Common-gain cancellation and flow-offset bias in expiratory ratios

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 givesThe 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 isFor small $b$, its sensitivity coefficient at zero offset is $(t_{1}B-TA)/(g{B}^2)$. 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 $-0.65\mathrm{s}{\mathrm{L}}^{-1}$. An offset $b=0.020\mathrm{L}{\mathrm{s}}^{-1}$ gives a first-order shift of $-0.013$; the exact ratio is $2.42/4.12\approx 0.58738$. 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.^[ref] The spirometry statement separately checks the acceptability of the constituent volumes.^[ref]