The Debiased Score Test: Hunt-and-test for Semiparametric Hypotheses

2026-07-30

AI Summary

This study addresses goodness-of-fit testing and effect modifier identification in semiparametric regression models—such as generalized additive models and partially linear models—by proposing a debiased score test. The method innovatively integrates a hunt-and-test strategy with debiasing correction: it first employs machine learning to identify sensitive directions in the empirical score function, then applies weighted least squares projection to remove bias, and finally tests whether the score along the selected direction is zero using an independent data split. Under relatively mild regularity conditions, the procedure effectively controls Type I error while enhancing statistical power. Extensive simulations and real-data analyses demonstrate its superior performance, with successful applications in detecting effect modification in HIV clinical trials and assessing additivity in insurance claim data. The method is implemented in the R package dScoreTest.

Abstract

The parametric score test assesses a hypothesis through derivatives of the log-likelihood, whose expectation vanishes under the null. When the parameter of interest is a regression function identified as a risk minimiser, we extend this idea to test whether it belongs to a given linear function class. This yields goodness-of-fit tests for common semiparametric regression models, including generalised additive and partially linear models. Suitably formulated, the framework also detects effect modifiers in observational studies. We propose a hunt-and-test strategy that splits the data into two: on one part, after fitting the null model, machine learning is used to identify a promising direction in the empirical scores; on the other, we test whether the score vanishes in that direction. To account for error in estimating the null model, we apply a debiasing correction based on a weighted least squares projection. We establish Type I error control under relatively mild conditions and show the test has power whenever the hunted direction is correlated with the true score. Simulations and real-data examples demonstrate favourable performance, including identifying effect modifiers in an HIV clinical trial and assessing an additive model for insurance claims. The methodology is implemented in the R package dScoreTest.

Problem

• semiparametric regression
• goodness-of-fit test
• effect modification
• score test
• hypothesis testing

Innovation

• Debiased Score Test
• hunt-and-test
• semiparametric hypothesis testing
• effect modification
• goodness-of-fit