Testing for spatial correlation and semiparametric spatial modeling of binary outcomes with application to aberrant crypt foci in colon carcinogenesis experiments

摘要

In an experiment to understand colon carcinogenesis, all animals were exposed to a carcinogen while half the animals were also exposed to radiation. Spatially, we measured the existence of aberrant crypt foci (ACF), namely morphologically changed colonic crypts that are known to be precursors of colon cancer development. The biological question of interest is whether the locations of these ACFs are spatially correlated: if so, this indicates that damage to the colon due to carcinogens and radiation is localized. Statistically, the data take the form of binary outcomes (corresponding to the existence of an ACF) on a regular grid. We develop score??type methods based upon the Matern and conditionally autoregression (CAR) correlation models to test for the spatial correlation in such data, while allowing for nonstationarity. Because of a technical peculiarity of the score??type test, we also develop robust versions of the method. The methods are compared to a generalization of Moran??s test for continuous outcomes, and are shown via simulation to have the potential for increased power. When applied to our data, the methods indicate the existence of spatial correlation, and hence indicate localization of damage. Assuming that there are correlations in the locations of the ACF, the questions are how great are these correlations, and whether the correlation structures di?er when an animal is exposed to radiation. To understand the extent of the correlation, we cast the problem as a spatial binary regression, where binary responses arise from an underlying Gaussian latent process. We model these marginal probabilities of ACF semiparametrically, using ?xed-knot penalized regression splines and single-index models. We ?t the models using pairwise pseudolikelihood methods. Assuming that the underlying latent process is strongly mixing, known to be the case for many Gaussian processes, we prove asymptotic normality of the methods. The penalized regression splines have penalty parameters that must converge to zero asymptotically: we derive rates for these parameters that do and do not lead to an asymptotic bias, and we derive the optimal rate of convergence for them. Finally, we apply the methods to the data from our experiment.