# Weibull model functions

### Description

'weibull' and 'weibull2' provide a very general way of specifying Weibull dose response functions, under various constraints on the parameters.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
weibull1(fixed = c(NA, NA, NA, NA),
names = c("b", "c", "d", "e"),
method = c("1", "2", "3", "4"),
ssfct = NULL,
fctName, fctText)
weibull2(fixed = c(NA, NA, NA, NA),
names = c("b", "c", "d", "e"),
method = c("1", "2", "3", "4"),
ssfct = NULL,
fctName, fctText)
weibull2x(fixed = rep(NA, 5),
names = c("b", "c", "d", "e", "t0"),
method = c("1", "2", "3", "4"),
ssfct = NULL,
fctName, fctText)
``` |

### 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 (see under 'Details'). |

`method` |
character string indicating the self starter function to use. |

`ssfct` |
a self starter function to be used. |

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

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

### Details

As pointed out in Seber and Wild (1989), there exist two different parameterisations of the Weibull model. They do not yield the same fitted curve for a given dataset (see under Examples).

One four-parameter Weibull model ('weibull1') is

* f(x) = c + (d-c) \exp(-\exp(b(\log(x)-\log(e)))).*

Another four-parameter Weibull model ('weibull2') is

* f(x) = c + (d-c) (1 - \exp(-\exp(b(\log(x)-\log(e))))).*

Both four-parameter functions are asymmetric with inflection point at the dose *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 function `drm`

.

### Author(s)

Christian Ritz

### References

Seber, G. A. F. and Wild, C. J (1989)
*Nonlinear Regression*,
New York: Wiley \& Sons (pp. 338–339).

### See Also

For convenience several special cases of the function 'weibull1' are available:
`W1.2`

, `W1.3`

and `W1.4`

.

Special cases of 'weibull2' are:
`W2.2`

, `W2.3`

and `W2.4`

.

These convenience functions should be used rather than the underlying functions
`weibull1`

and `weibull2`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
## Fitting two different Weibull models
ryegrass.m1 <- drm(ryegrass, fct = W1.4())
plot(ryegrass.m1, conLevel=0.5)
ryegrass.m2 <- drm(ryegrass, fct = W2.4())
plot(ryegrass.m2, conLevel=0.5, add = TRUE, type = "none", col = 2)
# you could also look at the ED values to see the difference
## A four-parameter Weibull model with b fixed at 1
ryegrass.m3 <- drm(ryegrass, fct = W1.4(fixed = c(1, NA, NA, NA)))
summary(ryegrass.m3)
## A four-parameter Weibull model with the constraint b>3
ryegrass.m4 <- drm(ryegrass, fct = W1.4(), lowerl = c(3, -Inf, -Inf, -Inf),
control = drmc(constr=TRUE))
summary(ryegrass.m4)
``` |