Benchmark Dose Probit
Invert p(d)=Φ(a+b·d) for a supplied extra-risk benchmark relative to p(0)=Φ(a); linear-dose probit is distinct from existing log-dose probit-5 LD50.
Description
Invert a supplied linear-dose probit model at a specified extra-risk benchmark.
A benchmark dose, or BMD, is a model-derived dose associated with a chosen response benchmark. Extra risk expresses the response increase relative to the fraction not already responding at baseline, so it differs from an unadjusted additional risk or an absolute response probability.1
This tool assumes response probability Φ(a + b × dose), where Φ is the standard normal cumulative distribution function. Dose enters linearly, not through its logarithm. You supply the intercept and a positive slope from a separately fitted model.
The tool returns the dose and the target absolute response probability. Dose units follow the fitted slope: if the fit used mg/kg/day, the calculated BMD has that unit. The selected risk fraction is an input, not a recommended default.
When to use Benchmark Dose Probit
Use it to check an analytical benchmark calculation from an existing quantal probit fit with an explicitly chosen extra-risk definition.
How to use Benchmark Dose Probit and interpret its calculation
- Enter finite intercept and positive slope from a linear-dose probit fit using the intended dose unit.
- Enter benchmarkExtraRisk strictly between 0 and 1, converting percentages to fractions.
- Run the calculation and read benchmarkDose with targetResponseProbability. Review model fit and parameter uncertainty before using the dose as a point of departure.
p0 = Φ(a); ptarget = p0 + BMR × (1 − p0); BMD = [Φ⁻¹(ptarget) − a]/b BMR is extra risk. The absolute target probability incorporates the modeled zero-dose baseline.
The implementation evaluates an approximate normal CDF and inverse normal quantile, then requires a positive finite resulting dose. Positive intercepts are allowed. Extreme baselines or risk fractions may make the target numerically indistinguishable from one and cause rejection.
{
"input": {
"intercept": 0,
"slope": 1,
"benchmarkExtraRisk": 0.5
},
"output": {
"benchmarkDose": 0.6744897501960817,
"targetResponseProbability": 0.75
}
}Changing the slope from one to two halves the BMD while leaving the target probability unchanged. Risk fractions zero and one, and a nonpositive slope, are rejected. This point estimate does not include a lower confidence bound.
Limitations and assumptions
No raw dose-response fitting, model selection, goodness-of-fit review, covariance estimation, or uncertainty interval is performed. A logarithmic-dose probit fit cannot be substituted into this formula unchanged.
The response benchmark and endpoint must be appropriate to the study. Numerical inversion does not establish a safe dose or an authoritative reference value.
Alternative or Complementary methods
For an approximate lower bound, supply compatible parameter covariance to the dedicated uncertainty tool. For a tested-concentration summary rather than a fitted response benchmark, use an explicitly justified NOEC analysis.
References
-
Benchmark Dose Technical Guidance (2012) — U.S. Environmental Protection Agency