Model Uncertainty in Environmental Dose–Response Risk Analysis

Figures & data

Table 1. Some common quantal dose–response models in environmental toxicology and carcinogenicity risk assessment, from West et al. (2012)

Model	Code	R(x)	BMD, ξq	Restrictions/notes
Logistic	M1			—
Probit	M2	Φ(β0 + β1x)		QBMR = Φ^−1{q[1 − Φ(β0)] + Φ(β0)}
Quantal-linear	M3	1 − exp { − β0 − β1x}		β0 ≥ 0, β1 ≥ 0
Quantal-quadratic	M4	γ0 + (1 − γ0)(1 − exp { − β1x^2})		0 ≤ γ0 ≤ 1, β1 ≥ 0
Two-stage	M5	1 − exp { − β0 − β1x − β2x^2}		βj ≥ 0, j = 0,1,2
				TBMR = − log(1 − q)
Log-logistic	M6			0 ≤ γ0 ≤ 1, β1 ≥ 0
Log-probit	M7	γ0 + (1 − γ0)Φ(β0 + β1log [x])		0 ≤ γ0 ≤ 1, β1 ≥ 0
Weibull	M8			0 ≤ γ0 ≤ 1, β1 ≥ 1
				WBMR = log{−log(1 − q)}

NOTE: BMD is the benchmark dose. The benchmark response is BMR = q. Φ(·) is the standard normal cdf. The quantal linear (M3) model is also referred to as the “one-stage” or “complementary-log” model, and may equivalently appear as γ₀ + (1 − γ₀)(1 − exp{−β₁x}), where γ₀ = 1 − exp{−β₀}.

Figure 1. Wald-type 95% BMDL coverage for correctly specified (solid black lines) and misspecified (dashed gray lines) models at q = 0.10. (Data from West et al. 2012.) Thick horizontal bar is 95% nominal level.

Table 2. Multi-model benchmark analysis of CH₂O carcinogenicity data in Section 4. Models, M_ℓ, from Table 1. Benchmark response (BMR) set to q = 0.10

Model, Mℓ		AICℓ	wℓ from (3.2)
Logistic, M1	7.2446	35.6226	0.0026	0.0191
Probit, M2	7.1004	31.2633	0.0233	0.1651
Quantal-linear, M3	1.5618	153.7791	5.79 × 10^−29	< 10^−5
Quantal-quadratic, M4	4.0334	61.9964	4.93 × 10^−9	< 10^−5
Two-stage, M5	4.0347	63.9963	—	—
Log-logistic, M6	6.8910	26.0357	0.3175	2.1878
Log-probit, M7	6.9002	24.6198	0.6444	4.4468
Weibull, M8	6.7693	32.5570	0.0122	0.0824

West, R.W., Piegorsch, W.W., Peña, E.A., An, L., Wu, W., Wickens, A.A., Xiong, H., and Chen, W. (2012), “The Impact of Model Uncertainty on Benchmark Dose Estimation,” Environmetrics, 23, 706–716.

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