FEV1 to FVC ratio ($R={\mathrm{FEV}}_1/\mathrm{FVC}$)

The FEV1 to FVC ratio is the dimensionless fraction obtained by dividing forced expiratory volume in one second by forced vital capacity:Under an ideal complete-expiration model with nonnegative outward flow, the first-second expelled volume cannot exceed the total, so $0\le R\le 1$. The ratio describes early emptying relative to the measured total; it is not itself a volume or an instantaneous volumetric flow rate.^[ref]

An original comparison makes the information loss clear. FEV1 of 3 L with FVC of 4 L gives the same ratio, 0.75, as 1.5 L with 2 L. Equal ratios do not imply equal absolute lung size, equal total expired volume or equal complete flow histories. Expressing 0.75 as 75% changes the display, not the quantity.

Common-gain cancellation and flow-offset bias in expiratory ratios distinguishes two instrument errors. If both measured volumes are multiplied by the same positive gain $g$, their ratio is unchanged. A constant flow offset $b$ instead accumulates for different durations. With true first-second volume $A$, total volume $B$, first-second duration $t_1$ and complete observation duration $T$,The cancellation of a common gain does not cancel the duration-dependent offset.

For stipulated $A=2.4$ L, $B=4$ L, $t_1=1$ s and $T=6$ s, the true ratio is 0.60. A common gain of 1.05 still gives 0.60. With unit gain and $b=0.020\mathrm{L}{\mathrm{s}}^{-1}$, however, the measured ratio becomes $2.42/4.12\approx 0.587$. These are constructed measurement errors, not observed patient data.

Early termination bias in forced vital capacity acts differently again: losing late expired volume can lower the denominator while leaving a valid first second intact, increasing the ratio. This is why a ratio needs valid constituent measurements. The technical statement treats FEV1 and FVC acceptability separately.^[ref] A clinical interpretation additionally requires the appropriate reference framework; the arithmetic alone does not identify a disease mechanism.

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]