Hypothesis test of arbitrary parametric structure in a generalized additive model

Before applying a nonparametric model such as a generalized additive model (GAM), it is natural to ask whether a simpler parametric model suffices to capture the variation in the data. To address this fundamental question, we propose a new methodology named Test for Arbitrary Parametric Structure (TAPS), which provides estimation and inference tools to determine whether a parametric structure sufficiently describes a target function in a GAM. The key strategy of TAPS is to translate the test of parametric structure into the test of variance of the random effect, and the novelty of TAPS lies in showing how to construct this translation for an arbitrary parametric structure, including the linearity, piecewise linearity with slope changes, and linearity discontinuity with jumps. We illustrate the utility of TAPS across diverse scientific domains by using the UK Biobank data. Specifically, we applied TAPS to reveal widespread nonlinearity in polygenic risk score effects, though prediction improvement over the linear model was limited for most traits. Alternatively, we employed TAPS to identify the causal effects of retirement that change various health and lifestyle traits, using regression discontinuity and kink designs.