Table 1.

Five single toxin models and their four special points, D = (5p + 1) (p + 1).

		logit/Hill	probit	Weibull	1HMT logit	AHMT logit
1	Mortality function formula	$\frac{e^{a+b\text{ ln }x}}{1+e^{a+b\text{ ln }x}}$	Φ(a + b ln x)	1 − e−ea+b ln x	$1-\frac{1}{{(1+e^{a+b\text{ ln }x})}^p}$	$\frac{{(e^{a+b\text{ ln }x})}^p}{{(1+e^{a+b\text{ ln }x})}^p}$
2	Inverse mortality function	$\text{ln}\frac{m}{1-m}$	Φ−1(m)	ln(− ln(1 − m))	$\text{ln }(\frac{1}{{(1-m)}^{1/p}}-1)$	$\text{ln}\frac{m^{1/p}}{1-m^{1/p}}$
3	LC50 on the L = a + b ln x scale	0	0	ln ln 2	ln(21/p − 1)	− ln(21/p − 1)
4	IRMR on the L scale	b	$b\frac{\varphi (L)}{\mathrm{\Phi }(L)(1-\mathrm{\Phi }(L))}$	$b\frac{e^L}{1-e^{-e^L}}$	$b\frac{p(1+e^L)}{1-{(1+e^L)}^{-p}}$	$b\frac{p{(1+e^L)}^{p-1}}{{(1+e^L)}^P-e^{p^L}}$
5	Efficacy function on L scale	$\frac{b{e}^L}{{(1+e^L)}^2}$	bϕ(L)	beL−eL	$\frac{bp{e}^L}{{(1+e^L)}^{p+1}}$	$\frac{pb{e}^{pL}}{{(1+e^L)}^{p+1}}$
6	Efficacy function on m scale	bm(1 − m)	bϕ(Φ−1(m))	−b(1 − m) ln(1 − m)	$bp(1-m)(1-{(1-m)}^{\frac{1}{p}})$	$pbm(1-m^{\frac{1}{p}})$
7	Maximum efficacy on L scale	0	0	0	− ln p	0
8	Maximum efficacy value	b/4	$b/\sqrt{2\pi }$	b/e	$\frac{b{p}^{p+1}}{{(p+1)}^{p+1}}$	$\frac{bp}{2^{p+1}}$
9	Maximum efficacy on m scale	1/2	1/2	(e − 1)/e ≃ 0.632	$1-\frac{p^p}{{(1+p)}^p}$	$\frac{p^p}{{(1+p)}^p}$
10	1st inflection point on L scale	$\text{ln}(2-\sqrt{3})\simeq -1.32$	−1	$\text{ln}\frac{3-\sqrt{5}}{2}\simeq -0.962$	$\text{ln }\left(\frac{3p+1-\sqrt{D}}{2p^2}\right)$	$\text{ln }\left(\frac{3p+1-\sqrt{D}}{2}\right)$
11	1st inflection point on m scale	$\frac{2-\sqrt{3}}{3-\sqrt{3}}\simeq 0.211$	Φ(−1) ≃ 0.159	$e^{-(3+\sqrt{5})/2}\simeq 0.073$	$1-{\left(\frac{2p^2}{2p^2+3p+1-\sqrt{D}}\right)}^p$	${\left(\frac{3p+1-\sqrt{D}}{3+3p-\sqrt{D}}\right)}^p$
12	2nd inflection point on L scale	$\text{ln}(2+\sqrt{3})\simeq 1.32$	1	$\text{ln}\frac{3+\sqrt{5}}{2}\simeq 0.962$	$\text{ln }\left(\frac{3p+1+\sqrt{D}}{2p^2}\right)$	$\text{ln }\left(\frac{3p+1+\sqrt{D}}{2}\right)$
13	2nd inflection point on m scale	$\frac{2+\sqrt{3}}{3+\sqrt{3}}\simeq 0.789$	Φ(1) ≃ 0.841	$e^{-(3-\sqrt{5})/2}\simeq 0.683$	$1-{\left(\frac{2p^2}{2p^2+3p+1+\sqrt{D}}\right)}^p$	${\left(\frac{3p+1+\sqrt{D}}{3+3p+\sqrt{D}}\right)}^p$