Prognosis-equivalent mapping of clinical measurements via survival analysis
A continuous clinical measurement recorded in fixed physical units may have different prognostic meaning across patients when its effect depends on a patient-level modifier. For example, the same tumor diameter may imply markedly different prognosis in an infant and an adult, because patient size can modify its prognostic effect. We formalize this problem through prognosis-equivalent mapping for time-to-event outcomes. The resulting estimand maps a measurement value observed at one modifier level to the value under a reference modifier level that yields the same conditional prognostic quantity. The basic operation is to equate a monotone conditional prognostic score and invert the reference-side curve. In observational data, however, treatment may be selected according to the modifier and the measurement, so a direct equate-and-invert procedure may reflect differences in treatment assignment as well as the prognostic meaning of the measurement itself. We therefore define, in addition to a direct approach, a policy-standardized mapping that standardizes treatment under a common policy using the g-formula. We also introduce an origin-referenced mapping that compares changes in prognostic score from a common anchor, thereby separating modifier-specific prognostic levels from modifier dependence in the measurement--prognosis relationship. Using Cox models, we develop linear and flexible inversion estimators with analytic and bootstrap confidence bands. Simulations evaluate finite-sample performance and illustrate how treatment standardization and origin referencing clarify what is being mapped.