Abstract
Clustering analysis is one of the most widely used statistical tools in many emerging areas such as microarray data analysis. For microarray and other high-dimensional data, the presence of many noise variables may mask underlying clustering structures. Hence removing noise variables via variable selection is necessary. For simultaneous variable selection and parameter estimation, existing penalized likelihood approaches in model-based clustering analysis all assume a common diagonal covariance matrix across clusters, which however may not hold in practice. To analyze high-dimensional data, particularly those with relatively low sample sizes, this article introduces a novel approach that shrinks the variances together with means, in a more general situation with cluster-specific (diagonal) covariance matrices. Furthermore, selection of grouped variables via inclusion or exclusion of a group of variables altogether is permitted by a specific form of penalty, which facilitates incorporating subject-matter knowledge, such as gene functions in clustering microarray samples for disease subtype discovery. For implementation, EM algorithms are derived for parameter estimation, in which the M-steps clearly demonstrate the effects of shrinkage and thresholding. Numerical examples, including an application to acute leukemia subtype discovery with microarray gene expression data, are provided to demonstrate the utility and advantage of the proposed method.
Highlights
Clustering analysis is perhaps the most widely used analysis method for microarray data: it has been used for gene function discovery (Eisen et al 1998 [10]) and cancer subtype discovery (Golub et al 1999 [15])
The basic statistical models of these approaches are all the same: informative variables are assumed to come from a mixture of Normals, corresponding to clusters, while noise variables coming from a single Normal distribution; they differ in how they are implemented
The Bayesian approaches are more flexible than the penalized methods, but they are computationally more demanding because of their use of MCMC for stochastic search; penalized methods enjoy the flexibility of the use of penalty functions, such as to accommodate grouped parameters or variables as to be discussed later
Summary
Clustering analysis is perhaps the most widely used analysis method for microarray data: it has been used for gene function discovery (Eisen et al 1998 [10]) and cancer subtype discovery (Golub et al 1999 [15]). Other recent efforts include the following: Raftery and Dean (2006) [41] considered a sequential, stepwise approach to variable selection in model-based clustering; as acknowledged by the authors, “when the number of variables is vast (e.g., in microarray data analysis when thousands of genes may be the variables being used), the method is too slow to be practical as it stands”. The aforementioned clustering methods did not allow for variable selection directly, while it is our main aim to consider variable selection, possibly assisted with biological knowledge This is in line with the currently increasing interest in incorporating biological information on gene functional groups into analysis of detecting differential gene expression (e.g. Pan 2006 [37]; Efron and Tibshirani 2007 [8]; Newton et al 2007 [36]).
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