The four-parameter log-logistic function

'LL.4' and 'LL2.4' provide the four-parameter log-logistic function, self starter function, names of the parameters and, optionally, first and second derivatives for a faster estimation.

LL.4(fixed = c(NA, NA, NA, NA), names = c("b", "c", "d", "e"), ...)

  l4(fixed = c(NA, NA, NA, NA), names = c("b", "c", "d", "e"), ...)

  LL2.4(fixed = c(NA, NA, NA, NA), names = c("b", "c", "d", "e"), ...)

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. The default is reasonable.
...	Additional arguments (see llogistic).

Details

The four-parameter log-logistic function is given by the expression $$ f(x) = c + \frac{d-c}{1+\exp(b(\log(x)-\log(e)))}$$

or in another parameterisation (converting the term \(\log(e)\) into a parameter) $$ f(x) = c + \frac{d-c}{1+\exp(b(\log(x)-\tilde{e}))}$$

The function is symmetric about the inflection point (\(e\)).

Value

See llogistic.

References

Seber, G. A. F. and Wild, C. J (1989) Nonlinear Regression, New York: Wiley \& Sons (p. 330).

Note

This function is for use with the function drm.

See also

Setting \(c=0\) yields LL.3. See also LL.5.

Examples

spinach.m1 <- drm(SLOPE~DOSE, CURVE, data = spinach, fct = LL.4())
spinach.m1
#> 
#> A 'drc' model.
#> 
#> Call:
#> drm(formula = SLOPE ~ DOSE, curveid = CURVE, data = spinach,     fct = LL.4())
#> 
#> Coefficients:
#>       b:1        b:2        b:3        b:4        b:5        c:1        c:2  
#>  0.519519   0.800796   0.681913   1.844809   1.650758  -0.016595   0.132589  
#>       c:3        c:4        c:5        d:1        d:2        d:3        d:4  
#>  0.146406   0.079552  -0.009066   1.879553   0.946000   1.090321   2.153578  
#>       d:5        e:1        e:2        e:3        e:4        e:5  
#>  1.806283   1.794955   0.945530   1.373023   0.197326   0.210793  
#>