Propagating Assay Error Into Occupancy Estimates

Why Assay Error Matters for Occupancy

An occupancy estimate is calculated from measurements, not observed directly. A small error in a tracer signal may have little effect when the reference signal is precise and well separated from nonspecific binding. The same error can have a large effect when the corrected reference is small or uncertain. Reporting an occupancy point estimate without accounting for these sources of uncertainty can therefore make weak evidence appear precise.

Consider an assay with a tracer signal measured under a treatment condition and a reference condition. The reference represents the signal expected without drug displacement. In each condition, some measured signal is nonspecific rather than target-related. Subtracting nonspecific binding gives the estimated specific signals; comparing those specific signals gives the estimated fraction of tracer binding remaining. Occupancy is one minus that fraction.

The ratio is a measurement model, not a guarantee that the assay is valid. It assumes that the signals are comparable across conditions, that subtracting \(N\) isolates the relevant specific signal, and that the reference is appropriate for the treated measurement. The calculations below quantify uncertainty conditional on those assumptions; they do not establish that the assumptions hold.

Propagating Uncertainty Through the Corrected Ratio

Suppose \(S\), \(R\), and \(N\) are estimated from assay data, with estimated standard errors and a covariance matrix \(\Sigma\). The first-order delta method approximates the variance of a smooth function of these measurements by multiplying their covariance matrix by the gradient of that function. For \(q=(S-N)/(R-N)\), the gradient, in the order \(S,R,N\), is:

The corresponding approximate variance is \(\widehat{\mathrm{Var}}(q)=\nabla q^{\mathsf T}\Sigma\nabla q\), evaluated at the measured or estimated values. Because \(O=1-q\), its gradient is \(-\nabla q\), and its approximate variance is the same. This calculation includes uncertainty from all three measurements and any covariance between them. If the measurements are independent, \(\Sigma\) is diagonal and the variance reduces to the sum of each squared derivative multiplied by the corresponding measurement variance.

The derivative with respect to \(N\) is especially informative: nonspecific signal affects both the numerator and denominator. Its uncertainty must not be counted as though it affected only the treated signal. If the same background estimate is subtracted from both conditions, its influence may partly cancel or may amplify uncertainty, depending on the observed signals and covariance structure.

Confidence Intervals on the Ratio Scale

A symmetric interval around \(q\) may be a poor approximation when the ratio is positive and imprecise: it can include negative ratios or fail to reflect the ratio’s asymmetry. If \(q>0\), one alternative is to work with \(\log q\). When the estimated log-ratio is approximately normal with standard error \(SE(\log q)\), a nominal 95% interval for \(q\) is formed on the log scale and exponentiated. The occupancy interval must then reverse the ratio endpoints because occupancy decreases as \(q\) increases.

This approach keeps the interval for \(q\) positive, but does not ensure that \(q\leq1\) or that occupancy lies between zero and one. The normal approximation on the log scale also needs adequate information and a well-behaved measurement model. A narrow interval does not compensate for a biased reference, an inappropriate nonspecific estimate, or other violations of the assay assumptions.

How Reference and Nonspecific Uncertainty Change the Result

The reference measurement can be especially influential because it is the denominator of \(q\). As \(R-N\) becomes small, the derivative with respect to \(S\), \(1/(R-N)\), grows in magnitude, as does the derivative with respect to \(R\) when the ratio is not near zero. Consequently, even modest measurement errors can produce a large standard error. If \(R-N\) is close to zero, the ratio may be highly unstable and a first-order delta-method interval may be unreliable.

Correlation also matters. For instance, if \(S\) and \(R\) share a calibration measurement, their errors may move together. The variance formula then includes covariance terms; treating the estimates as independent may overstate or understate uncertainty. The same principle applies if \(N\) is estimated from data that overlap with the treated or reference measurements. Use the covariance structure implied by how the assay data were collected and analyzed, rather than assuming independence for convenience.

Assumptions, Diagnostics, and Alternatives

The delta method is most defensible when the signal estimators are approximately unbiased, their covariance is estimated adequately, and the occupancy function is smooth in the region supported by the data. For the corrected-ratio calculation, the denominator \(R-N\) must be safely separated from zero. The normal approximation for the estimate must also be reasonable; strong skewness, small replicate counts, or a ratio near a boundary can make a symmetric interval inaccurate. For a log-ratio interval, the ratio must be positive and the sampling distribution of \(\log q\) must be adequately approximated by a normal distribution.

Useful checks include inspecting replicate variability for each signal, checking that corrected reference signal is well above zero relative to its standard error, and calculating intervals rather than relying on point estimates alone. Compare the interval with the physical range \(0\leq O\leq1\). An interval far outside that range is a warning about precision or approximation, not evidence that occupancy can take impossible values. Also verify that the covariance calculation matches the assay design and that uncertainty in calibration and background estimation has not been omitted.

Several alternatives can be preferable in difficult cases. A parametric bootstrap simulates complete assay measurements using an explicit error model, recalculates occupancy for each simulated data set, and uses the resulting distribution to form an interval. It can represent skewness and nonlinear propagation, but depends on the simulation model. Profile-likelihood intervals can be useful when the ratio likelihood is asymmetric or a denominator is uncertain; they require a specified likelihood and suitable parameter constraints. Bayesian intervals can incorporate prior information and constraints such as occupancy between zero and one, but their results depend on the likelihood and priors. None of these methods repairs an inappropriate binding or measurement model.

The practical value of uncertainty propagation is not merely a standard error attached to a percentage. It shows which measurements drive the uncertainty, whether the assay can distinguish plausible occupancy levels, and whether the reported interval is compatible with the model and physical interpretation. If the reference or nonspecific signal dominates the uncertainty, improving those measurements may be more useful than increasing precision in the treated tracer signal alone.