The purpose of this project is development of the statistical theory for possibly misspecified stochastic regression models and their applications in assessing the association between disease and explanatory variables. The current research is focused on the effect of possible misspecification such as nonlinearity and/or heteroscedasticity on statistical properties of mean square linear regression models. Theory has been developed that provides formulas for the asymptotic bias of the estimated effect of the main exposure variable and the asymptotic relevant efficiency of the corresponding test of significance compared to the correctly specified model. The theory is being used to assess the effect of measurement errors in the explanatory variables and to evaluate methods of adjusting for this effect. The case of generally structured measurement error is being considered when errors can have non-normal distribution, be heteroscedastic, and correlate among themselves and with the true values of the covariates, as is often encountered with self-reported variables such as nutritional intakes. The results of this work have been applied to evaluate the impact of dietary measurement error on estimating and testing the addition and substitution effects of the main exposure variable in energy adjustment methods in nutritional epidemiology. The developed theory is also being used to evaluate for these models the impact of bias correction on the variance of the corrected estimates and the power of the corresponding significance test relative to the size of the calibration/validation data sets necessary for this correction. The results have been presented at the ASA meeting and a paper is being prepared for publication.
