Additional Capabilities for Bioassay Quantal Response

Bioassay — Quantal Response Method

Capability added: Optional Validity Test Method ("Likelihood-Ratio Deviance" vs. "Finney's Classical Chi-Square") and Automatic Extreme-Response Diagnostic Note

What Was Added

The Quantal Response Method gains a second, optional way of computing its three validity tests (Significance of Regression, Parallelism, Linearity), alongside a new automatic diagnostic note that appears only when the data contains a specific condition that can make the default calculation numerically unstable. The point estimate of relative potency and its confidence interval are unaffected by either addition — both concern only the validity tests and how much users can trust them.

Regulatory and Statistical Background

A quantal response assay is used when the recorded outcome is a proportion — for example, the fraction of test subjects responding at each dose — rather than a continuous measurement. A model (commonly probit or logit) is fitted linking log-dose to response probability, and the same three validity checks as the other bioassay methods (regression significance, parallelism, linearity) are required before potency can be reported.

Two distinct, independently valid statistical procedures exist for computing those three validity tests. The modern default fits a series of nested generalised linear models by maximum likelihood and compares them via likelihood-ratio deviance tests. The classical alternative, described in full by D.J. Finney in "Statistical Method in Biological Assay" (a foundational reference this entire family of bioassay tools follows), instead uses a minimum chi-square procedure built on a linearized "working" variate. European Pharmacopoeia (Ph. Eur.), Chapter 5.3 — "Statistical Analysis of Results of Biological Assays and Tests"-compliant commercial software conventionally uses the latter.

The Mathematics

Deviance method (this tool's existing default). A fully saturated model — one free parameter for every individual preparation/dose group — is fitted by maximum likelihood, alongside progressively more restricted nested models (a common slope across preparations; a fully pooled single line). Each validity test is the likelihood-ratio deviance difference between two of these nested models:

Deviance difference = -2 x (log-likelihood of the simpler model - log-likelihood of the more general model)

which is compared to a chi-square distribution with degrees of freedom equal to the difference in the number of fitted parameters between the two models. This is the standard modern maximum-likelihood approach, but it depends on the fully saturated model converging to a finite estimate for every single group — which fails whenever a group's observed response rate is exactly 0% or exactly 100%, since no finite parameter value can reproduce that outcome under a probit or logit model (a condition statisticians call "perfect separation").

Finney's classical minimum chi-square method (the new option). Rather than fitting a saturated model, the procedure starts from the already-converged common-slope model and, for each dose group, computes a "working" probit or logit value — the fitted linear predictor adjusted by a correction term proportional to how far the observed proportion responding differs from the model's predicted proportion, weighted by the reciprocal of the binomial variance expected at that point:

working variate (per group) = fitted linear predictor + (observed proportion - fitted proportion) / (local slope of the link function)

weight (per group) = group size x (local slope of the link function)^2 / (fitted proportion x (1 - fitted proportion))

An ordinary weighted-least-squares regression and sequential analysis of variance is then run on these working variates in place of the raw proportions, giving Regression, Non-Parallelism and Non-Linearity chi-square statistics that never require any individual group's own parameter to be separately estimated — so a 0% or 100% group never causes a non-finite fit.

Reasoning: Why Two Conventions, and When to Use Each

Both methods are asymptotically equivalent for well-behaved data and, in practice, produce nearly identical chi-square statistics whenever every dose group has an observed response rate safely away from 0% or 100%. The deciding factor for which to use is therefore the data, not a matter of preference: whenever a dose group happens to show a genuinely extreme observed proportion — a common, unremarkable occurrence at the tails of a dose-response curve, especially with modest group sizes — the deviance method's validity statistics can become unstable or inflated, while Finney's method remains numerically well-behaved because it never requires that extreme group's own parameter to be fitted in isolation.

To make this actionable without requiring a user to understand the underlying numerical issue, the tool now also checks the raw data directly for this exact condition (any dose group where the number responding is exactly zero or exactly equal to the number of subjects tested) and, if found, displays an automatic note explaining the risk and recommending the Finney chi-square method — regardless of which validity method is currently selected. This detection is deliberately based on the observed proportions themselves, rather than on any numerical warning the underlying computation may or may not raise, since a broader class of technical warnings can otherwise fire even on well-behaved data with no extreme group present, which would be a false alarm.

Selecting Finney's method is recommended whenever the note appears, and is also useful for reconciling validity-test statistics with EP 5.3-compliant commercial software, which defaults to it. Where no extreme group is present, the tool's existing deviance-based default remains a fully valid, standard choice.

What Changes in the Output, and What Does Not

Affected:

• The Chi-Square (or F-value) and P-value columns of the three validity tests (Significance of Regression, Parallelism, Linearity)
• A new automatic diagnostic note (shown regardless of the selected method) when any dose group has an exactly 0% or 100% observed response

Not affected:

• The estimated relative potency for each test preparation
• The potency confidence interval
• The fitted regression coefficients (slope and intercepts)

Where to Find This Capability

On the Quantal Response Method tool's input form, a field labelled "Validity test method" offers a choice between "Likelihood-ratio deviance (this tool's default)" and "Finney's classical minimum chi-square (EP 5.3 default)". The automatic diagnostic note, when triggered, appears in the output alongside the validity results regardless of which method is selected.

Verification Against Published Reference Data

Both additions were checked against a published worked example that Finney himself used to illustrate the method — a probit-model assay of insulin potency measured by mouse convulsion response, at nine dose levels for the standard and five for the test preparation, drawn from Finney's own textbook. The standard preparation's lowest dose group (dose 3.4, n=33) has zero responders — a genuine 0% observation, and exactly the condition the new automatic note is designed to detect.

Source: D.J. Finney, "Statistical Method in Biological Assay," 3rd Edition (1978), Griffin, London, worked example on p. 376 — as reproduced in Stegmann Systems' PLA 3.0 published example report (Document-2077, "D.J. Finney, p. 376... Insulin, Mouse convulsion - with Suitability tests")

With the default (deviance-based) method active on this same dataset, the tool's automatic diagnostic note correctly appeared, identifying the standard preparation's dose-3.4 group as the source of risk and recommending the Finney chi-square method. With the Finney chi-square method selected, the note correctly did not appear, since that method is not subject to the underlying instability. A separate check confirmed the note does not appear on well-behaved datasets containing no extreme group, ruling out false alarms.

References

• European Pharmacopoeia (Ph. Eur.), Chapter 5.3 — "Statistical Analysis of Results of Biological Assays and Tests" (5th Edition, 2005, and successive editions).
• United States Pharmacopeia, General Chapter <111> — "Design and Analysis of Biological Assays".
• Finney, D.J. (1978). "Statistical Method in Biological Assay," 3rd Edition. Charles Griffin & Company, London. Worked example and data: p. 376.
• Stegmann Systems GmbH, PLA 3.0 published worked example (Document-2077) — https://www.bioassay.de/media/document/document-2077-o130_permanent_pla_307.pdf