drc (version 1.5-2)

weibull1: The Weibull functions

Description

'weibull' and 'weibull2' provide a very general way of specifying Weibull dose response functions, under various constraints on the parameters.

Usage

weibull1(lowerc = c(-Inf, -Inf, -Inf, -Inf),
           upperc = c(Inf, Inf, Inf, Inf), 
           fixed = c(NA, NA, NA, NA), 
           names = c("b", "c", "d", "e"),
           scaleDose = TRUE, fctName, fctText)
 
  weibull2(lowerc = c(-Inf, -Inf, -Inf, -Inf),
           upperc = c(Inf, Inf, Inf, Inf), 
           fixed = c(NA, NA, NA, NA), 
           names = c("b", "c", "d", "e"),
           scaleDose = TRUE, fctName, fctText)

Arguments

lowerc

numeric vector. The lower bound on parameters. Default is minus infinity.

upperc

numeric vector. The upper bound on parameters. Default is plus infinity.

fixed

numeric vector. Specifies which parameters are fixed and at what value they are fixed. NAs for parameter that are not fixed.

names

a vector of character strings giving the names of the parameters (should not contain ":"). The default is reasonable (see under 'Usage'). The order of the parameters is: b, c, d, e (see under 'Details').

scaleDose

logical. If TRUE dose values are scaled around 1 during estimation; this is required for datasets where all dose values are small.

fctName

optional character string used internally by convenience functions.

fctText

optional character string used internally by convenience functions.

Value

• The value returned is a list containing the non-linear function, the self starter function and the parameter names.

Details

As pointed out in Seber and Wild (1989), there exist two different parameterisations of the Weibull model. They do not yield the same fitted curve for a given dataset (see under Examples). One four-parameter Weibull model ('weibull1') is $$f(x) = c + (d-c) \exp(-\exp(b(\log(x)-\log(e)))).$$ Another four-parameter Weibull model ('weibull2') is $$f(x) = c + (d-c) (1 - \exp(-\exp(b(\log(x)-\log(e))))).$$ Both four-parameter functions are asymmetric about the inflection point, that is the parameter $\exp(e)$.

References

Seber, G. A. F. and Wild, C. J (1989) Nonlinear Regression, New York: Wiley & Sons (pp. 338--339).

See Also

For convenience several special cases of the function 'weibull1' are available: W1.2, W1.3 and W1.4. Special cases of 'weibull2' are: W2.2, W2.3 and W2.4.

Examples

## Fitting two different Weibull models
ryegrass.m1 <- drm(ryegrass, fct = weibull1())
plot(ryegrass.m1, conLevel=0.5, legend=FALSE)
ryegrass.m2 <- drm(ryegrass, fct = weibull2())
plot(ryegrass.m2, conLevel=0.5, add = TRUE, type = "fit", col=2)
# you could also look at the ED values to see the difference

## A four-parameter Weibull model with b fixed at 1
ryegrass.m3 <- drm(ryegrass, fct = weibull1(fixed=c(1, NA, NA, NA)))
summary(ryegrass.m3)

## A four-parameter Weibull model with the constraint b>3
ryegrass.m4 <- drm(ryegrass, fct = weibull1(), lowerl = c(3, -Inf, -Inf, -Inf), 
control=drmc(constr=TRUE))
summary(ryegrass.m4)