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

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  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 llogistic).

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

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## 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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