Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis

Abstract

In 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.

Figure 1

(a) The vertical axes are the individual rats and the horizontal axes are the distances in microns and circle represent the physical location of the crypts for all rats assayed at 24-hour time point. (b) Histogram of the mutual crypt distances (|△|) for all rats assayed at 24-hour time point. Plotted are the distances less than 1000 microns.

Figure 2

Posterior correlations as a function of crypt distance with 95% error bars. The vertical axis is the correlation and the horizontal axis is the distance between the crypts (△).

Figure 3

(a) Shown here are the posterior marginal mean functions for the four diet groups. The horizontal axis is the relative cell depth. CO is corn oil and FO is fish oil with or without (±) butyrate supplement. (b) Posterior mean along with 90% credible intervals of the pointwise difference between the diet functions.

Figure 4

Posterior interaction function between diet and butyrate as a function of relative cell depth using (log)p27 response (a) accounting for correlation and (b) assuming independence between crypts. Also shown are the 95% Bayesian credible intervals.

Figure 5

Plot of correlation function estimates using pseudolikelihood analysis. All the data (solid line), using only the toptertile of the crypts (dashed line), the middle tertile (dash dotted) and bottom tertile (dotted line).