drc (version 4.0-0)

cedergreen: The Cedergreen-Ritz-Streibig model

Description

'cedergreen' provides a very general way of specifying the Cedergreen-Ritz-Streibig modified log-logistic model for describing inverse U-shaped and U-shaped 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"), 
  method = c("1", "2", "3", "4"), ssfct = NULL,
  alpha, fctName, fctText)
  
  CRS.6(fixed = c(NA, NA, NA, NA, NA, NA), 
  names = c("b","c","d","e","f","g"),
  method = c("1", "2", "3", "4"), ssfct = NULL)
  
  ucedergreen(fixed = c(NA, NA, NA, NA, NA), 
  names = c("b", "c", "d", "e", "f"), 
  method = c("1", "2", "3", "4"), ssfct = NULL,
  alpha, fctName, fctText)

Value

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

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 order of the parameters is: b, c, d, e, f (see under 'Details').

method

character string indicating the self starter function to use.

ssfct

a self starter function to be used.

alpha

numeric value between 0 and 1, reflecting the steepness of the hormesis peak.

fctName

optional character string, used internally for naming convenience functions.

fctText

optional character string, used internally for naming convenience functions.

Author

Christian Ritz

Details

The inverse U-shaped 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 log-logistic model function to take hormesis into account.

The corresponding U-shaped model is given by the expression

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

References

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

Sweeney, C. J., Bottoms, M., Hutcheson, K., Lennart Schulz, L. (2026) Dose-response OECD 216 studies for agrochemicals, Integrated Environmental Assessment and Management, vjag057. https://doi.org/10.1093/inteam/vjag057

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 4 indicates a lower limit fixed at 0 (not estimated), 5 indicates lower and upper limits being estimated (no constraints), and a, b and c indicate pre-specified alpha values 1, 0.5 and 0.25, respectively.

Examples

## Estimating CRS model with a lower limit equal to 0 and alpha fixed at 1
lettuce.crs.4a <- drm(weight ~ conc, data = lettuce, fct = CRS.4a())

summary(lettuce.crs.4a)

predict(lettuce.crs.4a, se.fit = TRUE)
predict(lettuce.crs.4a, interval = "confidence")
predict(lettuce.crs.4a, interval = "prediction")

plot(lettuce.crs.4a)

# Example from Sweeney et al. (2026) with U-shaped hormesis data
concVec <- c(0,0,0,1,1,1,2,2,2,4,4,4,10,10,10, 50, 50, 50)
respVec <- c(2.1, 2.0, 1.8, 1.5, 1.2, 1.1, 2.2, 1.8, 2.1, 
4.6, 4.8, 5.2, 9.8, 8.8, 10.1, 11.1, 11.3, 10.9)

ex.ucrs.5b <- drm(respVec ~ concVec, fct = UCRS.5b())
summary(ex.ucrs.5b)
# estimated parameter f: 2.62248 (0.88486), p = 0.01098

plot(ex.ucrs.5b, bp = 0.001)
fitted(ex.ucrs.5b)

predict(ex.ucrs.5b, se.fit = TRUE)
predict(ex.ucrs.5b, interval = "confidence")