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Motivated by the current COVID-19 health crisis, we consider data analysis for quantitative polymerase chain-reaction (qPCR) measurements. We derive a theoretical result specifying the conditions under which all qPCR amplification curves (including their plateau phases) are identical up to an affine transformation, i.e. a multiplicative factor and horizontal shift. We use this result to develop a data analysis procedure for determining when an amplification curve exhibits characteristics of a true signal. The main idea behind this approach is to invoke a criterion based on constrained optimization that assesses when a measurement signal can be mapped to a master reference curve. We demonstrate that this approach: (i) can decrease the fluorescence detection threshold by up to a decade; and (ii) simultaneously improve confidence in interpretations of late-cycle amplification curves. Moreover, we demonstrate that the master curve is transferable reference data that can harmonize analyses between different labs and across several years. Application to reverse-transcriptase qPCR measurements of a SARS-CoV-2 RNA construct points to the usefulness of this approach for improving confidence and reducing limits of detection in diagnostic testing of emerging diseases. [ABSTRACT FROM AUTHOR]