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Statistical Inference in High-dimensional Poisson Regression with Applications to Mediation Analysis

Prabrisha Rakshit, Zijian Guo

Abstract

Large-scale datasets with count outcome variables are widely present in various applications, and the Poisson regression model is among the most popular models for handling count outcomes. This paper considers the high-dimensional sparse Poisson regression model and proposes bias-corrected estimators for both linear and quadratic transformations of high-dimensional regression vectors. We establish the asymptotic normality of the estimators, construct asymptotically valid confidence intervals, and conduct related hypothesis testing. We apply the devised methodology to high-dimensional mediation analysis with count outcome, with particular application of testing for the existence of interaction between the treatment variable and high-dimensional mediators. We demonstrate the proposed methods through extensive simulation studies and application to real-world epigenetic data.

Statistical Inference in High-dimensional Poisson Regression with Applications to Mediation Analysis

Abstract

Large-scale datasets with count outcome variables are widely present in various applications, and the Poisson regression model is among the most popular models for handling count outcomes. This paper considers the high-dimensional sparse Poisson regression model and proposes bias-corrected estimators for both linear and quadratic transformations of high-dimensional regression vectors. We establish the asymptotic normality of the estimators, construct asymptotically valid confidence intervals, and conduct related hypothesis testing. We apply the devised methodology to high-dimensional mediation analysis with count outcome, with particular application of testing for the existence of interaction between the treatment variable and high-dimensional mediators. We demonstrate the proposed methods through extensive simulation studies and application to real-world epigenetic data.

Paper Structure

This paper contains 25 sections, 18 theorems, 120 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Main Results and Contributions
  3. Related Work
  4. Organization and Notation
  5. Inference for Linear and Quadratic Functionals
  6. Inference for Linear Functional
  7. Inference for Quadratic Functional
  8. Theoretical Properties
  9. Guarantees for Inference for TEXT
  10. Guarantees for Inference for TEXT
  11. Application to Mediation Analysis
  12. Numerical Studies
  13. Linear Functional
  14. Group Significance
  15. Mediation Analysis
  16. ...and 10 more sections

Key Result

Proposition 1

Suppose that Condition (C1) holds. Then, if we choose $\lambda_0 := \left\|\frac{1}{n}\sum_{i=1}^{n}\epsilon_iX_{i\cdot}\right\|_{\infty} \asymp \sqrt{\frac{\log p}{n}}$ and assume that $\max_{i,j}\left|X_{ij}\right|k \lambda_0 \leq c$ for some constant $c>0$, then with probability greater than $1- where $S$ denotes the support of $\beta$ and $C>0$ is a positive constant.

Theorems & Definitions (18)

  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Proposition 2
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • ...and 8 more