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Significance testing for canonical correlation analysis in high dimensions.

Ian W McKeague1, Xin Zhang2

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Summary

This study introduces a post-selection inference method for canonical correlation analysis to detect linear relationships among many variables. It offers a reliable way to test for these connections while accounting for variable selection, enhancing statistical accuracy.

Keywords:
Efficient one-step estimatorGreedy search algorithmLarge-scale testingPost-selection inference

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Area of Science:

  • Statistics
  • Multivariate Analysis
  • Machine Learning

Background:

  • Canonical correlation analysis (CCA) identifies linear relationships between variable sets.
  • Post-selection inference is crucial for accurate statistical testing after variable selection.
  • Large datasets pose challenges for traditional CCA due to the curse of dimensionality and selection bias.

Purpose of the Study:

  • To develop a statistically sound method for testing linear relationships in large variable sets using CCA.
  • To address the challenge of adjusting for variable subset selection in CCA.
  • To provide a computationally tractable omnibus test for global null hypotheses in high-dimensional data.

Main Methods:

  • Post-selection inference approach to CCA.
  • Construction of a stabilized one-step estimator for the Euclidean norm of canonical correlations.
  • Development of a greedy search algorithm for estimator computation.
  • Asymptotic normality and consistency analysis of the estimator.

Main Results:

  • The proposed estimator is consistent and asymptotically normal under specified conditions.
  • A computationally tractable omnibus test is developed for detecting subset linear relationships.
  • A confidence interval accounting for variable selection is derived.

Conclusions:

  • The method provides a robust framework for testing linear relationships in high-dimensional data with variable selection.
  • The developed techniques enhance the reliability of statistical inference in complex multivariate settings.
  • This approach offers a valuable tool for exploratory data analysis and hypothesis testing in large-scale datasets.