Bayesian Hierarchical Spatially Correlated Functional Data Analysis with Application to Colon Carcinogenesis

Abstract

SummaryIn this article, we present new methods to analyze data from an experiment using rodent models to investigate the role of p27, an important cell-cycle mediator, in early colon carcinogenesis. The responses modeled here are essentially functions nested within a two-stage hierarchy. Standard functional data analysis literature focuses on a single stage of hierarchy and conditionally independent functions with near white noise. However, in our experiment, there is substantial biological motivation for the existence of spatial correlation among the functions, which arise from the locations of biological structures called colonic crypts: this possible functional correlation is a phenomenon we term crypt signaling. Thus, as a point of general methodology, we require an analysis that allows for functions to be correlated at the deepest level of the hierarchy. Our approach is fully Bayesian and uses Markov chain Monte Carlo methods for inference and estimation. Analysis of this data set gives new insights into the structure of p27 expression in early colon carcinogenesis and suggests the existence of significant crypt signaling. Our methodology uses regression splines, and because of the hierarchical nature of the data, dimension reduction of the covariance matrix of the spline coefficients is important: we suggest simple methods for overcoming this problem.

Statistics for spatial functional data: some recent contributions

“…Baladandayuthapani et al (2008) show an alternative for analyzing an experimental design with a spatially correlated functional response. They use Bayesian hierarchical models allowing to include spatial dependence among curves into standard FDA techniques, such as functional multiple regression and functional analysis of variance.…”

A Statistical Investigation to Monitor and Understand Atmospheric CFC Decline with the Spatial-longitudinal Bent-cable Model

“…This process may lead to investigating many models, and, due to the smooth transition phase, at least for the CFC data, may lead to too many parameters, which might diminish interpretability as compared to the bent-cable model. Another modeling approach for such data is the functional mixed effects model (e.g., Baladandayuthapani et al, 2008). So, further investigation is necessary to compare our methodology with other available statistical techniques to model this type of longitudinal changepoint data.…”

Spatial autoregressive functional plug-in prediction of ocean surface temperature

“…Such an approximation allows the construction of least squares estimators of the regression function with or without a penalization term. In Baladandayuthapani et al (2008), a fully Bayesian and Markov Chain Monte Carlo based approach is derived for the analysis of the spatial correlation between functional data arising from different spatial locations of biological structures called colonic crypts (i.e., it allows the analysis of crypt signaling phenomenon). In the non-parametric context, the statistical properties of kernel-based density estimators, formulated in the context of spatial functional random variables, are studied in Basse et al (2008).…”