Discovering high-order patterns of gene expression levels

Exploiting identifiability and intergene correlation for improved detection of differential expression

Accurate differential analysis of microarray data strongly depends on effective treatment of intergene correlation. Such dependence is ordinarily accounted for in terms of its effect on significance cutoffs. In this paper, it is shown that correlation can, in fact, be exploited to share information across tests and reorder expression differentials for increased statistical power, regardless of the threshold. Significantly improved differential analysis is the result of two simple measures: (i) adjusting test statistics to exploit information from identifiable genes (the large subset of genes represented on a microarray that can be classified a priori as nondifferential with very high confidence], but (ii) doing so in a way that accounts for linear dependencies among identifiable and nonidentifiable genes. A method is developed that builds upon the widely used two-sample t-statistic approach and uses analysis in Hilbert space to decompose the nonidentified gene vector into two components that are correlated and uncorrelated with the identified set. In the application to data derived from a widely studied prostate cancer database, the proposed method outperforms some of the most highly regarded approaches published to date. Algorithms in MATLAB and in R are available for public download.

Nonlinear dependence in the discovery of differentially expressed genes

Microarray data are used to determine which genes are active in response to a changing cell environment. Genes are “discovered” when they are significantly differentially expressed in the microarray data collected under the differing conditions. In one prevalent approach, all genes are assumed to satisfy a null hypothesis, ℍ₀, of no difference in expression. A false discovery (type 1 error) occurs when ℍ₀ is incorrectly rejected. The quality of a detection algorithm is assessed by estimating its number of false discoveries, 𝔉. Work involving the second-moment modeling of the z-value histogram (representing gene expression differentials) has shown significantly deleterious effects of intergene expression correlation on the estimate of 𝔉. This paper suggests that nonlinear dependencies could likewise be important. With an applied emphasis, this paper extends the “moment framework” by including third-moment skewness corrections in an estimator of 𝔉. This estimator combines observed correlation (corrected for sampling fluctuations) with the information from easily identifiable null cases. Nonlinear-dependence modeling reduces the estimation error relative to that of linear estimation. Third-moment calculations involve empirical densities of 3 × 3 covariance matrices estimated using very few samples. The principle of entropy maximization is employed to connect estimated moments to 𝔉 inference. Model results are tested with BRCA and HIV data sets and with carefully constructed simulations.

Unsupervised fuzzy pattern discovery in gene expression data

BACKGROUND

Discovering patterns from gene expression levels is regarded as a classification problem when tissue classes of the samples are given and solved as a discrete-data problem by discretizing the expression levels of each gene into intervals maximizing the interdependence between that gene and the class labels. However, when class information is unavailable, discovering gene expression patterns becomes difficult.

METHODS

For a gene pool with large number of genes, we first cluster the genes into smaller groups. In each group, we use the representative gene, one with highest interdependence with others in the group, to drive the discretization of the gene expression levels of other genes. Treating intervals as discrete events, association patterns of events can be discovered. If the gene groups obtained are crisp gene clusters, significant patterns overlapping different gene clusters cannot be found. This paper presents a new method of "fuzzifying" the crisp gene clusters to overcome such problem.

RESULTS

To evaluate the effectiveness of our approach, we first apply the above described procedure on a synthetic data set and then a gene expression data set with known class labels. The class labels are not being used in both analyses but used later as the ground truth in a classificatory problem for assessing the algorithm's effectiveness in fuzzy gene clustering and discretization. The results show the efficacy of the proposed method. The existence of correlation among continuous valued gene expression levels suggests that certain genes in the gene groups have high interdependence with other genes in the group. Fuzzification of a crisp gene cluster allows the cluster to take in genes from other clusters so that overlapping relationship among gene clusters could be uncovered. Hence, previously unknown hidden patterns resided in overlapping gene clusters are discovered. From the experimental results, the high order patterns discovered reveal multiple gene interaction patterns in cancerous tissues not found in normal tissues. It was also found that for the colon cancer experiment, 70% of the top patterns and most of the discriminative patterns between cancerous and normal tissues are among those spanning across different crisp gene clusters.

CONCLUSIONS

We show that the proposed method for analyzing the error-prone microarray is effective even without the presence of tissue class information. A unified framework is presented, allowing fast and accurate pattern discovery for gene expression data. For a large gene set, to discover a comprehensive set of patterns, gene clustering, gene expression discretization and gene cluster fuzzification are absolutely necessary.