R: Rank - Abundance or Dominance / Diversity Models

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

    ## Default S3 method:
    radfit(x, ...)
    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'
    predict(object, newdata, total, ...)
    ## S3 method for class 'radfit'
    plot(x, BIC = FALSE, legend = TRUE, ...)
    ## S3 method for class 'radfit.frame'
    plot(x, order.by, BIC = FALSE, model, legend = TRUE,
         as.table = TRUE, ...)
    ## S3 method for class 'radline'
    plot(x, xlab = "Rank", ylab = "Abundance", type = "b", ...)
    radlattice(x, BIC = FALSE, ...)
    ## S3 method for class 'radfit'
    lines(x, ...)
    ## S3 method for class 'radfit'
    points(x, ...)
    as.rad(x)
    ## S3 method for class 'rad'
    plot(x, xlab = "Rank", ylab = "Abundance", log = "y", ...)

### Arguments

`x` | Data frame, matrix or a vector giving species abundances, or an object to be plotted.
`family` | Error distribution (passed to `glm`). All alternatives accepting `link = "log"` in `family` can be used, although not all make sense.
`object` | A fitted result object.
`newdata` | Ranks used for ordinations. All models can interpolate to non-integer “ranks” (although this may be approximate), but extrapolation may fail
`total` | The new total used for predicting abundance. Observed total count is used if this is omitted.
`order.by` | A vector used for ordering sites in plots.
`BIC` | Use Bayesian Information Criterion, BIC, instead of Akaike's AIC. The penalty in BIC 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 `Null`, `Preemption`, `Lognormal`, `Zipf`, `Mandelbrot`.
`legend` | Add legend of line colours.
`as.table` | Arrange panels starting from upper left corner (passed to `xyplot`).
`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 of 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.

### 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).

Functions `rad.null`, `rad.preempt`, `rad.lognormal`, `rad.zipf` and `zipfbrot` fit the individual models (described below) for a single vector (row of data frame), 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.

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 γ 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.

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`. Preemption model is fitted as a 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. Families `Gamma` or `gaussian` 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.

All these functions have their own `plot` functions. When `radfit` was applied for 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. Alternatively, `radlattice` uses `Lattice` graphics to display each `radfit` model of a single site in a separate panel together with their AIC or BIC values.

Function `as.rad` is a base function to construct ordered RAD data. Its `plot` is used by other RAD `plot` functions which pass extra arguments (such as `xlab` and `log`) to this function.

### Value

Functions `rad.null`, `rad.preempt`, `rad.lognormal`, `zipf` and `zipfbrot` fit each a single RAD model to a single site. The result object has class `"radline"` and inherits from `glm`, and can be handled by some (but not all) `glm` methods.

Function `radfit` fits all models either to a single site or to all rows of a data frame or a matrix. When fitted to a single site, the function returns an object of class `"radfit"` with items `y` (observed values), `family`, and `models` which is a list of fitted `"radline"` models. When applied for a data frame or matrix, `radfit` function returns an object of class `"radfit.frame"` which is a list of `"radfit"` objects, each item names by the corresponding row name.

All result objects (`"radline"`, `"radfit"`, `"radfit.frame"`) can be accessed with same method functions. The following methods are available: `AIC`, `coef`, `deviance`, `logLik`. In addition the fit results can be accessed with `fitted`, `predict` and `residuals` (inheriting from `residuals.glm`). The graphical functions were discussed above in Details.

### 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[5,])
    mod
    plot(mod)
    mod <- radfit(BCI[1,])
    ## Standard plot overlaid for all models
    ## Pre-emption model is a line
    plot(mod)
    ## log for both axes: Zipf model is a line
    plot(mod, log = "xy")
    ## Lattice graphics separately for each model
    radlattice(mod)
    # Take a subset of BCI to save time and nerves
    mod <- radfit(BCI[3:5,])
    mod
    plot(mod, pch=".")

[Package vegan version 2.4-3 Index]