= Response-range calibration of a bioassay equivalence margin
\b[Response-range calibration of a bioassay equivalence margin] chooses a curve-shape tolerance from the permitted change in <relative biological potency> over a specified response interval. It makes explicit why the same slope-ratio limit need not suit different response ranges. Here calibration means setting an assay-design tolerance, rather than <measurement calibration> of an instrument.
For an original local <dose-response relationship>, write $y_R=a_R+b_Rz$ and $y_T=a_T+b_Tz$, where $z$ is the natural logarithm of concentration and both slopes are positive. Equal-effect log-concentration separation is $\Delta z(y)=(y-a_R)/b_R-(y-a_T)/b_T$. Across a response interval of width $W$, its change is $W(1/b_R-1/b_T)$. Requiring the equal-effect concentration ratio to change by no more than a stipulated multiplicative tolerance $Q>1$ gives
$$
W\left|\frac1{b_R}-\frac1{b_T}\right|\le\log Q.
$$
Set $s=b_T/b_R$ and $\eta=b_R\log Q/W$. For $0\le\eta<1$, the resulting <bioassay parallelism equivalence margin> is
$$
\boxed{\frac1{1+\eta}\le s\le\frac1{1-\eta}}.
$$
If $\eta\ge1$, this particular constraint supplies no finite upper bound on $s$; the lower bound remains $1/(1+\eta)$. The intercepts affect the potency level but cancel from its drift across the interval.
An original design example takes $W=0.8$ response units, $b_R=0.4$ response units per natural-log concentration, and $Q=1.1$. Then $\eta\approx0.04766$, giving approximately $0.9545\le s\le1.0500$. Halving the response interval doubles $\eta$ and relaxes these slope bounds. Rescaling the response measurement multiplies $W$ and $b_R$ together and leaves $\eta$ unchanged: the result depends on the experimental interval, not the arbitrary response units.
An <equivalence test> must still account for uncertainty in the estimated slope ratio. These local linear bounds do not establish matching plateaus in a nonlinear <dose-response relationship>. In the primary rituximab-standard collaborative study, analysts assessed multiple curve parameters and obtained study-specific limits from coded duplicates; those limits were explicitly not routine laboratory validity criteria.<Report on a Collaborative Study for Proposed Candidate 1st International Standard for the biological activities of Rituximab by Sandra Prior and others>[WHO/BS/2017.2309, printed pp. 8 and 10–11]{ref} Select the response interval and tolerated potency drift before inspecting the test preparation's disagreement.
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