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 linear stochastic regression and the effect of possible misspecification such as nonlinearity and heteroscedasticity. Theory is being developed that provides the asymptotic unconditional distribution of the least squares estimators of the regression coefficients in such models. In particular, it is shown that the expected values of the estimators are asymptotically the same as in conventional regression analysis, but the formulas for the standard errors of the estimated regression coefficients differ from those for the correctly specified models. The theory is being used to derive some new results in two applied areas. Firstly, to investigate the consequences of discretizing continuous explanatory variables by categorizing them into a small number (3-5) of groups, e.g., quartiles, or ordering them so as to investigate trend over these groups, as is routinely done in epidemiologic studies. The second problem under investigation is to assess the effect of measurement errors in the explanatory variables. Theory has been developed to evaluate the general case when measurement errors correlate among themselves and with the true values of the covariates, as is often the case with self-reported variables such as nutritional intakes. The results of this work are being applied to evaluate the statistical properties of three alternative energy-adjustment models in nutritional epidemiology.
