The three-parameter log-logistic function

'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.

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.

References

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

Note

This function is for use with the function drm.

Examples


## Fitting model with lower limit equal 0
ryegrass.model1 <- drm(rootl ~ conc, data = ryegrass, fct = LL.3())
summary(ryegrass.model1)
#> 
#> Model fitted: Log-logistic (ED50 as parameter) with lower limit at 0 (3 parms)
#> 
#> Parameter estimates:
#> 
#>               Estimate Std. Error t-value   p-value    
#> b:(Intercept)  2.47033    0.34168  7.2299 4.011e-07 ***
#> d:(Intercept)  7.85543    0.20438 38.4352 < 2.2e-16 ***
#> e:(Intercept)  3.26336    0.19641 16.6154 1.474e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error:
#> 
#>  0.5615802 (21 degrees of freedom)

## 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))
#> Warning: NaNs produced

summary(fe19.model1)
#> 
#> Model fitted: Log-logistic (ED50 as parameter) with upper limit at 1 (3 parms)
#> 
#> Parameter estimates:
#> 
#>                Estimate Std. Error t-value   p-value    
#> b:w213        -2.183042   0.232540 -9.3878 < 2.2e-16 ***
#> b:w214        -2.117258   0.257816 -8.2123 < 2.2e-16 ***
#> c:(Intercept)  0.171725   0.033821  5.0775 3.824e-07 ***
#> e:w213        16.599749   1.754358  9.4620 < 2.2e-16 ***
#> e:w214         9.553766   1.412122  6.7655 1.328e-11 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
modelFit(fe19.model1)
#> Goodness-of-fit test
#> 
#>             Df Chisq value p value
#>                                   
#> DRC model    5      8.9976  0.1092
plot(fe19.model1, ylim = c(0, 1.1), bp = -1, broken = TRUE, legendPos = c(0, 1))
abline(h = 1, lty = 2)