RAD
1
RAD – a FORTRAN program
for fitting species—rank order distributions
Version 2.1
Werner Ulrich
Nicolaus Copernicus University in Toruń
Chair of Ecology and Biogeography
Lwowska 1, 87-100 Toruń; Poland
e-mail: ulrichw @ umk.pl
Latest update: 08.06.2015
1. Introduction
Many SAD models have been developed to
Since its introduction to ecology by Raunkiær
mimic resource division and stochastic processes
(1909), species - abundance distributions (SADs) have
(Tokeshi 1999, McGill et al. 2007), although nearly all
been extensively studied (reviewed in McGill et al.
of them are now out of date. Interestingly, two of the
2007, Matthew and Whittaker 2014, 2015). They pro-
oldest models, the log-series (Fisher et al. 1943) and
vide an exhaustive description of the distribution of
the lognormal (Preston 1948) are still discussed and
abundances within an ecological community
the log-series received even a renaissance as two
(Magurran 2004, McGill et al. 2007, Morlon et al.
neutral models, the zero sum multinomial (Hubbell
2009, Matthews and Whittaker 2015) and have been
2001) and the maximum entropy approach (Harte
linked to differential resource use and competitive
2011) predict log-series. According to the multinomial
strength (Sugihara 1980, Tokeshi 1998, Pueyo, 2006),
neutral model the degree of dispersal should cause a
disturbance regimes (Gray and Mirza 1979), stochas-
gradient from lognormal to log-series type relative
tic processes (May 1975, Šizling et al. 2009), or spe-
abundance distributions (Hubbell 2001). The maxi-
cies-specific dispersal rates (Hubbell 2001, Zillio and
mum entropy based neutral model of Harte (2011)
Condit, 2007).
predicts the prevalence of log-series due to constraint
random assembly of local communities. Particularly
Relative abundance
1
the zero sum multinomial neutral model might serve
0.1
as an ecological standard (Rosindell et al. 2012).
0.01
0.001
0.0001
0
10
20
30
Species rank order
40
50
Figure 1. Species rank order—log abundance
(Whittaker) plots generated by RAD of the power
fraction model (yellow), the lognormal model (light
blue), the geometric series (dark blue), the log-series
(brown), and the power function (fractal) model
(grey).
2
RAD
The present RAD 2.1 software is the follow-up in a straight line when using a double log SAD plot.
of the older RAD software (Ulrich 2001) that fitted a They contain all models that are based on allometric
large number of relative abundance distributions. (power) functions (Frontier 1985, Moulliot et al. 2000,
RAD is focused on five models, the lognormal (Preston Pueyo 2006).
1948), the log-series (Fisher et al. 1943), the geomet-
While it is often possible to classify observed
ric (Motomura 1932), the power function (Frontier SADs into one of these three classes, it is frequently
1985, Mouillot et al. 2000), and the power fraction challenging or even impossible to discriminate within
(Tokeshi 1996). The latter represents a general niche a given class. Further, at low species richness it bedivision model.
comes increasingly difficult to discriminate between
RAD fits species rank order—log abundance SAD classes. Ulrich et al. (2010) advised not to fit asplots (often called Whittaker plots) only. It does not semblages of less than ten species.
deal with species—log abundance (Preston) distributions. Therefore RAD fits avoid possible biases intro-
Lognormal model
RAD generates lognormally distributed rank-
duces by grouping of species into abundance classes abundance distributions with S species from a stoand by the associated reduction of data.
chastic process with variance (shape parameter) z:
Reliable model identification needs fully censused assemblages. As most models differ by their
predictions of the relative abundances of the rarest
species, samples are often fitted equally by different
models. At low sample size model discrimination becomes increasingly challenging. Samples are most
often best fitted by log-series (Ulrich et al. 2010).
pi  ae z norm
where pi denotes the relative abundance of species i
and norm is a normally distributed random variate. a is
a normalizing constant that ensures that the sum of
all relative abundances adds to one.
Power fraction model
The relative abundance pi of species i is calcu-
2. Models
In general, we can subsume nearly all models
described fo far (McGill et al. 2007) into three classes.
One class of models results in SAD shapes similar to
simple exponential functions. The geometric series
(Motomura 1932), random assortment (Tokeshi 1990)
and dominance pre-emption (Tokeshi 1990) models,
and, as a descriptor of samples, the log-series (Fisher
et al. 1943) belong to this class (Fig.1). The second
class contains all models that generate S-shaped SAD
plots. To this class belong the lognormal (Preston
1948) and broken stick (McArthur 1957) models, and
the random (Tokeshi 1990), power (Tokeshi 1996),
MacArthur (Tokeshi 1990) and Sugihara (Sugihara
1980) fractions. The third class contains models result
lated iteratively by a two step process as developed in
Tokeshi (1996). First a species is chosen from an exponential probability distribution with parameter z (-∞ <
z < ∞) applied to the total abundance space. Next the
part of relative abundance occupied by this species is
dived into two parts using an equal random distribution. The power fraction is a very flexible model able
to mimic the whole range of S-shaped SAD distributions including the lognormal, the Sugihara fraction,
and the broken stick.
Geometric model
The geometric series is computed as
pi  az (1  z ) i
where pi denotes the relative abundance of species i.
The value of the parameter z may range between 0
RAD
3
Relative abundance
1
Si  z
Xi
i
with 0 < X < 1 and 0 < z < S.
Power function (fractal) model
0.1
In a stochastic version of the power function the relative abundance pi of species i is given by
0.01
pi  a ( lin S  1)  z
with lin being a linear random number and with the
shape parameter z > 0. a is again a normalizing con-
0.001
0
10
20
30
Species rank order
Figure 2. A typical log-series fit (black dots) of a hypothetical relative abundance distribution (blue
dots).
stant that ensures that the sum of all relative abundances adds to one.
3. Fitting algorithms
The fitting algorithm of RAD 2.1 is essentially
similar to the older RAD versions. Single parameter
and 1. a is a normalizing constant that ensures that
the sum of all relative abundances adds to one.
models (lognormal, power fraction, power function,
geometric) are fitted by an iterative encapsulation
Log-series model
process. Initially of each model five SADs are calculatIn the two parametric log-series the expected ed with parameters x equally spanning the whole
number of species Si with i individuals is given by
range of possible values (the maximum and minimum
possible values xmin and xmax, x1 = 0.25(xmax - xmin)
Figure 3. The SAD screen
4
RAD
Convergence is very fast
for all models and generally needs less than 20 iterations. Fitting very large
assemblages (> 500 speFigure 5. A batch file for
multiple analysis.
cies) may take considerable CPU time.
4. Program run
RAD first askes about the input file (Fig. 3).
Figure 4. A space delimited data file containing six
This should be a space delimited text file as shown in
communities to be analysed. Zero counts are eliminated prior to fitting.
Fig. 4. The first column has to contain species names,
the first line plot names. These names must not con+xmin, x2 = 0.5(xmax - xmin) +xmin, and x3 = 0.75(xmax xmin) +xmin. Of the three best fitting (see below) parameters two are retained as being the new xmin and xmax
This process is repeated until |xmax - xmin| < 0.001, a
convergence value that proved to be fully sufficient
tain spaces. Zero counts are eliminated prior to fitting.
Fitting is done with observed species only. After a
blank input (carriage return) the software expects the
name of a file containing multiple input files as shown
in Fig. 5.
for the present models.
The log-series is a two parameter model (z
and X). The parameter z is first iteratively estimated
as described above from
Next the software asks about the maximum
allowed abundance range (range = maximum/
minimum) for fitting the log-series. As RAD will create
a respective vector of size range, the value is limited
N

by the available working memory. The default value
S  z ln1  
z

is 50,000. If the observed community has a much largwith S being the total number of species and N the
total abundance. Next X is estimated accordingly. This
procedure proved to give robust results (Fig. 2), while
a simultaneous parameter encapsulation sometimes
Figure 6. Three methods for fitting (red: observed,
black: predicted), equivalent to the OLS (a), RMA (b),
and MA (c) approach.
1
be a count of individuals. In practise most abundance
data are given as density = counts / number of sample
units. RAD treats such data differentially. If all species
abundances of the input data are above or equal unity the total abundance is estimated as the sum of
these values. If the least abundance species has an
abundance below unity all values are first divided by
this value making the least abundant species to have
an abundance of one.
Relative abundance
failed. In the log-series the total abundance N should
(a)
(b)
0.1
(c)
0.01
0
1
2
3
4
5
6
Species rank order
7
8
RAD
5
Figure 7. The major output file of SAD.
er abundance range than predefined fits might be- where pi,obs, pi,exp denote the observed and fitted relacome increasingly worse as numbers of species with tive abundances of species i. Minimum values min
very low abundances will be underrepresented.
obtained with species j refer to the respective mini-
RAD proposes three target functions (Fig. 6) to mum when calculated over all species S.
assess goodness of fit. It uses similar approaches as in
linear regression. Ordinary least squares fits (OLS) use
d 
2
 (ln p
i , obs
 ln pi , exp )
2
RAD produces two output files. The first
S
(default = Output.txt) contains the summary statistics
S
reduced major axis (RMA) fits
sum of all species abundances (counts or densities). If
S
these are given as relative abundances RAD first di-
d 
2
S
and major axis (MA) fits
2
(Fig. 7). The total abundance (TotalA) refers to the
(  min[(ln pi ,obs  ln pi ,exp )  ( j  i ) ]
2
d2 
5. Output
(  min(ln pi ,obs  ln p j ,exp ) 2
vides all values by the value of the least abundant
species. Abundance difference (DiffA) provides the
S
S
Figure 8. The distribution file of SAD.
6
range of abundances (DiffA = most / least abundant
species). Curvature refers to the relative skewness of
the empirical distribution with respect to the skew-
RAD
7. System requirements
RAD is written in FORTRAN 95, has been
ness of the lognormal. In theory the lognormal has a compiled under a 64 bit architecture, and runs under
skewness of zero, but as the present lognormal is cal- Windows 8, 7, XP, and Vista. The present version is
culated from a stochastic process small differences only limited by the computer’s memory.
occur. Therefore
  (log pobs  log pobs )3  (log plnorm  log p ln lnorm )3 
 S

S
Curvature 
 S


3
3
( S  1)( S  2) 
 obs
 lnorm



In most cases curvature will be similar to the skewness of the observed SAD.
Next RAD prints the shape parameters z (and
8. Acknowledgements
Development of this program was support-
X in the case of the log-series) and the associated fits ed by a grant from the Polish Science Centre (grant
d2. The log-series parameter X is always below one 2014/13/B/NZ8/04681 ).
and nearly always above 0.9. In many cases it is very
close to one. In this case RAD prints a value of
9. References
1.0000000. Finally, RAD prints the Shannon diversity Connolly, S.R. and Dornelas, M. 2011. Fitting and emH and the respective evenness E = H/ln(S).
pirical evaluation of models for species abunThe second output file (Fig. 8) contains the
dance distributions. In: Biological Diversity:
empirical and the fitted relative abundance distribu-
Frontiers in Measurement and Assessment (eds
tions of the five models. The latter relative abundanc-
A.E. Magurran and B.J. McGill), pp. 123-140.
es are averages from 20 replicates using the best fit
Oxford University Press, Oxford.
parameter values. This number has proven to be suffi- Fisher, A.G., Corbet S.A. and Williams, S.A. 1943. The
cient for a smooth plot. In the case of the stochastic
relation between the number of species and the
models RAD prints for each species the standard devi-
number of individuals in a random sample of an
ation of the average abundance derived from 20 repli-
animal population. J. Anim. Ecol. 12: 42-58.
cates. Only the deterministic geometric model does Frontier, S. 1985. Diversity and structure in aquatic
not have a variance. In the case of serval plots
ecosystems. In: Oceanography and Marine Bio(columns) in the input file, RAD fits automatically the
logy—An Annual Review ( ed. M. Barnes). Aber-
marginal total distribution of all of these plots.
deen, pp. 253-312.
Gray, J.S. and Mirza, F.B. 1979. A possible method for
6. Citing RAD
RAD is freeware but nevertheless if you use
RAD in scientific work you should cite SAS as follows:
the detection of pollution-induced disturbance
on marine benthic communities. Marine Poll.
Bull. 10: 142-146.
Ulrich W. 2015. RAD – a FORTRAN program for fitting Harte, J. 2011. Maximum entropy and ecology. Oxford, Univ. Press.
species—rank order distributions. Version 2.1.
www.keib.umk.pl.
Hubbell, S.P. 2001. The unified theory of biogeogra-
RAD
7
phy and biodiversity. Princeton (Univ. Press).
Magurran, A.E. 2004. Measuring biological diversity.
Blackwell, Oxford.
Pueyo, S. 2006. Self-similarity in species–area relationship and in species–abundance distribution.
Oikos 112: 156–162.
MacArthur, R.H. 1957. On the relative abundance of
Raunkiær, C. 1909. Livsformen hos Planter paa ny
bird species. Proc. Natl. Acad. Science USA 43:
Jord. Kongelige Danske Videnskabernes Sels-
293-294.
kabs Skrifter. Naturvidenskabelig og Mathema-
Matthews, T.J. and Whittaker, R.J. 2014. Fitting and
comparing competing models of the species
tisk Afdeling 7: 1-70.
Rosindell, J., Hubbell, S.P., He, F., Harmon, L.J. and
abundance distribution: assessment and pro-
Etienne, R.S. 2012. The case for ecological neu-
spect. Frontiers Biogeogr. 6: 67–82.
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Matthews, T.J. and Whittaker, R.J. 2015. On the spe-
Šizling, A.L., Storch, D., Šizlingova, E., Reif, R. and Ga-
cies abundance distribution in applied ecology
ston, K. 2009. Species abundance distribution
and biodiversity management. J. Appl. Ecol. 52:
results from a spatial analogy of central limit
443-454.
theorem. Proc. Natl. Acad. Sci. USA 106: 6691–
May, R.M. 1975. Patterns of species abundance and
6695.
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ties (eds. Cody, M. L. and Diamond, J.M.) Cam-
explanation of species–abundance patterns. Am.
bridge University Press, pp. 81–120.
Nat. 116: 770–787.
McGill, B.J. et al. 2007. Species abundance distribu-
Tokeshi, M. 1990. Niche apportionment or random
tions: moving beyond single prediction theories
assortment: species abundance patterns revisi-
to integration within an eco-logical framework.
ted. J. Animal Ecol. 59: 1129-1146.
Ecol. Lett. 10: 995–1015.
Morlon, H. et al. 2009. Taking species abundance distributions beyond individuals. Ecol. Lett. 12:
488-501.
Motomura, I. 1932. On the statistical treatment of
communities. Zool. Mag. Tokyo 44: 379-383 (in
Japanese).
Moulliot, D., Lepretre, A., Andrei-Ruiz, M.-C. and
Tokeshi, M. 1996. Power fraction: a new explanation
of relative abundance patterns in species-rich
assemblages. Oikos 75: 543-550.
Tokeshi, M. 1998. Species coexistence. Blackwell, Oxord.
Ulrich, W. 2001. RAD—a Fortran program for fitting
relative abundance distributions.
www.keib.umk.pl.
Viale, D. 2000. The fractal model: an new model Ulrich, W., Ollik, M. and Ugland, K.I. 2010. A metato describe the species accumulation process
analysis of species – abundance distributions.
and relative abundance distribution (RAD). Oikos
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90: 333-342.
Preston, F.W. 1948. The commonness and rarity of
species. Ecology 29: 254–283.
Zillio, T. and Condit, R. 2007. The impact of neutrality,
niche differentiation and species input on diversity and abundance distributions. Oikos 116: 931
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