Additional Capabilities for Slope Ratio Bioassay

· 4 min read

On this page
  1. Bioassay — Slope Ratio Method
  2. What Was Added
  3. Regulatory and Statistical Background
  4. The Mathematics
  5. Reasoning: Why Two Conventions, and When to Use Each
  6. What Changes in the Output, and What Does Not
    1. Affected:
    2. Not affected:
  7. Where to Find This Capability
  8. Verification Against Published Reference Data
  9. References

Bioassay — Slope Ratio Method

Capability added: Optional Potency Confidence Interval Error Term ("Regression Residual" vs. "Pure Error")

 

What Was Added

The Slope Ratio Method gains the same optional confidence-interval convention already added to the Parallel Line Method: a choice of which variance estimate feeds the potency confidence interval. As with Parallel Line, this is opt-in, changes nothing by default, and never affects the estimated potency itself — only the width of its confidence interval.

Regulatory and Statistical Background

A slope-ratio assay fits a straight line, on the dose scale itself (not log-dose), to each preparation's response, constrained so that all preparations share one common intercept. Before a potency estimate is trusted, the assay must pass validity checks confirming the dose-response relationship is significant, that the preparations genuinely share a common intercept, and that each preparation's relationship is linear over the tested range.

As with the Parallel Line Method, both European Pharmacopoeia (Ph. Eur.), Chapter 5.3 — "Statistical Analysis of Results of Biological Assays and Tests" and United States Pharmacopeia, General Chapter <111> — "Design and Analysis of Biological Assays" use Fieller's theorem to build the potency confidence interval, and the two conventions diverge only in which variance estimate is supplied to that theorem.

The Mathematics

Convention A — "Regression Residual" (this tool's existing default). The variance left over after fitting the common-intercept model, pooling in any (hopefully negligible) failure of the preparations to actually share an intercept, and any (hopefully negligible) non-linearity:

MS(regression residual) = SS(residual after common-intercept fit) / df(residual after common-intercept fit)

Convention B — "Pure Error" (the new option). The variance between replicate readings at the exact same preparation and dose only — the same "Residual (pure) error" already used as the denominator for the assay's validity F-tests:

MS(pure error) = SS(pure error) / df(pure error), summed over every preparation x dose cell's own replicate spread

The chosen mean square and its degrees of freedom feed the same, unchanged Fieller confidence interval calculation used for every relative-potency ratio in this method.

Reasoning: Why Two Conventions, and When to Use Each

The same reasoning applies here as for the Parallel Line Method: for a genuinely valid assay, both conventions should give similar results, since a valid assay by definition has little non-common-intercept or non-linearity variation for the pure-error convention to exclude. Selecting "Pure Error" is primarily useful for reconciling results with EP 5.3-compliant commercial software or with a legacy validated method built on that convention.

What Changes in the Output, and What Does Not

Affected:

  • Lower CL and Upper CL (potency confidence limits) for each test preparation
  • Fieller's g statistic
  • A new "CI Error Term" column showing which convention and degrees of freedom were used

Not affected:

  • The estimated relative potency for each test preparation
  • The separate-regression and common-intercept regression coefficient tables, and the full ANOVA table
  • The validity test results (Significance of Regression, Intercept/Common-Blank-Response, Linearity)

Where to Find This Capability

On the Slope Ratio Method tool's input form, a field labelled "Potency CI error term" offers the same two choices as Parallel Line: "Regression residual (this tool's default)" and "Pure error — within-cell replicate variance only (EP 5.3 default)". The pure-error option requires at least one preparation/dose combination with replicate observations.

sr-add-option-bioassay


Verification Against Published Reference Data

This option was verified against a separate published worked example: a completely randomised (0,4,4,4)-design slope-ratio assay of influenza vaccine potency, comparing a standard against two test preparations, at four common dose levels with two replicates each (24 observations total, no zero-dose blank in this particular design).

Source: European Pharmacopoeia, 5th Edition (2005), Chapter 5.3, worked example 5.2.2 — as reproduced in Stegmann Systems' PLA 3.0 published example report (Document-3580, "PLA_3.0.7_Example_Slope-Ratio_Quantitative_Response_Assay_Standard_Report.pdf")


Preparation

Metric

Published (PLA/EP 5.3)

Computed (Pure Error option)

T

Potency ratio (%)

95.280

95.2801

T

95% CI

89.121 – 101.807

89.1207 – 101.8072

U

Potency ratio (%)

64.863

64.8632

U

95% CI

59.028 – 70.725

59.0277 – 70.7252

 

The full ANOVA decomposition also matched exactly, including a detail specific to this method: this tool splits the published report's single "Regression (df=3)" row into two more granular rows — "Regression" (the pooled dose effect, df=1) and "Preparations (Slope Differences)" (df=2) — which, summed together, equal the published Regression row's sum of squares exactly (1087.6652 in both cases). The "Intercept" row matches the published "non-Similarity" row exactly, and the "Non-linearity" row — together with its optional per-preparation breakdown — matches the published "non-Linearity (LoF)" row and its per-preparation breakdown exactly. Both regression models (separate per-preparation, and common-intercept) also matched to the decimal.

References