The four-parameter Weibull (type 2) model

Provides a general framework for the four-parameter Weibull type 2 model given by the equation $$f(x) = c + (d - c)(1 - \exp(-\exp(b(\log(x) - \log(e)))))$$

Usage

weibull2(
  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, specifying fixed parameters (use NA for parameters that should be estimated).
names: character vector of length 4 giving the names of the parameters (default c("b", "c", "d", "e")).
method: character string indicating the self starter method to use for obtaining starting values. One of "1" (default), "2", "3", or "4". See Details.
ssfct: a self starter function. If NULL (default), a built-in self starter is used based on method.
fctName: optional character string used internally for the function name.
fctText: optional character string used internally for the function description.

Value

A list containing the nonlinear function, self starter function, and parameter names. The list has class "Weibull-2".

Details

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 complementary 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