Abstract
<title>Abstract</title> <p>Background. Exogenous spike-in controls are often used in calibrated Ct-scale workflows to correct sample-processing variation. When a control does not track the target perfectly, unit-coefficient subtraction can add more error than it removes. Although the limitations of imperfect controls are recognised, an explicit variance-component criterion separating beneficial from harmful ΔCt normalisation has not been established for Ct-scale quantification. Results. We modelled target and control threshold cycles as sharing a processing perturbation while retaining channel-specific variability and measurement error. Under this model, ΔCt normalisation reduces mean squared error relative to a target-only estimator if and only if σs² > σc² + τC²/β². The optimal control coefficient is θ* = σs²/(σs² + σc² + τC²/β²), the fraction of the control-channel variance attributable to shared-processing variation. Four estimators were compared across five representative regimes using repeated five-fold cross-validation, and the benefit boundary was evaluated over 256 parameter combinations, verifying the analytic result and quantifying finite-sample trade-offs. Misapplied unit-coefficient normalisation increased root-mean-square error by up to 291%, whereas the largest reduction was 73%. Across three sample-size scenarios, estimating the control coefficient generally outperformed fixing it at one, except at small sample sizes when θ* was already close to unity. The qualitative recommendation was preserved under seven prespecified departures from the baseline model, including unequal amplification efficiency, heavy-tailed errors, batch effects, variable spike-in input and censoring. In simulated validation designs of 24 processed aliquots, the shared processing component and the decision contrast were both recovered with little bias, and the recommendation was correct in 100.0% and 99.6% of studies at the favourable and unfavourable settings with no directionally incorrect call in 1000 studies; at a setting 0.08 standard deviations from the boundary, the interval was indeterminate in 89.6%, as it should be. Conclusions. The criterion can be evaluated in a replicate validation design that separates shared processing, control-specific and measurement variance. When processed calibrators are available and the target–control coupling is unknown in advance, estimating the control coefficient is a more robust default than imposing unit-coefficient ΔCt normalisation. The result applies to calibrated Ct-scale workflows; low-copy and sequencing-count data require separate observation models.</p>