# CRS.5a: Cedergreen-Ritz-Streibig dose-reponse model for describing... In drc: Analysis of Dose-Response Curves

## Description

'CRS.5a', 'CRS.5b' and 'CRS.5c' provide the Cedergreen-Ritz-Streibig modified log-logistic model for describing (inverse u-shaped or j-shaped) hormesis.

'UCRS.5a', 'UCRS.5b' and 'UCRS.5c' provide the Cedergreen-Ritz-Streibig modified log-logistic model for describing u-shaped hormesis.

## Usage

 1 2 3  CRS.5a(names = c("b", "c", "d", "e", "f"), ...) UCRS.5a(names = c("b", "c", "d", "e", "f"), ...) 

## Arguments

 names a vector of character strings giving the names of the parameters. ... additional arguments to be passed from the convenience functions.

## Details

The model function for inverse u-shaped hormetic patterns is

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

,

which is a five-parameter model. It is a modification of the four-parameter log-logistic curve to take hormesis into account.

The parameters have the following interpretations

• b: Not direct interpretation

• c: Lower horizontal asymptote

• d: Upper horizontal asymptote

• e: Not direct interpretation

• f: Size of the hormesis effect: the larger the value the larger is the hormesis effect. f=0 corresponds to no hormesis effect and the resulting model is the four-parameter log-logistic model. This parameter should be positive in order for the model to make sense.

The model function for u-shaped hormetic patterns is

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

This model also simplifies to the four-parameter log-logistic model in case f=0 (in a slightly different parameterization as compared to the one used in LL.4).

The models denoted a,b,c are obtained by fixing the alpha parameter at 1, 0.5 and 0.25, respectively.

## Value

See cedergreen.

## Note

This function is for use with the function drm.

Christian Ritz

## References

See the reference under cedergreen.

Similar functions are CRS.4a and UCRS.4a, but with the lower limit (the parameter c) fixed at 0 (one parameter less to be estimated).
  1 2 3 4 5 6 7 8 9 10 11 12 ## Modified logistic model lettuce.m1 <- drm(lettuce[,c(2,1)], fct=CRS.5a()) summary(lettuce.m1) ED(lettuce.m1, c(50)) lettuce.m2 <- drm(lettuce[,c(2,1)], fct=CRS.5b()) summary(lettuce.m2) ED(lettuce.m2, c(50)) lettuce.m3 <- drm(lettuce[,c(2,1)], fct=CRS.5c()) summary(lettuce.m3) ED(lettuce.m3, c(50))