# Multiple Roots in Logistic Regression With Errors-in-Covariates Open Access

## Chen, Jian (2009)

Permanent URL: https://etd.library.emory.edu/concern/etds/db78tc26g?locale=en
Published

## Abstract

The unbiased estimating function method is a flexible approach to estimate and make
inferences on the parameters of interest. However, special problems arise when covariates
are measured with error.
Measurement errors arise in public health studies when some covariates are not mea-
sured precisely. We focus on the important case where the outcome is a binary variable and
the interest is in coefficients from a logistic regression model. Two widely used estimating
function methods for logistic regression with errors-in-covariates are the conditional score
(Stefanski & Carroll 1987) and the parametric-correction estimation procedure (Huang &
Wang 2001). The conditional score can have multiple-roots and not all of them are consis-
tent, whereas the parametric-correction estimation only generate consistent roots. On the
other hand, the conditional score in theory has an efficiency advantage in that its consis-
tent estimator is asymptotically locally efficient. Despite the multiple-roots problem, the
conditional approach is regarded as the standard method.
In this dissertation research, we aim to resolve the multiple-roots problem of the condi-
tional score in logistic regression with errors-in-covariates. We investigate the root behav-
iors of the conditional score in finite samples and demonstrate the existence and seriousness
of the problem posed by multiple roots, which have not been studied adequately in litera-
ture.
We propose two methods to achieve our research goal. In the first approach, we develop
a weighted-correction estimating function that only yields consistent estimators and com-
bine it with the conditional score using empirical likelihood. We prove that, asymptotically,
the proposed approach admits only consistent estimators and is locally efficient.
In the second approach, we construct objective functions based on the weighted-correction
estimating function and use them to distinguish among multiple roots from the conditional score.
In addition to developing the large sample theories of the proposed methods, we investi-
gate their finite-sample properties through an extensive simulation. The simulation studies
show that the proposed methods work well in finite samples and outperform existing meth-
ods in many situations. Finally, the proposed methods are applied to data presented in
Hammer et al. (1996) and Pan et al. (1990).

1 Introduction
1.1 Overview
1.2 The problems of existing methods
1.3 Objectives

2 Background
2.1 Functional methods for logistic regression with errors-in-covariates
2.1.1 Introduction
2.1.2 Logistic regression with errors-in-covariates
2.1.3 The conditional score
2.1.4 The parametric-correction estimation procedure
2.2 Multiple Roots
2.2.1 Introduction
2.2.2 Unbiased estimating functions and multiple roots
2.2.3 Choosing from multiple roots
2.2.4 Artificial likelihood functions
2.3 Empirical Likelihood
2.3.1 Overview
2.3.2 EL for estimating equations

3 Root Behaviors
3.1 Introduction
3.2 Root Behaviors in finite samples
3.3 Discussions

4 The combined estimation procedure
4.1 Introduction
4.2 The weighted-correction estimating function
4.3 The combined estimation procedure
4.4 Simulations
4.5 Real studies
4.6 Discussions
4.7 Appendix
4.7.1 Proofs
4.7.2 Asymptotic relative e_ciency
4.7.3 The modified Newton-Raphson procedures
4.7.4 Computations of Empirical Likelihood

5 Building Objective Functions
5.1 Introduction
5.2 Conditional quasi-likelihood
5.3 The objective functions
5.4 Distinguish among multiple roots
5.4.1 Simulations
5.4.2 A High Blood Pressure study
5.5 Discussions
5.6 Proofs

6 Summary and future work
6.1 Summary
6.2 Future work