Deep non-parametric logistic model with case-control data and external summary information

TL;DR

This paper introduces a deep non-parametric logistic model for case-control data that incorporates external summary information, providing a two-step estimation process with theoretical error bounds and practical validation.

Contribution

It proposes a novel two-step estimation method combining external data with deep neural networks for non-parametric logistic regression in case-control studies.

Findings

The estimator achieves the optimal convergence rate.

Simulation studies validate the theoretical error bounds.

Real data example demonstrates practical applicability.

Abstract

The case-control sampling design serves as a pivotal strategy in mitigating the imbalanced structure observed in binary data. We consider the estimation of a non-parametric logistic model with the case-control data supplemented by external summary information. The incorporation of external summary information ensures the identifiability of the model. We propose a two-step estimation procedure. In the first step, the external information is utilized to estimate the marginal case proportion. In the second step, the estimated proportion is used to construct a weighted objective function for parameter training. A deep neural network architecture is employed for functional approximation. We further derive the non-asymptotic error bound of the proposed estimator. Following this the convergence rate is obtained and is shown to reach the optimal speed of the non-parametric regression estimation. Simulation studies are conducted to evaluate the theoretical findings of the proposed method. A real data example is analyzed for illustration.