Table 3.
Expressions of the recommended models for use in the BMD approach, with (mean) response (y) being a function of dose (x), both on the original scale. See Table A.2 in Appendix A for the equivalent model expressions used in BMDS software
| Model | Number of model parameters | Model expression mean response (y) as function of dose (x) | Constraints |
|---|---|---|---|
| Full modela | Number of dose groups including background | Set of observed means or incidences at each dose | |
| Null modelb | 1 | y = a |
a > 0 for continuous data 0 < a < 1 for quantal data |
| Continuous data | |||
| Exponential family | |||
| 3‐parameter modelc | 3 | y = a exp(bx d) | a > 0, d > 1 |
| 4‐parameter modeld | 4 | y = a [c−(c−1)exp(−bx d)] | a > 0, b > 0, c > 0, d > 1 |
| Hill family | |||
| 3‐parameter modelc | 3 | y = a [1−x d/(b d + x d)] | a > 0, d > 1 |
| 4‐parameter modeld | 4 | y = a [1 + (c−1)x d/(b d + x d)] | a > 0, b > 0, c > 0, d > 1 |
| Quantal data | |||
| Logistic | 2 | y = 1/(1 + exp(−a−bx)) | b > 0 |
| Probit | 2 | y = CumNorm(a + bx) | b > 0 |
| Log‐logistic | 3 | y = a + (1−a)/(1 + exp(−log(x/b)/c)) | 0 ≤ a ≤ 1, b > 0, c > 0 |
| Log‐probit | 3 | y = a + (1−a) CumNorm(log(x/b)/c) | 0 ≤ a ≤ 1, b > 0, c > 0 |
| Weibull | 3 | y = a + (1−a) exp((x/b)c) | 0 ≤ a ≤ 1, b > 0, c > 0 |
| Gamma | 3 | y = a + (1−a) CumGam(bx c) | 0 ≤ a ≤ 1, b >0, c > 0 |
| LMS (two‐stage) model | 3 | y = a + (1−a)(1−exp(−bx−cx 2)) | a > 0, b> 0, c > 0 |
| Latent variable models (LVMs) based on the continuous models abovee | Depends on underlying continuous model | These models assume an underlying continuous response, which is dichotomised into yes/no response based on a (latent) cut‐off value that is estimated from the data | See continuous models |
a, b, c, d: unknown parameters that are estimated by fitting the model to the data.
CumNorm: cumulative (standard) normal distribution function.
CumGam: cumulative Gamma distribution function.
The full model will result in the maximum possible value of the log‐likelihood (given the statistical assumptions) for the data set considered.
The null model can be regarded as a model that is nested within any dose–response model: it reflects the situation of no dose response (= horizontal line).
Called model 3 in PROAST, and similarly (for the exponential model) in BMDS.
Called model 5 in PROAST, and similarly (for the exponential model) in BMDS.
The latent variable models are implemented in PROAST.