Kernel-Based Partial Permutation Test for Detecting Heterogeneous Functional Relationship

Xinran Li, Bo Jiang, Jun S. Liu

Research output: Contribution to journalArticlepeer-review

Abstract

We propose a kernel-based partial permutation test for checking the equality of functional relationship between response and covariates among different groups. The main idea, which is intuitive and easy to implement, is to keep the projections of the response vector Y on leading principle components of a kernel matrix fixed and permute Y’s projections on the remaining principle components. The proposed test allows for different choices of kernels, corresponding to different classes of functions under the null hypothesis. First, using linear or polynomial kernels, our partial permutation tests are exactly valid in finite samples for linear or polynomial regression models with Gaussian noise; similar results straightforwardly extend to kernels with finite feature spaces. Second, by allowing the kernel feature space to diverge with the sample size, the test can be large-sample valid for a wider class of functions. Third, for general kernels with possibly infinite-dimensional feature space, the partial permutation test is exactly valid when the covariates are exactly balanced across all groups, or asymptotically valid when the underlying function follows certain regularized Gaussian processes. We further suggest test statistics using likelihood ratio between two (nested) Gaussian process regression models, and propose computationally efficient algorithms utilizing the EM algorithm and Newton’s method, where the latter also involves Fisher scoring and quadratic programming and is particularly useful when EM suffers from slow convergence. Extensions to correlated and non-Gaussian noises have also been investigated theoretically or numerically. Furthermore, the test can be extended to use multiple kernels together and can thus enjoy properties from each kernel. Both simulation study and application illustrate the properties of the proposed test.

Original languageEnglish (US)
Pages (from-to)1429-1447
Number of pages19
JournalJournal of the American Statistical Association
Volume118
Issue number542
Early online dateJan 5 2022
DOIs
StatePublished - 2023

Keywords

  • Gaussian kernel
  • Gaussian process regression
  • Permutation test
  • Polynomial kernel
  • Regression discontinuity design

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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