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Inference for change-plane regression

Chaeryon Kang, Hunyong Cho, Rui Song, Moulinath Banerjee, Eric B. Laber, Michael R. Kosorok

TL;DR

The proposed methods to latent subgroup identification in precision medicine using the ACTG175 AIDS study data are illustrated and a parametric bootstrap procedure for inference is provided.

Abstract

A key challenge in analyzing the behavior of change-plane estimators is that the objective function has multiple minimizers. Two estimators are proposed to deal with this non-uniqueness. For each estimator, an n-rate of convergence is established, and the limiting distribution is derived. Based on these results, we provide a parametric bootstrap procedure for inference. The validity of our theoretical results and the finite sample performance of the bootstrap are demonstrated through simulation experiments. We illustrate the proposed methods to latent subgroup identification in precision medicine using the ACTG175 AIDS study data.

Inference for change-plane regression

TL;DR

The proposed methods to latent subgroup identification in precision medicine using the ACTG175 AIDS study data are illustrated and a parametric bootstrap procedure for inference is provided.

Abstract

A key challenge in analyzing the behavior of change-plane estimators is that the objective function has multiple minimizers. Two estimators are proposed to deal with this non-uniqueness. For each estimator, an n-rate of convergence is established, and the limiting distribution is derived. Based on these results, we provide a parametric bootstrap procedure for inference. The validity of our theoretical results and the finite sample performance of the bootstrap are demonstrated through simulation experiments. We illustrate the proposed methods to latent subgroup identification in precision medicine using the ACTG175 AIDS study data.
Paper Structure (42 sections, 22 theorems, 136 equations, 10 figures, 5 tables, 6 algorithms)

This paper contains 42 sections, 22 theorems, 136 equations, 10 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Data and model
  3. Estimation
  4. Consistency and rates of convergence
  5. Weak convergence
  6. Some preliminary results for proving Theorems \ref{['convergence-meanmid']} and \ref{['convergence-modemid']}
  7. Proof of Theorems \ref{['convergence-meanmid']} and \ref{['convergence-modemid']}
  8. Convergence on compact sets---Proof of Theorems \ref{['compact.convergence']}
  9. Inference
  10. Numerical estimation
  11. A uniform-search approach
  12. Mixed integer programming approaches
  13. Simulation study
  14. Application to the ACTG175 AIDS study
  15. Discussion
  16. ...and 27 more sections

Key Result

Theorem 4.1

(consistency) Let $\tilde{\theta}_n$ be any sequence in $K_n^{\ast}$ satisfying $M_n(\tilde{\theta}_n)\leq M_n(\theta_0)$ for all $n$ large enough almost surely. Under Conditions C1, C2, C3, C4 and C5, $\tilde{\theta}_n\rightarrow\theta_0$ almost surely.

Figures (10)

  • Figure S1: Rate of convergence results for mean-argmin estimators for Model 2. Both standard errors ($y$-axis) and sample sizes ($x$-axis) are presented on the log scale (with base 2). The lines and the annotated numbers are the least squares regression, and the corresponding slope estimates represent the exponents of the rate of convergence. M1, Model 1; M2, Model 2; M3, Model 3
  • Figure S2: The mean-argmin estimated and limiting CDFs for Model 2. The random coefficients are $\varsigma_1 = (-0.63, 0.40, 0.15, -0.66, 0.89, 0.89, -0.74)'$ and $\varsigma_2 = ( 0.67, -0.06, 0.10, 0.11, -0.52, 0.52, -0.64)'$ for Model 2.
  • Figure S3: Partial of the unit circle for the rate of convergence. $\theta$ is the angle between $\omega_1$ and $\omega_2$ which should be small.
  • Figure S4: Rate of convergence results for mean-argmin estimators (based on the CDA-MIP algorithm). Both standard errors ($y$-axis) and sample sizes ($x$-axis) are presented on the log scale (with base 2). The lines and the annotated numbers are the least squares regression and the corresponding slope estimates representing the exponents of the rate of convergence. M1, Model 1; M2, Model 2; M3, Model 3
  • Figure S5: The mean-argmin estimated and limiting CDFs for Models 1 and 2. The random coefficients are $\varsigma_1 = (-0.47, -0.26, 0.15, 0.82, -0.60, 0.80)'$, $\varsigma_2 = (0.89, 0.32, 0.26, -0.88, -0.59, -0.65)'$ for Model 1 and $\varsigma_1 = (-0.63, 0.40, 0.15, -0.66, 0.89, 0.89, -0.74)'$, $\varsigma_2 = ( 0.67, -0.06, 0.10, 0.11, -0.52, 0.52, -0.64)'$ for Model 2.
  • ...and 5 more figures

Theorems & Definitions (41)

  • Remark 2.1
  • Theorem 4.1
  • Theorem 4.2
  • proof : Proof of Theorem \ref{['theorem:rateofconvergence1']}
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Lemma 5.4
  • ...and 31 more