= Common-gain cancellation and flow-offset bias in expiratory ratios
\b[Common-gain cancellation and flow-offset bias in expiratory ratios] distinguishes an error that rescales both <volumes> equally from one that accumulates differently over time. Let true <FEV1> be $A$, true <FVC> be $B$, and their ratio be $R=A/B$. A shared positive <volume> gain $g$ cancels exactly: $(gA)/(gB)=R$. The result requires the same gain for both quantities, not merely the same instrument name.
A constant error $b$ in measured <volumetric flow rate> has another effect. If the first-second interval lasts $t_1$ and the full recording lasts $T$, integration gives
$$
R_m=\frac{gA+bt_1}{gB+bT}
=\frac{A+kt_1}{B+kT},\qquad k=\frac bg.
$$
The ratio alone identifies neither $b$ nor $g$ separately; only their relative combination enters this model. Nor does the positive duration-dependent error necessarily raise the ratio.
The exact difference is
$$
R_m-R=\frac{b(t_1B-TA)}{B(gB+bT)}.
$$
For small $b$, its <sensitivity coefficient> at zero offset is $(t_1B-TA)/(gB^2)$. If average early <volumetric flow rate> $A/t_1$ exceeds average whole-record <volumetric flow rate> $B/T$, a positive <volumetric flow rate> offset lowers the ratio. If the averages agree, that offset leaves the ratio unchanged in this model.
An original example uses $A=2.4$ L, $B=4$ L, $t_1=1$ s and $T=6$ s. The true ratio is 0.60. At $g=1$, the zero-offset sensitivity is $-0.65\,\mathrm{s\,L^{-1}}$. An offset $b=0.020\,\mathrm{L\,s^{-1}}$ gives a first-order shift of $-0.013$; the exact ratio is $2.42/4.12\approx0.58738$. A common 5% gain error without the offset would leave 0.60 unchanged.
Independent numerator and denominator gains would instead multiply $R$ by $g_A/g_B$. Clipping, early termination and different timing conventions are additional measurement effects, so gain cancellation is not a general guarantee of ratio accuracy. The GUM requires the full measurement model and shared-input effects to be retained.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, measurement sensitivity and correlated inputs]{ref} The spirometry statement separately checks the acceptability of the constituent <volumes>.<Standardization of Spirometry 2019 Update by Brian L. Graham and others>[Table 7, FEV1 and FVC criteria]{ref}
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