'W1.4' and 'W2.4' provide the four-parameter Weibull functions, self starter function and names of the parameters.

1 2 3 |

`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 to be passed from the convenience functions. |

The equations for the mean functions are given at `weibull1`

.

See `weibull1`

.

This function is for use with the model fitting function `drm`

.

Christian Ritz

Seber, G. A. F. and Wild, C. J (1989) *Nonlinear Regression*, New York: Wiley \& Sons (pp. 330–331).

Ritz, C (2009)
Towards a unified approach to dose-response modeling in ecotoxicology
*To appear in Environ Toxicol Chem*.

Setting *c=0* yields `W1.3`

. A more flexible function, allowing
fixing or constraining parameters, is available through `weibull1`

.

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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | ```
## Fitting a four-parameter Weibull (type 1) model
terbuthylazin.m1 <- drm(rgr~dose, data = terbuthylazin, fct = W1.4())
summary(terbuthylazin.m1)
## Fitting a first-order multistage model
## to data from BMDS by EPA
## (Figure 3 in Ritz (2009))
bmds.ex1 <- data.frame(ad.dose=c(0,50,100), dose=c(0, 2.83, 5.67),
num=c(6,10,19), total=c(50,49,50))
bmds.ex1.m1<-drm(num/total~dose, weights=total, data=bmds.ex1,
fct=W2.4(fixed=c(1,NA,1,NA)), type="binomial")
modelFit(bmds.ex1.m1) # same as in BMDS
summary(bmds.ex1.m1) # same background estimate as in BMDS
logLik(bmds.ex1.m1)
## BMD estimate identical to BMDS result
## BMDL estimate differs from BMDS result (different method)
ED(bmds.ex1.m1, 10, ci="delta")
## Better fit
bmds.ex1.m2<-drm(num/total~dose, weights=total, data=bmds.ex1,
fct=W1.4(fixed=c(-1,NA,1,NA)), type="binomial")
modelFit(bmds.ex1.m2)
summary(bmds.ex1.m2)
ED(bmds.ex1.m2, 50, ci = "delta")
## Creating Figure 3 in Ritz (2009)
bmds.ex1.m3 <- drm(num/total~dose, weights=total, data=bmds.ex1,
fct=LL.4(fixed=c(-1,NA,1,NA)), type="binomial")
plot(bmds.ex1.m1, ylim = c(0.05, 0.4), log = "", lty = 3, lwd = 2,
xlab = "Dose (mg/kg/day)", ylab = "",
cex=1.2, cex.axis=1.2, cex.lab=1.2)
mtext("Tumor incidence", 2, line=4, cex=1.2) # tailored y axis label
plot(bmds.ex1.m2, ylim = c(0.05, 0.4), log = "", add = TRUE, lty = 2, lwd = 2)
plot(bmds.ex1.m3, ylim = c(0.05, 0.4), log = "", add = TRUE, lty = 1, lwd = 2)
arrows(2.6 , 0.14, 2, 0.14, 0.15, lwd=2)
text(2.5, 0.14, "Weibull-1", pos=4, cex=1.2)
``` |

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