statforbiology (version 0.9.9)

SSbeta: Beta equation

Description

These functions provide the beta equation, a threshold model that was derived from the beta density function and it was adapted to describe phenomena taking place only within a minimum and a maximum threshold value (threshold model), for example to describe the germination rate (GR, i.e. the inverse of germination time) as a function of temperature. These functions provide the beta equation (beta.fun), the self-starters for the nls function (NLS.beta) and the self-starters for the drm function in the drc package (DRC.beta)

Usage

beta.fun(X, b, d, Xb, Xo, Xc)
NLS.beta(X, b, d, Xb, Xo, Xc)
DRC.beta()

Value

beta.fun, NLS.beta return a numeric value, while DRC.beta returns a list containing the nonlinear function and the self starter function

Arguments

X

a numeric vector of values at which to evaluate the model

b

model parameter

d

model parameter

Xb

model parameter (base threshold level)

Xo

model parameter (optimal threshold level)

Xc

model parameter (ceiling threshold level)

Author

Andrea Onofri

Details

This equation is parameterised as:

$$ f(x) = max\left( d \left\{ \left( \frac{X - Xb}{Xo - Xb} \right) \left( \frac{Xc - X}{Xc - Xo} \right) ^{\frac{Xc - Xo}{Xo - Xb}} \right\}^b , 0 \right)$$

It depicts a curve that is equal to 0 for X < Xb, grows up to a maximum, that is attained at X = Xo and decreases down to 0, that is attained at X = Xc and mantained for X > Xc.

References

Ratkowsky, DA (1990) Handbook of nonlinear regression models. New York (USA): Marcel Dekker Inc.

Onofri, A. (2020). A collection of self-starters for nonlinear regression in R. See: https://www.statforbiology.com/2020/stat_nls_usefulfunctions/

Examples

X <- c(1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50)
Y <- c(0, 0, 0, 7.7, 12.3, 19.7, 22.4, 20.3, 6.6, 0, 0)

model <- nls(Y ~ NLS.beta(X, b, d, Xb, Xo, Xc))
summary(model)
modelb <- drm(Y ~ X, fct = DRC.beta())
summary(modelb)
plot(modelb, log = "")