A semiparametric mixture approach to case-control studies with errors in covariables

Methods are devised for estimating the parameters of a prospective logistic model in a case-control study with dichotomous response D that depends on a covariate X. For a portion of the sample, both the gold standard X and a surrogate covariate W are available; however, for the greater portion of the data, only the surrogate covariable W is available. By using a mixture model, the relationship between the true covariable and the response can be modeled appropriately for both types of data. The likelihood depends on the marginal distribution of X and the measurement error density (W\X,D). The latter is modeled parametrically based on the validation sample. The marginal distribution of the true covariable is modeled using a nonparametric mixture distribution. In this way we can improve the efficiency and reduce the bias of the parameter estimates, The results also apply when there is no validation data provided the error distribution is known or estimated from an independent data source. Many of the results also apply to the easier case of prospective sampling.