Covariate adjusted regression and correlation

For many statistical applications, a regression model is a standard tool for analyzing data. An implicit assumption is that the predictors and response are directly observable. However, in some situations both the response and predictor variables are influenced by a confounding covariate. In this work, we develop covariate adjusted regression for situations where both the predictors and response are observed after being multiplied by an unknown function of a common observable confounding covariate. This model is motivated mainly by regression problems in the life sciences. One example is the analysis of the regression of plasma fibrinogen concentration as response on serum transferrin level as predictor for a sample of hemodialysis patients. In this example, both the response and predictor are thought to be influenced in a multiplicative fashion by the body mass index of the patients.; The contamination of the predictors and response by the same covariate can alter the regression and correlation relations of interest between the original response and predictors completely. Thus a new method needs to be developed to recover the original regression coefficients from the regression of the underlying response and predictors. We demonstrate how these regression coefficients can be estimated consistently by establishing a connection to varying coefficient regression. Under the same setting, we also propose covariate adjusted correlation analysis to consistently estimate the correlation between the underlying response and the predictor, adjusting for the confounding covariate. The asymptotic distributions of the resulting regression coefficients and correlation estimates are established. The distribution results combined with proposed consistent estimates of the asymptotic variance can be used for the construction of approximate confidence intervals for the regression coefficients and correlation.