Table A.2.

Dose–response models for continuous and quantal data in BMDS (US EPA, 2012)

Continuous models
Hill model $\mathit{\mu }\left(X\right)=\mathit{\gamma }+\mathit{\nu }\frac{X^{\mathit{\eta }}}{{\mathit{\kappa }}^{\mathit{\eta }}+X^{\mathit{\eta }}}$	Exponential models (a set of nested models) Model 3: $\mathit{\mu }\left(X\right)=\mathit{\gamma }+e^{{\left(kX\right)}^d}$ Model 5:$\mathit{\mu }\left(X\right)=\mathit{\gamma }\left(c-(c-1)e^{-{\left(kX\right)}^d}\right)$
Quantal models
Logistic model $p\left(X\right)=\frac{1}{1+e^{-\left(\mathit{\alpha }+\mathit{\beta }X\right)}}$ Log‐logistic model $p\left(X\right)=\mathit{\gamma }+\frac{1-\mathit{\gamma }}{1+e^{-\left(\mathit{\alpha }+\mathit{\beta }lnX\right)}}$ Probit model a $p\left(X\right)=\frac{1}{\sqrt{2\mathit{\pi }}}\underset{-\infty }{\overset{\mathit{\alpha }+\mathit{\beta }X}{\int }}{e}^{\frac{-x^2}{2}}dx$ Log‐probit model a $p\left(X\right)=\mathit{\gamma }+\frac{1-\mathit{\gamma }}{\sqrt{2\mathit{\pi }}}\underset{-\infty }{\overset{\mathit{\alpha }+\mathit{\beta }lnX}{\int }}{e}^{\frac{-x^2}{2}}dx$	Weibull model $p\left(X\right)=\mathit{\gamma }+\left(1-\mathit{\gamma }\right)\left(1-e^{-\mathit{\beta }{\left(X\right)}^{\mathit{\alpha }}}\right)$ Gamma model b $p\left(X\right)=\mathit{\gamma }+\left(1-\mathit{\gamma }\right)\frac{1}{\mathrm{\Gamma }\left(\mathit{\alpha }\right)}\underset{0}{\overset{\mathit{\beta }X}{\int }}{x}^{\left(\mathit{\alpha }-1\right)}{e}^{-x}dx$ Multistage model $p\left(X\right)=\mathit{\gamma }+\left(1-\mathit{\gamma }\right)\left(1-e^{-\sum _{j=1}^n{\mathit{\beta }}_j{X}^j}\right)$

For continuous models, the variance across dose group may either be assumed to be constant or non‐constant (a power function of the mean response).

^aIn the model, $\frac{1}{\sqrt{2\mathit{\pi }}}{e}^{\frac{-x^2}{2}}$is the standard normal density function.

^bIn the model, $\mathrm{\Gamma }\left(\mathit{\alpha }\right)=\underset{0}{\overset{\infty }{\int }}{x}^{\left(\mathit{\alpha }-1\right)}{e}^{-x}dx$ is the gamma function.