A generalized Shapiro–Wilk statistic for testing high-dimensional normality

Shapiro and Wilk’s [Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52, 591–611] W-statistic was found to have competitive power performance in testing univariate normality. Generalizations of the W-statistic to the multivariate case have been proposed by many researchers. In this paper, we propose a family of generalized W-statistics for testing high-dimensional normality by using the theory of spherical distributions. The proposed statistics apply to the case that the sample size is smaller than the dimension. Monte Carlo studies demonstrate feasible performance of the proposed tests in controlling type I error rates and power against some non-normal data. It is concluded that the proposed statistics are superior to existing generalizedW-statistics and show competitive benefits in testing high-dimensional normality with small sample size.     

A flexible approximate likelihood ratio test for detecting differential expression in microarray data

Identifying differentially expressed genes in microarray data has been studied extensively and several methods have been proposed. Most popular methods in the study of gene expression microarray data analysis rely on normal distribution assumption and are based on a Wald statistic. These methods may be inefficient when expression levels follow a skewed distribution. To deal with possible violations of the normality assumption, we propose a method based on Generalized Logistic Distribution of Type II (GLDII). The motivation behind this distributional assumption is to allow longer tails than normal distribution. This is important in analyzing gene expression data since extreme values are common in such experiments. The shape parameter for GLDII allows flexibility in modeling a wide range of distributions. To simplify the computational complexity involved in carrying out Likelihood Ratio (LR) tests for several thousands of genes, an Approximate LR Test (ALRT) is proposed. We also generalize the two-class ALRT method to multi-class microarray data. The performance of the ALRT method under the GLDII assumption is compared to methods based on Wald-type statistics using simulation. The results from the simulations show that our method performs quite well compared to the significance analysis of microarrays (SAM) approach using standardized Wilcoxon rank statistics and the empirical Bayes (E-B) t-statistics. Our method is also less sensitive to extreme values. We illustrate our method using two publicly available gene expression data sets.     

Constructing gene regulatory networks from microarray data using GA/PSO with DTW

Recently, many methods have been proposed for constructing gene regulatory networks (GRNs). However, most of the existing methods ignored the time delay regulatory relation in the GRN predictions. In this paper, we propose a hybrid method, termed GA/PSO with DTW, to construct GRNs from microarray datasets. The proposed method uses test of correlation coefficient and the dynamic time warping (DTW) algorithm to determine the existence of a time delay relation between two genes. In addition, it uses the particle swarm optimization (PSO) to find thresholds for discretizing the microarray dataset. Based on the discretized microarray dataset and the predicted types of regulatory relations among genes, the proposed method uses a genetic algorithm to generate a set of candidate GRNs from which the predicted GRN is constructed. Three real-life sub-networks of yeast are used to verify the performance of the proposed method. The experimental results show that the GA/PSO with DTW is better than the other existing methods in terms of predicting sensitivity and specificity.     

Cluster analysis using multivariate normal mixture models to detect differential gene expression with microarray data

DNA microarrays make it possible to study simultaneously the expression of thousands of genes in a biological sample. Univariate clustering techniques have been used to discover target genes with differential expression between two experimental conditions. Because of possible loss of information due to use of univariate summary statistics, it may be more effective to use multivariate statistics. We present multivariate normal mixture model based clustering analyses to detect differential gene expression between two conditions.Deviating from the general mixture model and model-based clustering, we propose mixture models with specific mean and covariance structures that account for special features of two-condition microarray experiments. Explicit updating formulas in the EM algorithm for three such models are derived. The methods are applied to a real dataset to compare the expression levels of 1176 genes of rats with and without pneumococcal middle-ear infection to illustrate the performance and usefulness of this approach. About 10 genes and 20 genes are found to be differentially expressed in a six-dimensional modeling and a bivariate modeling, respectively. Two simulation studies are conducted to compare the performance of univariate and multivariate methods. Depending on data, neither method can always dominate the other. The results suggest that multivariate normal mixture models can be useful alternatives to univariate methods to detect differential gene expression in exploratory data analysis.     

A permutation test motivated by microarray data analysis

We introduce a nonparametric test intended for large-scale simultaneous inference in situations where the utility of distribution-free tests is limited because of their discrete nature. Such situations are frequently dealt with in microarray analysis where the number of tests is much larger than the sample size. The proposed test statistic is based on a certain distance between the distributions from which the samples under study are drawn. In a simulation study, the proposed permutation test is compared with permutation counterparts of the t-test and the Kolmogorov–Smirnov test. The usefulness of the proposed test is discussed in the context of microarray gene expression data and illustrated with an application to real datasets.     

Computing P-values for a class of permutation tests of equal survival functions

Dallas and Rao (Biometrics 56 (2000) 154) proposed a class of permutation tests for testing the equality of two survival distributions based on randomly right censored survival time data consisting of both paired and unpaired observations. Data sets of this type can occur frequently in medical settings. Two members of this class were advocated for use due to their generally high power for detecting scale and location shifts in the exponential and log-logistic distributions for the survival times, and improved power over paired data test procedures that disregard unpaired observations. Because the computations for the tests become quite laborious as the sample sizes increase, computing routines are required for practical implementation of these tests. This paper provides computing routines to execute the tests.     

A novel hybrid feature selection method for microarray data analysis

Recently, many methods have been proposed for microarray data analysis. One of the challenges for microarray applications is to select a proper number of the most relevant genes for data analysis. In this paper, we propose a novel hybrid method for feature selection in microarray data analysis. This method first uses a genetic algorithm with dynamic parameter setting (GADP) to generate a number of subsets of genes and to rank the genes according to their occurrence frequencies in the gene subsets. Then, this method uses the χ²-test for homogeneity to select a proper number of the top-ranked genes for data analysis. We use the support vector machine (SVM) to verify the efficiency of the selected genes. Six different microarray datasets are used to compare the performance of the GADP method with the existing methods. The experimental results show that the GADP method is better than the existing methods in terms of the number of selected genes and the prediction accuracy.     

A new framework for identifying differentially expressed genes

Microarrays have been widely used to classify cancer samples and discover the biological types, for example tumor versus normal phenotypes in cancer research. One of the challenging scientific tasks in the post-genomic epoch is how to identify a subset of differentially expressed genes from thousands of genes in microarray data which will enable us to understand the underlying molecular mechanisms of diseases, accurately diagnosing diseases and identifying novel therapeutic targets. In this paper, we propose a new framework for identifying differentially expressed genes. In the proposed framework, genes are ranked according to their residuals. The performance of the framework is assessed through applying it to several public microarray data. Experimental results show that the proposed method gives more robust and accurate rank than other statistical test methods, such as t-test, Wilcoxon rank sum test and KS-test. Another novelty of the method is that we design an algorithm for selecting a small subset of genes that show significant variation in expression (“outlier” genes). The number of genes in the small subset can be controlled via an alterable window of confidence level. In addition, the results of the proposed method can be visualized. By observing the residual plot, we can easily find genes that show significant variation in two groups of samples and learn the degrees of differential expression of genes. Through a comparison study, we found several “outlier” genes which had been verified in previous biological experiments while they were either not identified by other methods or had lower ranks in standard statistical tests.     

Testing equality of covariance matrices when data are incomplete

In the statistics literature, a number of procedures have been proposed for testing equality of several groups’ covariance matrices when data are complete, but this problem has not been considered for incomplete data in a general setting. This paper proposes statistical tests for equality of covariance matrices when data are missing. A Wald test (denoted by T₁), a likelihood ratio test (LRT) (denoted by R), based on the assumption of normal populations are developed. It is well-known that for the complete data case the classic LRT and the Wald test constructed under the normality assumption perform poorly in instances when data are not from multivariate normal distributions. As expected, this is also the case for the incomplete data case and therefore has led us to construct a robust Wald test (denoted by T₂) that performs well for both normal and non-normal data. A re-scaled LRT (denoted by R^*) is also proposed. A simulation study is carried out to assess the performance of T₁, T₂, R, and R^* in terms of closeness of their observed significance level to the nominal significance level as well as the power of these tests. It is found that T₂ performs very well for both normal and non-normal data in both small and large samples. In addition to its usual applications, we have discussed the application of the proposed tests in testing whether a set of data are missing completely at random (MCAR).     

Shrinkage-based regularization tests for high-dimensional data with application to gene set analysis

Traditional multivariate tests such as Hotelling’s test or Wilk’s test are designed for classical problems, where the number of observations is much larger than the dimension of the variables. For high-dimensional data, however, this assumption cannot be met any longer. In this article, we consider testing problems in high-dimensional MANOVA where the number of variables exceeds the sample size. To overcome the challenges with high dimensionality, we propose a new approach called a shrinkage-based regularization test, which is suitable for a variety of data structures including the one-sample problem and one-way MANOVA. Our approach uses a ridge regularization to overcome the singularity of the sample covariance matrix and applies a soft-thresholding technique to reduce random noise and improve the testing power. An appealing property of this approach is its ability to select relevant variables that provide evidence against the hypothesis. We compare the performance of our approach with some competing approaches via real microarray data and simulation studies. The results illustrate that the proposed statistics maintains relatively high power in detecting a wide family of alternatives.     

Identification of differentially expressed spatial clusters using humoral response microarray data

The protein microarray is a powerful chip-based technology for profiling hundreds of proteins simultaneously and is being increasingly used. To study humoral response in pancreatic cancers, scientists have developed a two-dimensional liquid separation technique and built a two-dimensional protein microarray. However, identifying regions of differential expression on the protein microarray requires the use of appropriate statistical methods to assess the large amounts of data generated. A permutation-based test is proposed that incorporates spatial information of the two-dimensional antibody microarray. By borrowing strength from neighboring differentially expressed spots, the procedure is able to detect differentially expressed regions with high power while controlling the familywise type I error at 0.05 in simulation studies. The proposed methodology is also applied to a real microarray dataset.     

Identifying differentially expressed genes in dye-swapped microarray experiments of small sample size

When using microarray analysis to determine gene dependence, one of the goals is to identify differentially expressed genes. However, the inherent variations make analysis challenging. We propose a statistical method (SRA, swapped and regression analysis) especially for dye-swapped design and small sample size. Under general assumptions about the structure of the channels, scanner, and target effects from the experiment, we prove that SRA removes bias caused by these effects. We compare our method with ANOVA, using both simulated and real data. The results show that SRA has consistent sensitivity for the identification of differentially expressed genes in dye-swapped microarrays, particularly when the sample size is small. The program for the proposed method is available at http://www.ibms.sinica.edu.tw/∼csjfann/firstflow/program.htm.     

Identifying cluster number for subspace projected functional data clustering

We propose a new approach, the forward functional testing (FFT) procedure, to cluster number selection for functional data clustering. We present a framework of subspace projected functional data clustering based on the functional multiplicative random-effects model, and propose to perform functional hypothesis tests on equivalence of cluster structures to identify the number of clusters. The aim is to find the maximum number of distinctive clusters while retaining significant differences between cluster structures. The null hypotheses comprise equalities between the cluster mean functions and between the sets of cluster eigenfunctions of the covariance kernels. Bootstrap resampling methods are developed to construct reference distributions of the derived test statistics. We compare several other cluster number selection criteria, extended from methods of multivariate data, with the proposed FFT procedure. The performance of the proposed approaches is examined by simulation studies, with applications to clustering gene expression profiles.     

Identification and prioritization of differentially expressed genes for time-series gene expression data

Identification of differentially expressed genes (DEGs) in time course studies is very useful for understanding gene function, and can help determine key genes during specific stages of plant development. A few existing methods focus on the detection of DEGs within a single biological group, enabling to study temporal changes in gene expression. To utilize a rapidly increasing amount of single-group time-series expression data, we propose a two-step method that integrates the temporal characteristics of time-series data to obtain a B-spline curve fit. Firstly, a flat gene filter based on the Ljung–Box test is used to filter out flat genes. Then, a B-spline model is used to identify DEGs. For use in biological experiments, these DEGs should be screened, to determine their biological importance. To identify high-confidence promising DEGs for specific biological processes, we propose a novel gene prioritization approach based on the partner evaluation principle. This novel gene prioritization approach utilizes existing co-expression information to rank DEGs that are likely to be involved in a specific biological process/condition. The proposed method is validated on the Arabidopsis thaliana seed germination dataset and on the rice anther development expression dataset.     

Towards improving fuzzy clustering using support vector machine: Application to gene expression data

Recent advancement in microarray technology permits monitoring of the expression levels of a large set of genes across a number of time points simultaneously. For extracting knowledge from such huge volume of microarray gene expression data, computational analysis is required. Clustering is one of the important data mining tools for analyzing such microarray data to group similar genes into clusters. Researchers have proposed a number of clustering algorithms in this purpose. In this article, an attempt has been made in order to improve the performance of fuzzy clustering by combining it with support vector machine (SVM) classifier. A recently proposed real-coded variable string length genetic algorithm based clustering technique and an iterated version of fuzzy C-means clustering have been utilized in this purpose. The performance of the proposed clustering scheme has been compared with that of some well-known existing clustering algorithms and their SVM boosted versions for one simulated and six real life gene expression data sets. Statistical significance test based on analysis of variance (ANOVA) followed by posteriori Tukey-Kramer multiple comparison test has been conducted to establish the statistical significance of the superior performance of the proposed clustering scheme. Moreover biological significance of the clustering solutions have been established.     

Bayesian models for two-sample time-course microarray experiments

A truly functional Bayesian method for detecting temporally differentially expressed genes between two experimental conditions is presented. The method distinguishes between two biologically different set ups, one in which the two samples are interchangeable, and one in which the second sample is a modification of the first, i.e. the two samples are non-interchangeable. This distinction leads to two different Bayesian models, which allow more flexibility in modeling gene expression profiles. The method allows one to identify differentially expressed genes, to rank them and to estimate their expression profiles. The proposed procedure successfully deals with various technical difficulties which arise in microarray time-course experiments, such as small number of observations, non-uniform sampling intervals and presence of missing data or repeated measurements. The procedure allows one to account for various types of error, thus offering a good compromise between nonparametric and normality assumption based techniques. In addition, all evaluations are carried out using analytic expressions, hence the entire procedure requires very little computational effort. The performance of the procedure is studied using simulated and real data.     

Cascade of genetic algorithm and decision tree for cancer classification on gene expression data

Abstract: Cancer classification, through gene expression data analysis, has produced remarkable results, and has indicated that gene expression assays could significantly aid in the development of efficient cancer diagnosis and classification platforms. However, cancer classification, based on DNA array data, remains a difficult problem. The main challenge is the overwhelming number of genes relative to the number of training samples, which implies that there are a large number of irrelevant genes to be dealt with. Another challenge is from the presence of noise inherent in the data set. It makes accurate classification of data more difficult when the sample size is small. We apply genetic algorithms (GAs) with an initial solution provided by t statistics, called t‐GA, for selecting a group of relevant genes from cancer microarray data. The decision‐tree‐based cancer classifier is built on the basis of these selected genes. The performance of this approach is evaluated by comparing it to other gene selection methods using publicly available gene expression data sets. Experimental results indicate that t‐GA has the best performance among the different gene selection methods. The Z‐score figure also shows that some genes are consistently preferentially chosen by t‐GA in each data set.     

基于聚类和微粒群优化的基因选择新方法

鉴于传统的基因选择方法会选出大量冗余基因从而导致较低的样本预测准确率,提出一种基于聚类和微粒群优化的基因选择算法。首先采用聚类算法将基因分成固定数目的簇;然后,采用极限学习机作为分类器进行簇中的特征基因分类性能评价,得到一个备选基因库;最后,采用基于微粒群优化和极限学习机的缠绕法从备选基因库中选择具有最大分类率、最小数目的基因子集。所选出的基因具有良好的分类性能。在两个公开的微阵列数据集上的实验结果表明,相对于一些经典的方法,新方法能够以较少的基因获得更高的分类性能。     

A heavy-tailed empirical Bayes method for replicated microarray data

DNA microarray has been recognized as being an important tool for studying the expression of thousands of genes simultaneously. These experiments allow us to compare two different samples of cDNA obtained under different conditions. A novel method for the analysis of replicated microarray experiments based upon the modelling of gene expression distribution as a mixture of α-stable distributions is presented. Some features of the distribution of gene expression, such as Pareto tails and the fact that the variance of any given array increases concomitantly with an increase in the number of genes studied, suggest the possibility of modelling gene expression distribution on the basis of α-stable density. The proposed methodology uses very well known properties of α-stable distribution, such as the scale mixture of normals. A Bayesian log-posterior odds is calculated, which allows us to decide whether a gene is expressed differentially or not. The proposed methodology is illustrated using simulated and experimental data and the results are compared with other existing statistical approaches. The proposed heavy-tail model improves the performance of other distributions and is easily applicable to microarray gene data, specially if the dataset contains outliers or presents high variance between replicates.     

Direct integration of microarrays for selecting informative genes and phenotype classification

The ability to provide thousands of gene expression values simultaneously makes microarray data very useful for phenotype classification. A major constraint in phenotype classification is that the number of genes greatly exceeds the number of samples. We overcame this constraint in two ways; we increased the number of samples by integrating independently generated microarrays that had been designed with the same biological objectives, and reduced the number of genes involved in the classification by selecting a small set of informative genes. We were able to maximally use the abundant microarray data that is being stockpiled by thousands of different research groups while improving classification accuracy. Our goal is to implement a feature (gene) selection method that can be applicable to integrated microarrays as well as to build a highly accurate classifier that permits straightforward biological interpretation. In this paper, we propose a two-stage approach. Firstly, we performed a direct integration of individual microarrays by transforming an expression value into a rank value within a sample and identified informative genes by calculating the number of swaps to reach a perfectly split sequence. Secondly, we built a classifier which is a parameter-free ensemble method using only the pre-selected informative genes. By using our classifier that was derived from large, integrated microarray sample datasets, we achieved high accuracy, sensitivity, and specificity in the classification of an independent test dataset.