Consistent Functional Methods for Logistic Regression With Errors in Covariates

Abstract

We propose consistent functional methods for logistic regression in which some covariates are not accurately ascertainable. Among existing methods for generalized linear models, the conditional-score approach to normal errors does not guarantee the convergence of its estimators, and the corrected-score method generally is not applicable to the logistic-regression score function. In this article, after constructing a correction-amenable estimation procedure with the true covariates, we formulate parametric- and nonparametric-correction estimation procedures in the presence of additive errors in covariates. The former procedure accommodates the situation with known (but not necessarily normal) error distribution, whereas the latter further relieves this distributional assumption requirement, given that additional replicated mismeasured covariates or instrumental variables are available. Large-sample theory is developed; the proposed estimators are consistent and asymptotically normal. We investigate their asymptotic relative efficiency and, through simulations, examine their finite-sample properties. Application to an acquired immunodeficiency syndrome study is provided to illustrate the proposed methods.

KEY WORDS:

• Conditional score
• Corrected score
• Estimating equation
• Generalized linear model
• Instrumental Variable
• Measurement error
• Nonparametric correction
• Parametric correction
• Replication data