An estimated score method for missing and auxiliary covariates in two stage studies

Abstract

We consider the following type of two stage studies. First, a full scale study is conducted by observing an outcome Y, imperfect surrogate measure T of the unobserved covariate of interest X, and covariates U for all members of a cohort, where the cohort is a random sample from some infinite, underlying population. Second, a validation study is conducted, in which we ascertain the previously unmeasured covariate X for a subset of the subjects selected at random within each category/stratum of the cohort. Stratum membership can depend on Y, T and U from the full scale study. Several methods of analysis are now available that one can use to estimate the relationship between Y and X in two stage studies, but each method has potential limitations, especially when both Y and X are continuous. We present a method of analysis for two stage study data with continuous and/or discrete variables. The proposed method generalizes the mean score method of Reilly and Pepe (Biometrika 82 (1995) 299) in two ways. First, it uses an estimating equations approach to estimate mean scores. Second, it applies even when the outcome and independent variables are continuous, provided sampling of subjects for the validation study is random within categories. Despite its greater applicability, this new approach retains many of the consistency properties of the mean score method. One can use the method of analysis for effect estimation with many study designs when error in measurement or misclassification of independent variables has occurred. Like the mean score method, this method of analysis uses an estimated score function. We assume some “regularity conditions”, but make no distributional assumptions about the misclassified variable or the measurement error. A limited simulation study was performed to evaluate the proposed method and two examples are presented to illustrate the proposed method.