 Selection bias from a noisy denominator threshold

Selection bias from a noisy denominator threshold occurs when accepting measurements according to the same noisy central signal used for normalization changes the distribution of the reported ratios. For side signal $S$, measured central signal $C$, positive cutoff $c$ and acceptance event $A=\{C\ge c\}$, the sample mean over accepted shots targets $\mathbb{E}[S/C\mid A]$. It need not target the true-signal ratio or the unfiltered mean ratio. This is a consequence of conditioning, even when every physical pulse has the same true brightness.

For an original two-state readout model, let the true central signal be5, let measured $C$ equal4 or6 with equal probabilities, and let $S=0.02$ be constant. Both denominator errors have zero mean. The noiseless ratio is0.004, whereas the unfiltered mean measured ratio isA cutoff $c=5$ accepts only $C=6$, giving $\mathbb{E}[S/C\mid A]=1/300\approx 0.003333$ and acceptance probability1/2. This separates the unfiltered nonlinear ratio bias from the further change introduced by selection. Both denominators were already safely positive; the selection effect is not merely a division-by-zero problem.

The direction is not universal when numerator and denominator errors are related. Keep $C=4$ or6 but let $S=0.02+k(C-5)$. ThenFor $k=0.01$, the two observed side signals are0.01 and0.03, with ratios0.0025 and0.005. The same cutoff now gives0.005, above the same true-signal ratio0.004. For $k=0.004$, the observed signals are exactly proportional and every ratio is0.004, so selection leaves it unchanged. The noise dependence and measurement model determine whether thresholding shifts the accepted ratios upward, downward or not at all.

When a joint density and integrable accepted ratio exist, the dependence is explicit:The conditional numerator response cannot generally be replaced by its unconditional mean. Report the acceptance rule and accepted fraction alongside the shot-resolved pulse-contrast estimator; increasing the number of accepted shots reduces sampling scatter but does not undo the conditioning.

A physical motivation comes from a single-shot correlator that spatially encoded the nonlinear mixing signal and imaged it onto a filtered CCD detector.^[ref] If central pixels decide whether a trace is retained, later normalization uses data already conditioned by that choice. The two-state examples are original readout models, not measurements attributed to that instrument. Finite ratio moments under a positive denominator cutoff establish mathematical moment control under stated assumptions; they do not establish absence of selection bias.