Parametric and nonparametric methods for understanding the relationship between carcinogen-induced DNA adduct levels in distal and proximal regions of the colon

An important problem in studying the etiology of colon cancer is understand ing the relationship between DNA adduct levels (broadly, DNA damage) in cel ls within colonic crypts in distal and proximal parts of the colon, followi ng treatment with a carcinogen and different types of diet. In particular, it is important to understand whether rats who have elevated adduct levels in particular positions in distal region crypts also have elevated levels i n the same positions of the crypts in proximal regions, and whether this re lationship depends on diet. We cast this problem as estimating the correlat ion function of two responses as a function of a covariate for studies wher e both responses are measured on the same experimental units but not the sa me subsampling units. Parametric and nonparametric methods are developed an d applied to a dataset from an ongoing study, leading to potentially import ant and surprising biological results. Theoretical calculations suggest tha t the nonparametric method, based on nonparametric regression, should in fa ct have statistical properties nearly the same as if the functions nonparam etrically estimated were known. The methodology used in this article can be applied to other settings when the goal of the study is to model the corre lation of two continuous repeated measurement responses as a function of a covariate, whereas the two responses of interest can be measured on the sam e experimental units but not on the same subsampling units, In our example, the two responses were measured in two different regions of the colon.