Detecting locations and regions with different expression levels using hybrid adaptive spline

High-resolution mapping of specific regions is of crucial importance for the subsequent discovery of the disease-associated clones and thus the genes they harbor. The copy number changes in cancer often span large regions of the genome (16), although losses or gains of smaller scale and micro-deletions or micro-gains (from 100 bp to 4 Mb and not detectable by standard cytogenetic methods) are also of importance. Several methods have been proposed to take into account the spatial correlations such as likelihood based on a fixed-width window correlation structure (8), moving averages (17), CGH-Plotter (16), break point model (18) and cluster along chromosomes (19). In reality, correlations may vary with chromosomal structures and thus locations, and the sizes of segments harboring genomic alterations also vary along chromosomes. Therefore, assumptions of fixed correlation structures and difference shapes may be too restrictive.

We use the hybrid adaptive spline (HAS) that has the ability to handle a wide variety of shapes and spatial inhomogeneities (20). It is an objective approach that allows data to dictate the shape of a function. Let yt be one of the z, t or u statistic defined in Equations 21.1, 21.2 and 21.4 respectively. We assume that y = f(x) + e„ i = !,...,!, (21.5)

where f is a function of xi and ei's are random errors with mean zero and variance a2. For simplicity we transformed the variable x into the interval [0,1].

Since copy number changes occur in local regions, the expectation of a ratio profile along the chromosome equals zero except in some regions that harbor the changes. Thus our goal is to detect locations or regions where f(x) * 0. Since f may have discontinuous points and is spatially inhomogeneous, common nonparametric regression methods such as smoothing spline do not provide a good estimate for f. Spatially adaptive, HAS was proposed to handle spatial inhomogeneity as in array CGH data. For simplicity, we introduce the HAS procedure using cubic spline bases. The HAS procedure for general spline bases can be found in Luo and Wahba (20).

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