drc (version 1.5-2)

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(lowerc = c(-Inf, -Inf, -Inf, -Inf, -Inf), 
  upperc = c(Inf, Inf, Inf, Inf, Inf), fixed = c(NA, NA, NA, NA, NA), 
  names = c("b", "c", "d", "e", "f"), alpha, scaleDose = TRUE)
  
  CRS.6(lowerc=c(-Inf, -Inf, -Inf, -Inf, -Inf, -Inf), 
  upperc=c(Inf, Inf, Inf, Inf, Inf, Inf), 
  fixed=c(NA, NA, NA, NA, NA, NA), 
  names=c("b","c","d","e","f","g"))
  
  ucedergreen(lowerc = c(-Inf, -Inf, -Inf, -Inf, -Inf), 
  upperc = c(Inf, Inf, Inf, Inf, Inf), fixed = c(NA, NA, NA, NA, NA), 
  names = c("b", "c", "d", "e", "f"), alpha, scaleDose = TRUE)

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, f (see under 'Details').

alpha

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

scaleDose

logical. If TRUE dose values are scaled around 1 during estimation; this is required for datasets where all dose values are small.

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

Special cases are CRS.4a, CRS.4a, UCRS.5a and UCRS.5a where a,b and c coresspond to the pre-specified alpha values 1, 0.5 and 0.25, respectively.

Examples

## Modified logistic model with the constraint f>0
lettuce.m1 <- multdrc(weight~conc,data=lettuce, 
fct=cedergreen(fixed=c(NA, NA, NA, NA, NA), 
lowerc=c(-Inf, -Inf, -Inf, -Inf, 0), alpha=1), 
control=mdControl(constr=TRUE))

summary(lettuce.m1)
ED(lettuce.m1, c(10, 50, 90))


## Estimating alpha
lettuce.m2<-drm(weight~conc,data=lettuce,fct=CRS.6())
summary(lettuce.m2)
plot(lettuce.m2)  # oops: not increasing until hormesis peak