The derivation of the likelihood function for binary data from two types of main study/validation study designs where model covariates are measured with error is elaborated. Rather than limiting consideration to a restricted family of models with convenient mathematical properties, we suggest that empirical considerations, customized to the data at hand, should drive model choices. The joint likelihood function for the main study, in which the covariates are measured with error, and the validation study, in which they are not, is maximized, and estimation and inference proceeds using standard theory. Although the choice of the measurement error model is driven by empirical considerations, the relatively small validation study sizes typically seen may lead to misspecification, resulting in bias in estimation and inference about exposure-disease relationships. By using a nonparametric form for the measurement error model, the resulting semi-parametric methods suggested by Robins, Rotnitzky, and Zhao (1994, Journal of the American Statistical Association 89, 864-866) and Robins, Hsieh, and Newey (1995, Journal of the Royal Statistical Society, Series B 57, 409-424) are free from bias due to misspecification of the measurement error model, trading efficiency for robustness as usual. These fully and semi-parametric methods are illustrated with a detailed example from a main study/validation study of the health effects of occupational exposure to chemotherapeutics among pharmacists (Valanis et al., 1993, American Journal of Hospital Pharmacy 50, 455-462). A constant, prevalence ratio model for common binary events, with gamma covariate measurement error, is derived and empirically verified by the available data. A careful reanalysis of the data, taking measurement error fully into account, leads to a threefold increase in the log relative risk and no loss of statistical power. The semi-parametric estimates are consistent with the parametric results, providing reassurance that important bias due to misspecification of the measurement error model is unlikely.
Department of Epidemiology, Harvard School of Public Health, Boston, Massachusetts 02115, USA
Harvard University, School of Public Health, Department of Epidemiology and Biostatistics, Boston, Massachusetts