drc (version 0.9-0)

weibull: The Weibull function

Description

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

Usage

weibull(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, useDer = FALSE)
 
  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, useDer = FALSE)

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.

useDer

logical. If TRUE derivatives are supplied, otherwise they are not supplied. Not yet implemented!

Value

• The value returned by the 'weibull' is a list with the following components
• fct
  The dose response function.
• ssfct
  The self starter function.
• deriv1
  The first derivative.
• deriv2
  The second derivative.
• lowerc
  The lower bounds on the parameters.
• upperc
  The upper bounds on the parameters.
• edfct
  The function used to calculate ED values.
• sifct
  The function used to calculate SI values.

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 ('weibull') 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 'weibull' are available: w2, w3 and w4.

Examples

## Fitting two different Weibull models
model1 <- multdrc(FA, fct = weibull())
plot(model1)
model2 <- multdrc(FA, fct = weibull2())
plot(model2, type="add", col=2)

## A four-parameter Weibull model with b fixed at 1
model3 <- multdrc(FA, fct = weibull(fixed=c(1, NA, NA, NA)))
summary(model3)

## A four-parameter Weibull model with the constraint b>3
model4 <- multdrc(FA, fct = weibull(lowerc=c(3, -Inf, -Inf, -Inf)), 
control=mdControl(constr=TRUE))
summary(model4)

rm(model1, model2, model3, model4)