A Nonparametric 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 which 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 covariate W is available. By using a mixture model, the relationship between the true covariate 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 jX; D). The latter is modeled para-metrically based on the validation sample. The marginal distribution of the true covariate is modeled using a nonparametric mixture distribution. In this way we can improve the eeciency and reduce the bias of the parameter estimates. The results are suuciently general that they allow us to provide the rst results which allow no validation, if the error distribution is modeled from independent data. Many of the results also apply to the easier case of prospective sampling.