The clustering of regression models method with applications in gene expression data

Identification of differentially expressed genes and clustering of genes are two important and complementary objectives addressed with gene expression data. For the differential expression question, many "per-gene" analytic methods have been proposed. These methods can generally be characterized as using a regression function to independently model the observations for each gene; various adjustments for multiplicity are then used to interpret the statistical significance of these per-gene regression models over the collection of genes analyzed. Motivated by this common structure of per-gene models, we proposed a new model-based clustering method--the clustering of regression models method, which groups genes that share a similar relationship to the covariate(s). This method provides a unified approach for a family of clustering procedures and can be applied for data collected with various experimental designs. In addition, when combined with per-gene methods for assessing differential expression that employ the same regression modeling structure, an integrated framework for the analysis of microarray data is obtained. The proposed methodology was applied to two microarray data sets, one from a breast cancer study and the other from a yeast cell cycle study.     

Multi-class clustering and prediction in the analysis of microarray data

DNA microarray technology provides tools for studying the expression profiles of a large number of distinct genes simultaneously. This technology has been applied to sample clustering and sample prediction. Because of a large number of genes measured, many of the genes in the original data set are irrelevant to the analysis. Selection of discriminatory genes is critical to the accuracy of clustering and prediction. This paper considers statistical significance testing approach to selecting discriminatory gene sets for multi-class clustering and prediction of experimental samples. A toxicogenomic data set with nine treatments (a control and eight metals, As, Cd, Ni, Cr, Sb, Pb, Cu, and AsV with a total of 55 samples) is used to illustrate a general framework of the approach. Among four selected gene sets, a gene set omega(I) formed by the intersection of the F-test and the set of the union of one-versus-all t-tests performs the best in terms of clustering as well as prediction. Hierarchical and two modified partition (k-means) methods all show that the set omega(I) is able to group the 55 samples into seven clusters reasonably well, in which the As and AsV samples are considered as one cluster (the same group) as are the Cd and Cu samples. With respect to prediction, the overall accuracy for the gene set omega(I) using the nearest neighbors algorithm to predict 55 samples into one of the nine treatments is 85%.     

Gene expression analysis with the parametric bootstrap

Recent developments in microarray technology make it possible to capture the gene expression profiles for thousands of genes at once. With this data researchers are tackling problems ranging from the identification of 'cancer genes' to the formidable task of adding functional annotations to our rapidly growing gene databases. Specific research questions suggest patterns of gene expression that are interesting and informative: for instance, genes with large variance or groups of genes that are highly correlated. Cluster analysis and related techniques are proving to be very useful. However, such exploratory methods alone do not provide the opportunity to engage in statistical inference. Given the high dimensionality (thousands of genes) and small sample sizes (often <30) encountered in these datasets, an honest assessment of sampling variability is crucial and can prevent the over-interpretation of spurious results. We describe a statistical framework that encompasses many of the analytical goals in gene expression analysis; our framework is completely compatible with many of the current approaches and, in fact, can increase their utility. We propose the use of a deterministic rule, applied to the parameters of the gene expression distribution, to select a target subset of genes that are of biological interest. In addition to subset membership, the target subset can include information about relationships between genes, such as clustering. This target subset presents an interesting parameter that we can estimate by applying the rule to the sample statistics of microarray data. The parametric bootstrap, based on a multivariate normal model, is used to estimate the distribution of these estimated subsets and relevant summary measures of this sampling distribution are proposed. We focus on rules that operate on the mean and covariance. Using Bernstein's Inequality, we obtain consistency of the subset estimates, under the assumption that the sample size converges faster to infinity than the logarithm of the number of genes. We also provide a conservative sample size formula guaranteeing that the sample mean and sample covariance matrix are uniformly within a distance epsilon > 0 of the population mean and covariance. The practical performance of the method using a cluster-based subset rule is illustrated with a simulation study. The method is illustrated with an analysis of a publicly available leukemia data set.     

Assessing reliability of gene clusters from gene expression data

The rapid development of microarray technologies has raised many challenging problems in experiment design and data analysis. Although many numerical algorithms have been successfully applied to analyze gene expression data, the effects of variations and uncertainties in measured gene expression levels across samples and experiments have been largely ignored in the literature. In this article, in the context of hierarchical clustering algorithms, we introduce a statistical resampling method to assess the reliability of gene clusters identified from any hierarchical clustering method. Using the clustering trees constructed from the resampled data, we can evaluate the confidence value for each node in the observed clustering tree. A majority-rule consensus tree can be obtained, showing clusters that only occur in a majority of the resampled trees. We illustrate our proposed methods with applications to two published data sets. Although the methods are discussed in the context of hierarchical clustering methods, they can be applied with other cluster-identification methods for gene expression data to assess the reliability of any gene cluster of interest. Electronic Publication     

Analyzing the similarity of samples and genes by MG-PCC algorithm,t-SNE-SS and t-SNE-SG maps

Background

For analyzing these gene expression data sets under different samples, clustering and visualizing samples and genes are important methods. However, it is difficult to integrate clustering and visualizing techniques when the similarities of samples and genes are defined by PCC(Person correlation coefficient) measure.

Results

Here, for rare samples of gene expression data sets, we use MG-PCC (mini-groups that are defined by PCC) algorithm to divide them into mini-groups, and use t-SNE-SSP maps to display these mini-groups, where the idea of MG-PCC algorithm is that the nearest neighbors should be in the same mini-groups, t-SNE-SSP map is selected from a series of t-SNE(t-statistic Stochastic Neighbor Embedding) maps of standardized samples, and these t-SNE maps have different perplexity parameter. Moreover, for PCC clusters of mass genes, they are displayed by t-SNE-SGI map, where t-SNE-SGI map is selected from a series of t-SNE maps of standardized genes, and these t-SNE maps have different initialization dimensions. Here, t-SNE-SSP and t-SNE-SGI maps are selected by A-value, where A-value is modeled from areas of clustering projections, and t-SNE-SSP and t-SNE-SGI maps are such t-SNE map that has the smallest A-value.

Conclusions

From the analysis of cancer gene expression data sets, we demonstrate that MG-PCC algorithm is able to put tumor and normal samples into their respective mini-groups, and t-SNE-SSP(or t-SNE-SGI) maps are able to display the relationships between mini-groups(or PCC clusters) clearly. Furthermore, t-SNE-SS(m)(or t-SNE-SG(n)) maps are able to construct independent tree diagrams of the nearest sample(or gene) neighbors, where each tree diagram is corresponding to a mini-group of samples(or genes).

     

Discriminant analysis to evaluate clustering of gene expression data

In this work we present a procedure that combines classical statistical methods to assess the confidence of gene clusters identified by hierarchical clustering of expression data. This approach was applied to a publicly released Drosophila metamorphosis data set [White et al., Science 286 (1999) 2179-2184]. We have been able to produce reliable classifications of gene groups and genes within the groups by applying unsupervised (cluster analysis), dimension reduction (principal component analysis) and supervised methods (linear discriminant analysis) in a sequential form. This procedure provides a means to select relevant information from microarray data, reducing the number of genes and clusters that require further biological analysis.     

Analyzing Large Gene Expression and Methylation Data Profiles Using StatBicRM: Statistical Biclustering-Based Rule Mining

Microarray and beadchip are two most efficient techniques for measuring gene expression and methylation data in bioinformatics. Biclustering deals with the simultaneous clustering of genes and samples. In this article, we propose a computational rule mining framework, StatBicRM (i.e., statistical biclustering-based rule mining) to identify special type of rules and potential biomarkers using integrated approaches of statistical and binary inclusion-maximal biclustering techniques from the biological datasets. At first, a novel statistical strategy has been utilized to eliminate the insignificant/low-significant/redundant genes in such way that significance level must satisfy the data distribution property (viz., either normal distribution or non-normal distribution). The data is then discretized and post-discretized, consecutively. Thereafter, the biclustering technique is applied to identify maximal frequent closed homogeneous itemsets. Corresponding special type of rules are then extracted from the selected itemsets. Our proposed rule mining method performs better than the other rule mining algorithms as it generates maximal frequent closed homogeneous itemsets instead of frequent itemsets. Thus, it saves elapsed time, and can work on big dataset. Pathway and Gene Ontology analyses are conducted on the genes of the evolved rules using David database. Frequency analysis of the genes appearing in the evolved rules is performed to determine potential biomarkers. Furthermore, we also classify the data to know how much the evolved rules are able to describe accurately the remaining test (unknown) data. Subsequently, we also compare the average classification accuracy, and other related factors with other rule-based classifiers. Statistical significance tests are also performed for verifying the statistical relevance of the comparative results. Here, each of the other rule mining methods or rule-based classifiers is also starting with the same post-discretized data-matrix. Finally, we have also included the integrated analysis of gene expression and methylation for determining epigenetic effect (viz., effect of methylation) on gene expression level.     

Differential gene expression detection and sample classification using penalized linear regression models

Differential gene expression detection and sample classification using microarray data have received much research interest recently. Owing to the large number of genes p and small number of samples n (p > n), microarray data analysis poses big challenges for statistical analysis. An obvious problem owing to the 'large p small n' is over-fitting. Just by chance, we are likely to find some non-differentially expressed genes that can classify the samples very well. The idea of shrinkage is to regularize the model parameters to reduce the effects of noise and produce reliable inferences. Shrinkage has been successfully applied in the microarray data analysis. The SAM statistics proposed by Tusher et al. and the 'nearest shrunken centroid' proposed by Tibshirani et al. are ad hoc shrinkage methods. Both methods are simple, intuitive and prove to be useful in empirical studies. Recently Wu proposed the penalized t/F-statistics with shrinkage by formally using the (1) penalized linear regression models for two-class microarray data, showing good performance. In this paper we systematically discussed the use of penalized regression models for analyzing microarray data. We generalize the two-class penalized t/F-statistics proposed by Wu to multi-class microarray data. We formally derive the ad hoc shrunken centroid used by Tibshirani et al. using the (1) penalized regression models. And we show that the penalized linear regression models provide a rigorous and unified statistical framework for sample classification and differential gene expression detection.     

Mixture modelling of gene expression data from microarray experiments

MOTIVATION: Hierarchical clustering is one of the major analytical tools for gene expression data from microarray experiments. A major problem in the interpretation of the output from these procedures is assessing the reliability of the clustering results. We address this issue by developing a mixture model-based approach for the analysis of microarray data. Within this framework, we present novel algorithms for clustering genes and samples. One of the byproducts of our method is a probabilistic measure for the number of true clusters in the data. RESULTS: The proposed methods are illustrated by application to microarray datasets from two cancer studies; one in which malignant melanoma is profiled (Bittner et al., Nature, 406, 536-540, 2000), and the other in which prostate cancer is profiled (Dhanasekaran et al., 2001, submitted).     

Analysis of temporal gene expression profiles: clustering by simulated annealing and determining the optimal number of clusters

MOTIVATION: Cluster analysis of genome-wide expression data from DNA microarray hybridization studies has proved to be a useful tool for identifying biologically relevant groupings of genes and samples. In the present paper, we focus on several important issues related to clustering algorithms that have not yet been fully studied. RESULTS: We describe a simple and robust algorithm for the clustering of temporal gene expression profiles that is based on the simulated annealing procedure. In general, this algorithm guarantees to eventually find the globally optimal distribution of genes over clusters. We introduce an iterative scheme that serves to evaluate quantitatively the optimal number of clusters for each specific data set. The scheme is based on standard approaches used in regular statistical tests. The basic idea is to organize the search of the optimal number of clusters simultaneously with the optimization of the distribution of genes over clusters. The efficiency of the proposed algorithm has been evaluated by means of a reverse engineering experiment, that is, a situation in which the correct distribution of genes over clusters is known a priori. The employment of this statistically rigorous test has shown that our algorithm places greater than 90% genes into correct clusters. Finally, the algorithm has been tested on real gene expression data (expression changes during yeast cell cycle) for which the fundamental patterns of gene expression and the assignment of genes to clusters are well understood from numerous previous studies.     

Fuzzy C-means method for clustering microarray data

MOTIVATION: Clustering analysis of data from DNA microarray hybridization studies is essential for identifying biologically relevant groups of genes. Partitional clustering methods such as K-means or self-organizing maps assign each gene to a single cluster. However, these methods do not provide information about the influence of a given gene for the overall shape of clusters. Here we apply a fuzzy partitioning method, Fuzzy C-means (FCM), to attribute cluster membership values to genes. RESULTS: A major problem in applying the FCM method for clustering microarray data is the choice of the fuzziness parameter m. We show that the commonly used value m = 2 is not appropriate for some data sets, and that optimal values for m vary widely from one data set to another. We propose an empirical method, based on the distribution of distances between genes in a given data set, to determine an adequate value for m. By setting threshold levels for the membership values, genes which are tigthly associated to a given cluster can be selected. Using a yeast cell cycle data set as an example, we show that this selection increases the overall biological significance of the genes within the cluster. AVAILABILITY: Supplementary text and Matlab functions are available at http://www-igbmc.u-strasbg.fr/fcm/     

Assessing gene significance from cDNA microarray expression data via mixed models.

The determination of a list of differentially expressed genes is a basic objective in many cDNA microarray experiments. We present a statistical approach that allows direct control over the percentage of false positives in such a list and, under certain reasonable assumptions, improves on existing methods with respect to the percentage of false negatives. The method accommodates a wide variety of experimental designs and can simultaneously assess significant differences between multiple types of biological samples. Two interconnected mixed linear models are central to the method and provide a flexible means to properly account for variability both across and within genes. The mixed model also provides a convenient framework for evaluating the statistical power of any particular experimental design and thus enables a researcher to a priori select an appropriate number of replicates. We also suggest some basic graphics for visualizing lists of significant genes. Analyses of published experiments studying human cancer and yeast cells illustrate the results.     

Strategies for identifying statistically significant dense regions in microarray data

We propose and study the notion of dense regions for the analysis of categorized gene expression data and present some searching algorithms for discovering them. The algorithms can be applied to any categorical data matrices derived from gene expression level matrices. We demonstrate that dense regions are simple but useful and statistically significant patterns that can be used to 1) identify genes and/or samples of interest and 2) eliminate genes and/or samples corresponding to outliers, noise, or abnormalities. Some theoretical studies on the properties of the dense regions are presented which allow us to characterize dense regions into several classes and to derive tailor-made algorithms for different classes of regions. Moreover, an empirical simulation study on the distribution of the size of dense regions is carried out which is then used to assess the significance of dense regions and to derive effective pruning methods to speed up the searching algorithms. Real microarray data sets are employed to test our methods. Comparisons with six other well-known clustering algorithms using synthetic and real data are also conducted which confirm the superiority of our methods in discovering dense regions. The DRIFT code and a tutorial are available as supplemental material, which can be found on the Computer Society Digital Library at http://computer.org/tcbb/archives.htm.     

Context-specific infinite mixtures for clustering gene expression profiles across diverse microarray dataset

MOTIVATION: Identifying groups of co-regulated genes by monitoring their expression over various experimental conditions is complicated by the fact that such co-regulation is condition-specific. Ignoring the context-specific nature of co-regulation significantly reduces the ability of clustering procedures to detect co-expressed genes due to additional 'noise' introduced by non-informative measurements. RESULTS: We have developed a novel Bayesian hierarchical model and corresponding computational algorithms for clustering gene expression profiles across diverse experimental conditions and studies that accounts for context-specificity of gene expression patterns. The model is based on the Bayesian infinite mixtures framework and does not require a priori specification of the number of clusters. We demonstrate that explicit modeling of context-specificity results in increased accuracy of the cluster analysis by examining the specificity and sensitivity of clusters in microarray data. We also demonstrate that probabilities of co-expression derived from the posterior distribution of clusterings are valid estimates of statistical significance of created clusters. AVAILABILITY: The open-source package gimm is available at http://eh3.uc.edu/gimm.     

Cell protection, resistance and invasiveness of two malignant mesotheliomas as assessed by 10K-microarray

Malignant pleural mesothelioma (MPM) is an aggressive serosal tumor, strongly associated with former exposure to asbestos fibers and for which there is currently no effective treatment available. In human, MPM is characterized by a high local invasiveness, poor prognosis and therapeutic outcomes. In order to assess molecular changes that specify this phenotype, we performed a global gene expression profiling of human MPM. Using a 10,000-element microarray, we analyzed mRNA relative gene expression levels by comparing a mesothelioma cell line to either a pleural cell line or tumor specimens. To analyze these gene expression data, we used various bioinformatics softwares. Hierarchical clustering methods were used to group genes and samples with similar expression in an unsupervised mode. Genes of known function were further sorted by enzyme, function and pathway clusters using a supervised software (IncyteGenomics). Taken together, these data defined a molecular fingerprint of human MPM with more than 700 up- or down-regulated genes related to several traits of the malignant phenotype, specially associated with MPM invasiveness, protection and resistance to anticancer defenses. This portrait is meaningful in disease classification and management, and relevant in finding new specific markers of MPM. These molecular markers should improve the accuracy of mesothelioma diagnosis, prognosis and therapy.     

Novel clustering algorithm for microarray expression data in a truncated SVD space

MOTIVATION: This paper introduces the application of a novel clustering method to microarray expression data. Its first stage involves compression of dimensions that can be achieved by applying SVD to the gene-sample matrix in microarray problems. Thus the data (samples or genes) can be represented by vectors in a truncated space of low dimensionality, 4 and 5 in the examples studied here. We find it preferable to project all vectors onto the unit sphere before applying a clustering algorithm. The clustering algorithm used here is the quantum clustering method that has one free scale parameter. Although the method is not hierarchical, it can be modified to allow hierarchy in terms of this scale parameter. RESULTS: We apply our method to three data sets. The results are very promising. On cancer cell data we obtain a dendrogram that reflects correct groupings of cells. In an AML/ALL data set we obtain very good clustering of samples into four classes of the data. Finally, in clustering of genes in yeast cell cycle data we obtain four groups in a problem that is estimated to contain five families. AVAILABILITY: Software is available as Matlab programs at http://neuron.tau.ac.il/~horn/QC.htm.     

Bagging to improve the accuracy of a clustering procedure

MOTIVATION: The microarray technology is increasingly being applied in biological and medical research to address a wide range of problems such as the classification of tumors. An important statistical question associated with tumor classification is the identification of new tumor classes using gene expression profiles. Essential aspects of this clustering problem include identifying accurate partitions of the tumor samples into clusters and assessing the confidence of cluster assignments for individual samples. RESULTS: Two new resampling methods, inspired from bagging in prediction, are proposed to improve and assess the accuracy of a given clustering procedure. In these ensemble methods, a partitioning clustering procedure is applied to bootstrap learning sets and the resulting multiple partitions are combined by voting or the creation of a new dissimilarity matrix. As in prediction, the motivation behind bagging is to reduce variability in the partitioning results via averaging. The performances of the new and existing methods were compared using simulated data and gene expression data from two recently published cancer microarray studies. The bagged clustering procedures were in general at least as accurate and often substantially more accurate than a single application of the partitioning clustering procedure. A valuable by-product of bagged clustering are the cluster votes which can be used to assess the confidence of cluster assignments for individual observations. SUPPLEMENTARY INFORMATION: For supplementary information on datasets, analyses, and software, consult http://www.stat.berkeley.edu/~sandrine and http://www.bioconductor.org.     

Prediction of group patterns in social mammals based on a coalescent model

This study describes a statistical model which assumes that mammal group patterns match with groups of genetic relatives. Given a fixed sample size, recursive algorithms for the exact computation of the probability distribution of the number of groups are provided. The recursive algorithms are then incorporated into a statistical likelihood framework which can be used to detect and quantify departure from the null-model by estimating a clustering parameter. The test is then applied to ecological data from social herbivores and carnivores. Our findings support the hypothesis that genetic relatedness is likely to predict group patterns when large mammals have few or no predators.