# 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

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
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)
``` |

### 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. This argument needs to be specified. |

`fctName` |
optional character string used internally by convenience functions. |

`fctText` |
optional character string used internally by convenience functions. |

### Details

The model is given by the expression

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

### Value

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

### Note

The functions are for use with the functions `drm`

.

### Author(s)

Christian Ritz

### 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

1 2 3 4 |