Table I.

Selected quantal dose-response models common in toxicological and carcinogenic risk assessment

Model	Code	R(x)	BMD	Restrictions/Notes
Logistic	1	$\frac{1}{1+\exp \{-{\beta }_0-{\beta }_{1}x\}}$	$\frac{1}{{\beta }_1}\log \left\{\frac{1+e^{-{\beta }_0}\mathrm{BMR}}{1-\mathrm{BMR}}\right\}$	−
Probit	2	Φ(β0 + β1x)	$\frac{{\mathrm{Q}}_{\mathrm{BMR}}-{\beta }_0}{{\beta }_1}$	QBMR = Φ−1{BMR[1 −Φ(β0)] + Φ(β0)}
Quantal-linear	3	1 − exp{−β0 − β1x}	$\frac{-\ln (1-\mathrm{BMR})}{{\beta }_1}$	β0 ≥ 0, β1 ≥ 0
Quantal-quadratic	4	γ0 + (1−γ0)(1−exp{−β1x2})	$\sqrt{\frac{-\ln (1-\mathrm{BMR})}{{\beta }_1}}$	0 ≤ γ0 ≤ 1, β1 ≥ 0
Two-stage	5	1 − exp{−β0 − β1x − β2x2}	$\frac{-{\beta }_1+\sqrt{{\beta }_{1^2}+4{\beta }_2{\mathrm{T}}_{\mathrm{BMR}}}}{2{\beta }_2}$	βj ≥ 0, j = 0,1,2 TBMR = −log(1 − BMR)
Log-logistic	6	${\gamma }_0+\frac{1-{\gamma }_0}{1+\mathit{exp}\{-{\beta }_0-{\beta }_1\log [x\left]\right\}}$	$\exp \left\{\frac{{\mathrm{L}}_{\mathrm{BMR}}-{\beta }_0}{{\beta }_1}\right\}$	0 ≤ γ0 ≤ 1, β1 ≥ 0 LBMR = log{BMR/(1−BMR)}
Log-probit	7	γ0 + (1−γ0)Φ(β0 + β1log[x])	$\exp \left\{\frac{{\Phi }^{-1}\left(\mathrm{BMR}\right)-{\beta }_0}{{\beta }_1}\right\}$	0 ≤ γ0 ≤ 1, β1 ≥ 0
Weibull	8	γ0 + (1−γ0)[1–exp{−eβ0xβ1}]	$\exp \left\{\frac{{\mathrm{W}}_{\mathrm{BMR}}-{\beta }_0}{{\beta }_1}\right\}$	0 ≤ γ0 ≤ 1, β1 ≥ 1 WBMR = log{−log(1−BMR)}

Notes: The quantal linear (Model 3) model is also referred to as the ‘one-stage’ model or as the ‘complementary-log’ model, and may equivalently appear as γ₀ + (1 − γ₀)(1 − exp{−β₁x}), where γ₀ = 1 − exp{−β₀}.

The two-stage model (Model 5) and the quantal-linear/one-stage model (Model 3) are special cases of the more general ‘Multi-stage’ model in carcinogenesis testing ^{(4, §4.2.1)}.