drc (version 1.6-0)

cedergreen: The Cedergreen-Ritz-Streibig model

Description

'cedergreen' provides a very general way of specifying then Cedergreen-Ritz-Streibig modified log-logistic model for describing hormesis, under various constraints on the parameters. CRS.6 is the extension of link{cedergreen} with freely varying alpha parameter. For u-shaped hormesis data 'ucedergreen' provides a very general way of specifying the Cedergreen-Ritz-Streibig modified log-logistic model, under various constraints on the parameters.

Usage

cedergreen(fixed = c(NA, NA, NA, NA, NA), 
  names = c("b", "c", "d", "e", "f"), alpha)
  
  CRS.6(fixed = c(NA, NA, NA, NA, NA, NA), 
  names = c("b","c","d","e","f","g"))
  
  ucedergreen(fixed = c(NA, NA, NA, NA, NA), names = c("b", "c", "d", "e", "f"), alpha)

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').

alpha

numeric. The degree of hormesis. Needs to be specified!

Value

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

concept

hormesis hormetic effect initial stimulation u-shaped

Details

The model is given by the expression $$f(x) = c + \frac{d-c+f exp(-1/(x^{\alpha}))}{1+exp(b(log(x)-log(e)))}$$ which is a five-parameter model (alpha is fixed or freely varying). Not all features (eg EC/ED calculation) are available for the model with freely varying alpha. It is a modification of the four-parameter logistic curve to take hormesis into account. The u-shaped model is given by the expression $$f(x) = cd - \frac{d-c+f \exp(-1/x^{\alpha})}{1+\exp(b(\log(x)-\log(e)))}$$

References

Cedergreen, N. and Ritz, C. and Streibig, J. C. (2005) Improved empirical models describing hormesis, Environmental Toxicology and Chemistry 24, 3166--3172.

See Also

For fixed alpha, several special cases are handled by the following convenience functions CRS.4a, CRS.4b, CRS.4c, CRS.5a, CRS.5b, CRS.5c, UCRS.4a, UCRS.4b, UCRS.4c, UCRS.5a, UCRS.5b, UCRS.5c where a, b and c correspond to the pre-specified alpha values 1, 0.5 and 0.25, respectively.

Examples

## Estimating CRS model with alpha unknown
lettuce.crsm1 <- drm(weight~conc, data = lettuce, fct = CRS.6())
summary(lettuce.crsm1)
plot(lettuce.crsm1)  # oops: not increasing until hormesis peak