Estimation of the disease-specific diagnostic marker distribution under verification bias

We consider the estimation of the parameters indexing a parametric model for the conditional distribution of a diagnostic marker given covariates and disease status. Such models are useful for the evaluation of whether and to what extent a marker's ability to accurately detect or discard disease depends on patient characteristics. A frequent problem that complicates the estimation of the model parameters is that estimation must be conducted from observational studies. Often, in such studies not all patients undergo the gold standard assessment of disease. Furthermore, the decision as to whether a patient undergoes verification is not controlled by study design. In such scenarios, maximum likelihood estimators based on subjects with observed disease status are generally biased. In this paper, we propose estimators for the model parameters that adjust for selection to verification that may depend on measured patient characteristics and additionally adjust for an assumed degree of residual association. Such estimators may be used as part of a sensitivity analysis for plausible degrees of residual association. We describe a doubly robust estimator that has the attractive feature of being consistent if either a model for the probability of selection to verification or a model for the probability of disease among the verified subjects (but not necessarily both) is correct.