No-interaction Model for Hump-backed Species Richness vs. Biomass
humpfit fits a no-interaction model for species
richness vs. biomass data (Oksanen 1996). This is a null model that
produces a hump-backed response as an artifact of plant size and
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humpfit(mass, spno, family = poisson, start) ## S3 method for class 'humpfit' summary(object, ...) ## S3 method for class 'humpfit' predict(object, newdata = NULL, ...) ## S3 method for class 'humpfit' plot(x, xlab = "Biomass", ylab = "Species Richness", lwd = 2, l.col = "blue", p.col = 1, type = "b", ...) ## S3 method for class 'humpfit' points(x, ...) ## S3 method for class 'humpfit' lines(x, segments=101, ...) ## S3 method for class 'humpfit' profile(fitted, parm = 1:3, alpha = 0.01, maxsteps = 20, del = zmax/5, ...)
Vector of starting values for all three parameters.
Family of error distribution. Any
Result object of
Axis labels in
Line and point colour in
Number of segments used for fitted lines.
Parameters for profiling range and density.
Other parameters to functions.
The no-interaction model assumes that the humped species richness
pattern along biomass gradient is an artifact of plant size and
density (Oksanen 1996). For low-biomass sites, it assumes that plants
have a fixed size, and biomass increases with increasing number of
plants. When the sites becomes crowded, the number of plants and
species richness reaches the maximum. Higher biomass is reached by
increasing the plant size, and then the number of plants and species
richness will decrease. At biomasses below the hump, plant number and
biomass are linearly related, and above the hump, plant number is
proportional to inverse squared biomass. The number of plants is
related to the number of species by the relationship (
function) from Fisher's log-series (Fisher et al. 1943).
The parameters of the model are:
hump: the location of the hump on the biomass gradient.
scale: an arbitrary multiplier to translate the biomass into virtual number of plants.
alpha: Fisher's alpha to translate the virtual number of plants into number of species.
alpha are intermingled and this
function should not be used for estimating Fisher's
alpha. Probably the only meaningful and interesting
parameter is the location of the
Function may be very
difficult to fit and easily gets trapped into local solutions, or
fails with non-Poisson families, and function
profile should be
used to inspect the fitted models. If you have loaded
MASS, you can use functions
pairs.profile for graphical inspection of the
confint.profile.glm for the profile
based confidence intervals.
The original model intended to show that there is no need to speculate about ‘competition’ and ‘stress’ (Al-Mufti et al. 1977), but humped response can be produced as an artifact of using fixed plot size for varying plant sizes and densities.
function returns an object of class
"humpfit" inheriting from
"glm". The result object has specific
methods. In addition, it can be accessed by the following methods
type = "partial"),
fitted, and perhaps some others.
In addition, function
ellipse) can be used to draw approximate confidence ellipses
for pairs of parameters, if the normal assumptions look appropriate.
The function is a replacement for the original
at the archive of Journal of Ecology. There the function was
represented as a mixed
glm with one non-linear
hump) and a special one-parameter link function from
Fisher's log-series. The current function directly applies non-linear
maximum likelihood fitting using function
expected problems with the current approach are:
The function is discontinuous at
humpand may be difficult to optimize in some cases (the lines will always join, but the derivative jumps).
The function does not try very hard to find sensible starting values and can fail. The user may supply starting values in argument
startif fitting fails.
The estimation is unconstrained, but both
alphashould always be positive. Perhaps they should be fitted as logarithmic. Fitting
familymodels might become easier, too.
Al-Mufti, M.M., Sykes, C.L, Furness, S.B., Grime, J.P & Band, S.R. (1977) A quantitative analysis of shoot phenology and dominance in herbaceous vegetation. Journal of Ecology 65,759–791.
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 of an animal population. Journal of Animal Ecology 12, 42–58.
Oksanen, J. (1996) Is the humped relationship between species richness and biomass an artefact due to plot size? Journal of Ecology 84, 293–295.
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## ## Data approximated from Al-Mufti et al. (1977) ## mass <- c(140,230,310,310,400,510,610,670,860,900,1050,1160,1900,2480) spno <- c(1, 4, 3, 9, 18, 30, 20, 14, 3, 2, 3, 2, 5, 2) sol <- humpfit(mass, spno) summary(sol) # Almost infinite alpha... plot(sol) # confint is in MASS, and impicitly calls profile.humpfit. # Parameter 3 (alpha) is too extreme for profile and confint, and we # must use only "hump" and "scale". library(MASS) plot(profile(sol, parm=1:2)) confint(sol, parm=c(1,2))
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