fisherfit {vegan} | R Documentation

## Fit Fisher's Logseries and Preston's Lognormal Model to Abundance Data

### Description

Function `fisherfit` fits Fisher's logseries to abundance data. Function `prestonfit` groups species frequencies into doubling octave classes and fits Preston's lognormal model, and function `prestondistr` fits the truncated lognormal model without pooling the data into octaves.

### Usage

    fisherfit(x, ...)
    ## S3 method for class 'fisherfit':
    confint(object, parm, level = 0.95, ...)
    ## S3 method for class 'fisherfit':
    profile(fitted, alpha = 0.01, maxsteps = 20, del = zmax/5,
        ...)
    prestonfit(x, ...)
    prestondistr(x, truncate = -1, ...)
    ## S3 method for class 'prestonfit':
    plot(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue",
        line.col = "red", lwd = 2, ...)
    ## S3 method for class 'prestonfit':
    lines(x, line.col = "red", lwd = 2, ...)
    veiledspec(x, ...)
    as.fisher(x, ...)

### Arguments

`x` | Community data vector for fitting functions or their result object for `plot` functions.
`object, fitted` | Fitted model.
`parm` | Not used.
`level` | The confidence level required.
`alpha` | The extend of profiling as significance.
`maxsteps` | Maximum number of steps in profiling.
`del` | Step length.
`truncate` | Truncation point for log-Normal model, in log2 units. Default value -1 corresponds to the left border of zero Octave. The choice strongly influences the fitting results.
`xlab, ylab` | Labels for `x` and `y` axes.
`bar.col` | Colour of data bars.
`line.col` | Colour of fitted line.
`lwd` | Width of fitted line.
`...` | Other parameters passed to functions.

### Details

In Fisher's logarithmic series the expected number of species f with n observed individuals is f_n = α x^n / n (Fisher et al. 1943). The estimation follows Kempton & Taylor (1974) and uses function `nlm`. The estimation is possible only for genuine counts of individuals. The parameter α is used as a diversity index, and α and its standard error can be estimated with a separate function `fisher.alpha`. The parameter x is taken as a nuisance parameter which is not estimated separately but taken to be n/(n+α). Helper function `as.fisher` transforms abundance data into Fisher frequency table.

Function `fisherfit` estimates the standard error of alpha. However, the confidence limits cannot be directly estimated from the standard errors, but you should use function `confint` based on profile likelihood. Function `confint` uses function `confint.glm` of the MASS package, using `profile.fisherfit` for the profile likelihood. Function `profile.fisherfit` follows `profile.glm` and finds the tau parameter or signed square root of two times log-Likelihood profile. The profile can be inspected with a `plot` function which shows the tau and a dotted line corresponding to the Normal assumption: if standard errors can be directly used in Normal inference these two lines are similar.

Preston (1948) was not satisfied with Fisher's model which seemed to imply infinite species richness, and postulated that rare species is a diminishing class and most species are in the middle of frequency scale. This was achieved by collapsing higher frequency classes into wider and wider ``octaves'' of doubling class limits: 1, 2, 3–4, 5–8, 9–16 etc. occurrences. Any logseries data will look like lognormal when plotted this way. The expected frequency f at abundance octave o is defined by f = S0 exp(-(log2(o)-mu)^2/2/sigma^2), where μ is the location of the mode and σ the width, both in log2 scale, and S0 is the expected number of species at mode. The lognormal model is usually truncated on the left so that some rare species are not observed. Function `prestonfit` fits the truncated lognormal model as a second degree log-polynomial to the octave pooled data using Poisson error. Function `prestondistr` fits left-truncated Normal distribution to log2 transformed non-pooled observations with direct maximization of log-likelihood. Function `prestondistr` is modelled after function `fitdistr` which can be used for alternative distribution models. The functions have common `print`, `plot` and `lines` methods. The `lines` function adds the fitted curve to the octave range with line segments showing the location of the mode and the width (sd) of the response.

The total extrapolated richness from a fitted Preston model can be found with function `veiledspec`. The function accepts results both from `prestonfit` and from `prestondistr`. If `veiledspec` is called with a species count vector, it will internally use `prestonfit`. Function `specpool` provides alternative ways of estimating the number of unseen species. In fact, Preston's lognormal model seems to be truncated at both ends, and this may be the main reason why its result differ from lognormal models fitted in Rank–Abundance diagrams with functions `rad.lognormal`.

### Value

The function `prestonfit` returns an object with fitted `coefficients`, and with observed (`freq`) and fitted (`fitted`) frequencies, and a string describing the fitting `method`. Function `prestondistr` omits the entry `fitted`. The function `fisherfit` returns the result of `nlm`, where item `estimate` is α. The result object is amended with the following items:

`df.residuals` | Residual degrees of freedom.
`nuisance` | Parameter x.
`fisher` | Observed data from `as.fisher`.

### Note

It seems that Preston regarded frequencies 1, 2, 4, etc.. as ``tied'' between octaves. This means that only half of the species with frequency 1 were shown in the lowest octave, and the rest were transferred to the second octave. Half of the species from the second octave were transferred to the higher one as well, but this is usually not as large number of species. This practise makes data look more lognormal by reducing the usually high lowest octaves, but is too unfair to be followed. Therefore the octaves used in this function include the upper limit. If you do not accept this, you must change the function yourself. Williamson & Gaston (2005) discuss alternative definitions in detail, and they should be consulted for a critical review of log-Normal model.

### Author(s)

Bob O'Hara bob.ohara@helsinki.fi (`fisherfit`) and Jari Oksanen.

### References

Fisher, R.A., Corbet, A.S. & Williams, C.B. (1943). The relation between the number of species and the number of individuals in a random sample of animal population. Journal of Animal Ecology 12: 42–58.

Kempton, R.A. & Taylor, L.R. (1974). Log-series and log-normal parameters as diversity discriminators for Lepidoptera. Journal of Animal Ecology 43: 381–399.

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

Williamson, M. & Gaston, K.J. (2005). The lognormal distribution is not an appropriate null hypothesis for the species–abundance distribution. Journal of Animal Ecology 74, 409–422.

### See Also

`diversity`, `fisher.alpha`, `radfit`, `specpool`. Function `fitdistr` of MASS package was used as the model for `prestondistr`. Function `density` can be used for smoothed ``non-parametric'' estimation of responses, and `qqplot` is an alternative, traditional and more effective way of studying concordance of observed abundances to any distribution model.

### Examples

    data(BCI)
    mod <- fisherfit(BCI[5,])
    mod
    plot(profile(mod))
    confint(mod)
    # prestonfit seems to need large samples
    mod.oct <- prestonfit(colSums(BCI))
    mod.ll <- prestondistr(colSums(BCI))
    mod.oct
    mod.ll
    plot(mod.oct)
    lines(mod.ll, line.col="blue3") # Different
    ## Smoothed density
    den <- density(log2(colSums(BCI)))
    lines(den$x, ncol(BCI)*den$y, lwd=2) # Fairly similar to mod.oct
    ## Extrapolated richness
    veiledspec(mod.oct)
    veiledspec(mod.ll)

[Package vegan version 1.8-6 Index]