Probit LOD95 Calculator

Why Probit Regression?

A common mistake is to report LOD as the lowest concentration where all replicates are positive, or to use simple interpolation between dilution levels. These approaches are statistically unreliable and ignore the binary nature of the data. Probit regression is the recommended method (CLSI EP17-A2, FDA guidance) because it properly models the probability of detection as a function of concentration using a cumulative normal distribution.

The Probit Model

Where Φ⁻¹ is the inverse normal CDF (probit function), p is the probability of detection, and α and β are fitted parameters. The model assumes that the log-concentration at which detection occurs follows a normal distribution across replicate measurements.

Minimum Data Requirements

The calculator needs at least 3 dilution levels with valid data, and the data must show a dose-response transition (not all 0% or all 100% detection). Beyond that, there is no hard minimum for replicates per level — the calculator will fit a probit model even with 3 replicates per concentration. However, fewer replicates means wider confidence intervals and less reliable point estimates.

For screening experiments (e.g., 10-fold dilutions × 3 replicates), the calculator will produce a result but will display warnings about the limited data. This is useful for getting a rough LOD estimate, but should not be used for regulatory submissions without further testing.

Experimental Design Tips

• Use at least 5–8 concentration levels spanning the expected LOD range
• Test at least 20 replicates per level (more is better for narrow CIs). With 3 replicates, the 95% CI can span several orders of magnitude
• Include concentrations that give 0%, partial, and 100% detection — the transition zone (where some replicates detect and some don't) is the most informative for fitting the curve
• Space concentrations in half-log or 2-fold steps (e.g., 1, 2, 5, 10, 20, 50, 100) near the expected LOD. 10-fold steps are fine for screening but give fewer data points in the transition zone
• Report LOD95 with confidence intervals — the point estimate alone is insufficient

Common Mistakes

• Reporting the lowest concentration with 100% detection as the LOD
• Using too few replicates (<10) which gives very wide confidence intervals
• Using 10-fold dilution steps and treating the result as definitive — this is fine for a preliminary estimate but the LOD could fall anywhere within a 10× range between adjacent dilutions
• Using logistic (logit) regression instead of probit when guidelines specify probit
• Ignoring goodness-of-fit — a poor fit indicates the probit model may be inappropriate

How This Calculator Works

The calculator fits a probit regression model to your binary (positive/negative) data using Iteratively Reweighted Least Squares (IRLS), the standard maximum likelihood estimation method for generalized linear models with a probit link. Your data is modelled as: Φ⁻¹(p) = α + β × log₁₀(concentration), where Φ⁻¹ is the inverse normal CDF.

From the fitted model, the LOD at any detection probability is calculated by inverting the equation: log₁₀(LOD) = (Φ⁻¹(target) − α) / β. For LOD95, the target is 0.95, so Φ⁻¹(0.95) = 1.645. Confidence intervals are computed using the delta method applied to the variance-covariance matrix of the fitted parameters.

The Pearson chi-squared goodness-of-fit statistic compares observed and expected counts at each dilution level. A low p-value (<0.05) suggests the probit model does not adequately fit your data — this can happen with outlier concentrations, non-monotonic data, or when the underlying dose-response does not follow a normal distribution.