Probit Detection Limit

Invert a fitted standard-normal probit versus log10 concentration model at a caller quantile (1.64485 for 95% hits); coefficients use probit zero, not probit five.

Description

Invert a fitted standard-normal probit model against log10 concentration at a supplied target response quantile.

A detection-probability model describes how often an assay returns a positive result at a concentration. In a probit model, that probability is transformed through the inverse standard-normal cumulative distribution. The transformed value is zero at 50% probability; a value near 1.644854 corresponds to 95% probability.^[1]

This tool assumes the transformed probability equals an intercept plus a positive slope times log10 concentration. Once those coefficients have been fitted elsewhere, algebra can find the concentration corresponding to a chosen response probability. A 95% hit probability is different from a 95% confidence bound on the estimated concentration.

Probit Detection Limit performs that inversion and returns both log10 concentration and ordinary concentration. Its coefficients use the zero-centered normal convention, not the older probit-5 convention.

When to invert an existing detection-probability model

Use the tool after obtaining compatible fitted coefficients and selecting a target probability. It helps check a point estimate and avoid mixing log bases, concentration units, or probit conventions.

Calculate the concentration for a chosen assay hit probability.
Compare target probabilities using the same fitted model.
Check the zero-centered versus probit-5 intercept convention.

How to supply coefficients and invert the probit relation

Enter Probit intercept and a positive Probit slope from a model fitted against log10 concentration.
Enter Target normal quantile as the standard-normal quantile of the desired probability, such as 1.6448536269514722 for 95%; do not enter 95 or 0.95.
Run the calculation and read Log 10 detection concentration and Detection concentration in the concentration unit used for the fit.

Φ⁻¹(Pdetect) = a + b × log10(C); log10DetectionConcentration = (zTarget − a)/b; detectionConcentration = 10^((zTarget−a)/b)

Φ is the standard-normal cumulative distribution, a is the intercept, and b is the positive slope. The tool receives zTarget rather than calculating it from probability.

The implementation validates a positive slope, subtracts the intercept from the supplied quantile, divides by slope, and exponentiates base ten. It does not fit coefficients from hit counts. A legacy probit-5 intercept must be converted to the zero-centered convention before entry.

json

{
  "input": {
    "probitIntercept": 0,
    "probitSlope": 1,
    "targetNormalQuantile": 1.6448536269514722
  },
  "output": {
    "log10DetectionConcentration": 1.6448536269514722,
    "detectionConcentration": 44.14216471165427
  }
}

Nonpositive slope, missing fields, or nonfinite inputs produce errors. Negative finite intercepts and target quantiles are accepted.

Limitations of a fitted-model inversion

No model fitting, replicate-count analysis, confidence interval, goodness-of-fit check, or laboratory validation is performed.
Units, log base, and probit convention must match the fitted coefficients. Natural-log or probit-5 coefficients cannot be used unchanged.
The target is a response probability, not an uncertainty level or an automatically selected regulatory detection limit.
Extreme exponents can overflow and cause an output error, or underflow to numeric zero. Extrapolation beyond fitted concentrations is not evaluated.

Alternative or Complementary dose-response inversions

Choose a model whose response and concentration convention match the analysis. The LD50 tool uses probit-5 and log-dose, while an extra-risk benchmark tool uses standard-normal probit on linear dose. Those distinctions change the meaning of the coefficients and target.

References

R statistics documentation: family objects and binomial probit links — R Core Team

Similar or alternative tools

Probit LD50 Estimate
Invert a probit-5 model against log10 dose at its 50% response endpoint.
Benchmark Dose Probit
Invert a standard-normal probit model on linear dose for a supplied extra-risk benchmark.