index]