Robust Permutation Tests for Penalized Splines

Penalized splines are frequently used in applied research for understanding functional relationships between variables. In most applications, statistical inference for penalized splines is conducted using the random effects or Bayesian interpretation of a smoothing spline. These interpretations can be used to assess the uncertainty of the fitted values and the estimated component functions. However, statistical tests about the nature of the function are more difficult, because such tests often involve testing a null hypothesis that a variance component is equal to zero. Furthermore, valid statistical inference using the random effects or Bayesian interpretation depends on the validity of the utilized parametric assumptions. To overcome these limitations, I propose a flexible and robust permutation testing framework for inference with penalized splines. The proposed approach can be used to test omnibus hypotheses about functional relationships, as well as more flexible hypotheses about conditional relationships. I establish the conditions under which the methods will produce exact results, as well as the asymptotic behavior of the various permutation tests. Additionally, I present extensive simulation results to demonstrate the robustness and superiority of the proposed approach compared to commonly used methods.