Evaluating the Assumption of Linearity: Testing the Adequacy of Parametric Models using Penalized Splines

Abstract

Ordinary least squares (OLS) linear regression is arguably the most widely utilized statistical method in research, and as the name implies uses a linear surface to model the relationship between the mean response and covariates. When the relationship between the response and covariate is linear or nearly linear, OLS regression is a robust an efficient method, providing a concise summary of the data. However, when the functional form deviates significantly from linearity, the linear model is insufficient and may miss important patterns in the data. In the case of nonlinearity, smoothing represents an appealing alternative that allow the data to “speak for itself.” One of the most important applications of semiparametric and nonparametric regression models is testing whether parametric regression models are well-specified. Penalized splines or P-splines represent a robust and flexible framework for evaluating general departures from linearity. The primary aim of this dissertation was to comparatively evaluate several P-spline-based methods for testing the assumption of linearity between a response and a single predictor. The smoothing-based methods assessed in this dissertation include Wald-type tests, approximate likelihood ratio tests (LRTs), and an exact restricted likelihood ratio test (RLRT). The exact RLRT showed the most robust and reliable statistical performance. Its strong performance is attributed to its use of simulation and spectral decomposition to derive an approximation of the exact finite-sample null distribution of the RLRT statistic. Although this test is readily implemented via the exactRLRT() function in the R package RLRsim, it is currently restricted to Gaussian models and hypothesis tests involving a single variance component. Future work should extend these methods to accommodate non-Gaussian continuous outcomes (e.g., log-normal, gamma). Future research should also focus on developing robust methods for multiple variance components, especially for group-specific smooths when the goal is to compare functional forms.