Statistical Methods for Biomarker Discovery and Inference on Causal Functionals

Abstract

This work addresses two statistical problems motivated by precision medicine: constructing clinically relevant decision rules and formally assessing heterogeneity in treatment response. The first project introduces a clinically motivated objective for cancer screening and constructs an optimal biomarker-based decision rule that balances statistical optimality with real-world implementability. The second and third projects focus on formal testing of homogeneity, with the second developing a general framework and the third focusing specifically on treatment effect heterogeneity under continuous treatment or exposure. The second project develops a nonparametric framework for testing whether a statistical parameter defined through conditional distributions is constant across strata defined by the conditioning variable, connecting our method to norm-based tests on function-valued parameters. Unlike many such tests, ours has a tractable limiting null distribution. The third project addresses inference for effect heterogeneity under continuous treatment, an area where flexible estimation methods exist but formal testing has received little attention. We characterize heterogeneity through a local parameter capturing how the dose-response curve varies with covariates and propose a Wald-type test based on the Hilbert norm of this parameter's projection onto a reproducing kernel Hilbert space. Using recent advances in semiparametric efficiency theory for Hilbert-valued objects, we construct a one-step estimator and establish the test's asymptotic validity under standard regularity conditions.