= Tail-volume loss and ratio inflation in truncated expiration
\b[Tail-volume loss and ratio inflation in truncated expiration] connects unrecorded late gas to a biased <FEV1 to FVC ratio>. Suppose a valid first-second <volume> $A$ is retained, the recorded total is $M$, and additional uncollected <volume> is $L\ge0$. The complete ratio is $A/(M+L)$, while using $M$ as <FVC> gives $A/M$.
If independent information bounds $L\le L_{\max}$, then
$$
\frac{A}{M+L_{\max}}\le R_{\rm true}\le\frac AM.
$$
For an original example with $A=2.4$ L, $M=3$ L and $L_{\max}=1$ L, the complete ratio lies between 0.60 and 0.80 under this bound. This is a deterministic interval conditional on the tail bound, not automatically a statistical <confidence interval>.
<Bounding uncollected expiratory volume from terminal flow> adds a model. Let <volumetric flow rate> after the stop time be $Q(T+s)=q_Te^{-s/\tau}$. The missing <volume> is
$$
L=\int_0^\infty q_Te^{-s/\tau}\,ds=q_T\tau.
$$
If $\tau\le\tau_{\max}$, this gives $L\le q_T\tau_{\max}$. For stipulated terminal <volumetric flow rate> $q_T=0.050\,\mathrm{L\,s^{-1}}$ and $\tau_{\max}=4$ s, the bound is 0.20 L. With the same $A$ and $M$, the ratio is then between 0.75 and 0.80.
The bound extends to a positive sum of exponential tails: if each component has terminal <volumetric flow rate> $q_i$ and <time constant> at most $\tau_{\max}$, then $L=\sum_iq_i\tau_i\le\tau_{\max}\sum_iq_i$. Without a justified upper time scale, however, very small terminal <volumetric flow rate> can coexist with a large slowly emptying <volume>. A visually flat trace alone does not supply that missing assumption.
The relative inflation from using the truncated ratio is especially simple:
$$
\frac{A/M}{A/(M+L)}-1=\frac LM,
$$
for positive $A$. In the bounded 0.20 L example it is at most $0.20/3\approx6.67\%$. The result assumes termination after a valid first second; an invalid numerator needs another error model.
The primary spirometry standard evaluates end of forced expiration and distinguishes it from the validity of <FEV1>.<Standardization of Spirometry 2019 Update by Brian L. Graham and others>[End of forced expiration; Table 7]{ref} The calculations explain sensitivity to lost tail <volume> rather than replace the standard's actual maneuver criteria.
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