# The three-parameter log-logistic function

### Description

'LL.3' and 'LL2.3' provide the three-parameter log-logistic function where the lower limit is equal to 0.

'LL.3u' and 'LL2.3u' provide three-parameter logistic function where the upper limit is equal to 1, mainly for use with binomial/quantal response.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 | ```
LL.3(fixed = c(NA, NA, NA), names = c("b", "d", "e"), ...)
LL.3u(upper = 1, fixed = c(NA, NA, NA), names = c("b", "c", "e"), ...)
l3(fixed = c(NA, NA, NA), names = c("b", "d", "e"), ...)
l3u(upper = 1, fixed = c(NA, NA, NA), names = c("b", "c", "e"), ...)
LL2.3(fixed = c(NA, NA, NA), names = c("b", "d", "e"), ...)
LL2.3u(upper = 1, fixed = c(NA, NA, NA), names = c("b", "c", "e"), ...)
``` |

### Arguments

`upper` |
numeric value. The fixed, upper limit in the model. Default is 1. |

`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. The default is reasonable. |

`...` |
Additional arguments (see |

### Details

The three-parameter log-logistic function with lower limit 0 is

* f(x) = 0 + \frac{d-0}{1+\exp(b(\log(x)-\log(e)))}*

or in another parameterisation

* f(x) = 0 + \frac{d-0}{1+\exp(b(\log(x)-e))}*

The three-parameter log-logistic function with upper limit 1 is

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

or in another parameterisation

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

Both functions are symmetric about the inflection point (*e*).

### Value

See `llogistic`

.

### Note

This function is for use with the function `drm`

.

### Author(s)

Christian Ritz

### References

Finney, D. J. (1971) *Probit Analysis*, Cambridge: Cambridge University Press.

### See Also

Related functions are `LL.2`

, `LL.4`

, `LL.5`

and the more general
`llogistic`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | ```
## Fitting model with lower limit equal 0
ryegrass.model1 <- drm(rootl ~ conc, data = ryegrass, fct = LL.3())
summary(ryegrass.model1)
## Fitting binomial response
## with non-zero control response
## Example dataset from Finney (1971) - example 19
logdose <- c(2.17, 2,1.68,1.08,-Inf,1.79,1.66,1.49,1.17,0.57)
n <- c(142,127,128,126,129,125,117,127,51,132)
r <- c(142,126,115,58,21,125,115,114,40,37)
treatment <- factor(c("w213","w213","w213","w213",
"w214","w214","w214","w214","w214","w214"))
# Note that the control is included in one of the two treatment groups
finney.ex19 <- data.frame(logdose, n, r, treatment)
## Fitting model where the lower limit is estimated
fe19.model1 <- drm(r/n~logdose, treatment, weights = n, data = finney.ex19,
logDose = 10, fct = LL.3u(), type="binomial",
pmodels = data.frame(treatment, 1, treatment))
summary(fe19.model1)
modelFit(fe19.model1)
plot(fe19.model1, ylim = c(0, 1.1), bp = -1, broken = TRUE, legendPos = c(0, 1))
abline(h = 1, lty = 2)
``` |

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