llogistic: The log-logistic function

Description

'llogistic' provides a very general way of specifying log-logistic models, under various constraints on the parameters.

Usage

llogistic(fixed = c(NA, NA, NA, NA, NA), 
  names = c("b", "c", "d", "e", "f"),
  ss = c("1", "2", "3"), ssfct = NULL,
  fctName, fctText)

  llogistic2(fixed = c(NA, NA, NA, NA, NA), 
  names = c("b", "c", "d", "e", "f"),
  ss = c("1", "2", "3"), ssfct = NULL,
  fctName, fctText)

Arguments

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, f (see under 'Details').

character string indicating the self starter function to use.

ssfct

a self starter function to be used.

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 nonlinear function, the self starter function and the parameter names.

concept

Hill

Details

The default arguments yields the five-parameter log-logistic function given by the expression

f (x) = c + \frac{d - c}{(1 + \exp (b (\log (x) - \log (e))))^{f}}

If the parameter $f$ differs from 1 then the function is asymmetric; otherwise it is symmetric (on log scale). This function is fitted using llogistic. The log-logistic function with log(e) rather than e as a parameter, that is using the parameterisation

f (x) = c + \frac{d - c}{(1 + \exp (b (\log (x) - e)))^{f}}

is fitted using llogistic2. Sometimes the log-logistic models are also called Hill models.

References

Finney, D. J. (1979) Bioassay and the Practise of Statistical Inference, Int. Statist. Rev., 47, 1--12. Seber, G. A. F. and Wild, C. J. (1989) Nonlinear Regression, New York: Wiley & Sons (p. 330).

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