Bias Reduction in Logistic Regression with Estimated Variance Predictors

Abstract

We study the problem of modeling a response as a function of baseline covariates and a primary predictor of interest that is a noisy measurement of a subject-specific variance. The problem arises naturally in biostatistical joint models wherein the subjects’ primary endpoints are related to the features of subject-specific longitudinal risk processes or profiles. Often the longitudinal process features of interest are parameters of a longitudinal mean function. However, there is a relatively recent and growing interest in relating primary endpoints to longitudinal process variances. In the application motivating our work longitudinal processes consist of 30-day blood pressure trajectories measured between 91 and 120 days post dialysis therapy, with the primary endpoints being short-term mortality. Often the longitudinal risk processes are adequately characterized in terms of trends such as the slopes and intercepts identified with the subject-specific biomarker processes. Modeling of the trend lines results in subject-specific estimated intercepts and slopes, thus inducing a heteroscedastic measurement-error model structure where the estimated trend parameters play the role of measurements of the “true” subject-specific trend parameters that appear as predictors in the primary endpoint model. Our interest lies in models in which the residual variances of the longitudinal processes feed into the model for the primary endpoint. These subject-specific variance parameters are estimated in the course of trend-line fitting creating a measurement error model scenario where variances are predictors and mean squared errors are their noisy measurements. Background literature is reviewed and several methodological approaches for addressing the resulting errors-in-variances problem are studied.