Wavelet-based nonparametric modeling of hierarchical functions in colon carcinogenesis

In this article we develop new methods for analyzing the data from an experiment using rodent models to investigate the effect of type of dietary fat on O-6-methylguanine-DNA-methyltransferase (MGMT), an important biomarker in early colon carcinogenesis. The data consist of observed profiles over a spatial variable contained within a two-stage hierarchy, a structure that we dub hierarchical functional data. We present a new method providing a unified framework for modeling these data, simultaneously yielding estimates and posterior samples for mean, individual, and subsample-level profiles, as well as covariance parameters at the various hierarchical levels. Our method is nonparametric in that it does not require the prespecification of parametric forms for the functions and involves modeling in the wavelet space, which is especially effective for spatially heterogeneous functions as encountered in the MGMT data. Our approach is Bayesian; the only informative hyperparameters in our model are effectively smoothing parameters. Analysis of this dataset yields interesting new insights into how MGMT operates in early colon carcinogenesis, and how this may depend on diet. Our method is general, so it can be applied to other settings where hierarchical functional data are encountered.