TY - JOUR
T1 - Kernel-Based Partial Permutation Test for Detecting Heterogeneous Functional Relationship
AU - Li, Xinran
AU - Jiang, Bo
AU - Liu, Jun S.
N1 - This research is partly supported by the NSF grant DMS-1712714. We thank the Associate Editor and two reviewers for constructive comments. The views expressed herein are the authors’ alone and are not necessarily the views of Two Sigma Investments LP, or any of its affiliates.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Gaussian kernel
KW - Gaussian process regression
KW - Permutation test
KW - Polynomial kernel
KW - Regression discontinuity design
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U2 - 10.1080/01621459.2021.2000867
DO - 10.1080/01621459.2021.2000867
M3 - Article
AN - SCOPUS:85122404629
SN - 0162-1459
VL - 118
SP - 1429
EP - 1447
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 542
ER -