The four-parameter Weibull type 1 model

The general Weibull type 1 model for fitting dose-response data.

Usage

weibull1(
  fixed = c(NA, NA, NA, NA),
  names = c("b", "c", "d", "e"),
  method = c("1", "2", "3", "4"),
  ssfct = NULL,
  fctName,
  fctText
)

Arguments

fixed: numeric vector of length 4. Specifies which parameters are fixed and at what value. Use NA for parameters that are not fixed.
names: character vector of length 4 giving the names of the parameters b, c, d, and e.
method: character string indicating the self starter function to use for obtaining starting values ("1" (default), "2", "3", or "4"). See Details.
ssfct: a self starter function to be used. If NULL (default), the built-in self starter is used.
fctName: optional character string used internally for the function name.
fctText: optional character string used internally for the function text description.

Value

A list of class Weibull-1 containing the nonlinear function, self starter function, and parameter names.

Details

The four-parameter Weibull type 1 model is given by the expression $$f(x) = c + (d - c) \exp(-\exp(b(\log(x) - \log(e))))$$

The model is sometimes also called the Gompertz model.

The method argument determines how starting values for the parameters b and e are estimated (the starting values for c and d are always based on the range of the response values). Four methods are available:

"1" (default): Linear regression on transformed data. Applies a log-log transformation to the response and a log transformation to the dose, then fits a linear regression to estimate starting values for b and e.
"2": Anke's procedure. Estimates e by finding the dose at which the response crosses the midpoint between c and d, then estimates b as the median of back-calculated values.
"3": Stepwise approach. Identifies where the mean response crosses the midpoint between c and d and uses the corresponding dose as the starting value for e. The starting value for b is based on the sign of the slope at that point.
"4": Normolle's procedure. Uses the mean of the dose range as an initial estimate for e, then estimates b and e using median-based back-calculations.

References

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

Author

Christian Ritz