BMDL

Propagate supplied linear-dose probit parameter covariance through extra-risk BMD and a log-delta lower bound; approximate, not a likelihood-profile BMDL or regulatory certification.

Description

Approximate a lower benchmark dose by propagating probit parameter covariance on a logarithmic dose scale.

A benchmark dose, or BMD, is the dose associated with a selected modeled response. A lower benchmark-dose bound additionally accounts for statistical uncertainty. Benchmark response and uncertainty methods must be specified when comparing analyses.^[1]

This tool uses a linear-dose probit model p(d) = Φ(a + b × d) and an extra-risk target relative to modeled background. It receives fitted parameters and their covariance, then applies a local delta approximation. Its lower estimate is not a profile-likelihood result or a fit to experimental observations.

When to use BMDL

Use it to reproduce the stated log-delta approximation from an externally fitted probit model with a suitable covariance estimate.

How to use BMDL and understand its results

Enter finite intercept, positive slope, and benchmarkExtraRisk strictly between zero and one. The slope unit is the reciprocal of the intended dose unit.
Enter a symmetric positive-definite 2 × 2 parameterCovariance matrix in intercept, slope order and a positive oneSidedCriticalZ appropriate to the chosen confidence convention.
Run the tool and review estimatedBenchmarkDose, doseStandardError, and approximateLowerBenchmarkDose in the same dose unit.

p0 = Φ(a); p* = p0 + r(1 − p0); z* = Φ⁻¹(p*); D = (z* − a)/b

Φ is the standard normal distribution function; r is extra risk, not an absolute response probability.

g_a = [(1 − r)φ(a)/φ(z*) − 1]/b; g_b = −D/b; SE = sqrt(gᵀΣg); lower = D × exp(−criticalZ × SE/D)

φ is the standard normal density. The gradient includes the dependence of modeled background on the intercept.

The implementation evaluates normal-distribution approximations and propagates both diagonal and off-diagonal covariance terms. No fixed confidence level is selected automatically.

json

{
  "input": {
    "intercept": 0,
    "slope": 1,
    "benchmarkExtraRisk": 0.5,
    "parameterCovariance": [
      [
        0.01,
        0
      ],
      [
        0,
        0.01
      ]
    ],
    "oneSidedCriticalZ": 1.645
  },
  "output": {
    "estimatedBenchmarkDose": 0.6744897502234195,
    "doseStandardError": 0.07704136462325434,
    "approximateLowerBenchmarkDose": 0.5589509850534436
  }
}

A zero or singular covariance matrix is rejected. Off-diagonal entries must match exactly, and the determinant must be positive. Extreme parameters that make inversion or the finite dose calculation fail are rejected; a finite-precision exponential may underflow to zero.

Limitations and assumptions

The approximation assumes that local parameter uncertainty describes the dose adequately. It does not assess fit, model selection, small-sample coverage, or extrapolation suitability.

It does not select a benchmark response or covariance estimator. A lower result alone is not an authoritative exposure limit, and this method may differ materially from other BMDL procedures.

Alternative or Complementary methods

Use the unbounded probit calculator to inspect the underlying target dose. For a continuous endpoint, choose a suitable continuous-response model rather than treating a mean change as extra probability risk.

References

Benchmark Dose Technical Guidance (2012) — U.S. Environmental Protection Agency

Similar or alternative tools

Benchmark Dose Probit
Calculate the same extra-risk target without a covariance-based lower estimate.
Continuous BMD
Invert a specified continuous exponential mean-response model.