Microarray data analysis using rival penalized EM algorithm in normal mixture models

Microarrays are part of a new class of biotechnologies, which allow the monitoring of expression levels of thousands of genes simultaneously. In microarray data analysis, the comparison of gene expression profiles with respect to different conditions and the selection of biologically interesting genes are crucial tasks. Multivariate statistical methods have been applied to analyze these large data sets. To identify genes with altered expression under two experimental conditions, we propose a nonparametric statistical approach. Specifically, we propose estimating the distributions of a t-type statistic and its null statistic, using kernel methods. A comparison of these two distributions by means of a likelihood ratio test can identify genes with significantly changed expressions. A new method to provide more stable estimates of tail probabilities is proposed, as well as a method for the calculation of the cut-off point and the acceptance region. The methodology is applied to a leukaemia data set containing expression levels of 7129 genes, and is compared with normal mixture model and the traditional t-test.

SummaryPrivacy is becoming increasingly important in collaborative data analysis, especially those involving personal or sensitive information commonly arising from health and commercial settings. The aim of privacy preserving statistical algorithms is to allow inference to be drawn on the joint data without disclosing private data held by each party. This paper presents a privacy‐preserving expectation–maximization (PPEM) algorithm for carrying out maximum likelihood estimation of the parameters of mixture models. We address the scenario of horizontally partitioned data distributed among three or more parties. The PPEM algorithm is a two‐cycle iterative distributed algorithm for fitting mixture models under privacy‐preserving requirements. A distinct advantage of PPEM is that it does not require a trusted third party for cooperative learning, unlike most existing schemes that implement a master/slave hierarchy. By adopting a ring topology and adding random noises to messages before encryption, PPEM helps prevent information leakage in the case of corrupted parties. Furthermore, in contrast to existing works, which typically assume a Honest‐but‐Curious adversary, we consider the much stronger case of a Malicious adversary. For illustration, PPEM is applied to two of the most popular mixture models, namely, the normal mixture model (NMM) andt‐mixture model (tMM), and their effectiveness is analyzed through a security analysis. A real data example is also presented to evaluate the computational complexity and accuracy of PPEM relative to its non‐privacy‐preserving version.

Spotted cDNA microarrays are emerging as a powerful and cost-effective tool for large-scale analysis of gene expression. Microarrays can be used to measure the relative quantities of specific mRNAs in two or more tissue samples for thousands of genes simultaneously. While the power of this technology has been recognized, many open questions remain about appropriate analysis of microarray data. One question is how to make valid estimates of the relative expression for genes that are not biased by ancillary sources of variation. Recognizing that there is inherent "noise" in microarray data, how does one estimate the error variation associated with an estimated change in expression, i.e., how does one construct the error bars? We demonstrate that ANOVA methods can be used to normalize microarray data and provide estimates of changes in gene expression that are corrected for potential confounding effects. This approach establishes a framework for the general analysis and interpretation of microarray data.

Application of the S-score algorithm for analysis of oligonucleotide microarrays

An exciting biological advancement over the past few years is the use of microarray technologies to measure simultaneously the expression levels of thousands of genes. The bottleneck now is how to extract useful information from the resulting large amounts of data. An important and common task in analyzing microarray data is to identify genes with altered expression under two experimental conditions. We propose a nonparametric statistical approach, called the mixture model method (MMM), to handle the problem when there are a small number of replicates under each experimental condition. Specifically, we propose estimating the distributions of a t -type test statistic and its null statistic using finite normal mixture models. A comparison of these two distributions by means of a likelihood ratio test, or simply using the tail distribution of the null statistic, can identify genes with significantly changed expression. Several methods are proposed to effectively control the false positives. The methodology is applied to a data set containing expression levels of 1,176 genes of rats with and without pneumococcal middle ear infection.

In this paper, we propose a finite mixture model of generalized Gaussian distributions (GDD) for robust segmentation and data modeling in the presence of noise and outliers. The model has more flexibility to adapt the shape of data and less sensibility for over-fitting the number of classes than the Gaussian mixture. In a first part of the present work, we propose a derivation of the maximum-likelihood estimation of the parameters of the new mixture model and we propose an information-theory based approach for the selection of the number of classes. In a second part, we propose some applications relating to image, motion and foreground segmentation to measure the performance of the new model in image data modeling with comparison to the Gaussian mixture.

In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.

A challenging task in time series microarray data analysis is to identify co-expressed groups of genes from a large input space. The overall objective of this study is to obtain knowledge about the most important genes and clusters related to production and growth rate in a real-world microarray data analysis task. Various measures are engaged to evaluate the importance of each gene and to group genes based on their correlation with the output and each other. Some strategies for grouping and selecting genes are integrated resulting in several models tested for real biological data. All proposed models are tested on a real microarray data analysis problem and the results obtained are throughtly presented as well as interpreted from a biological perspective.

This paper addresses the problem of identifying groups that satisfy the specific conditions for the means of feature variables. In this study, we refer to the identified groups as “target clusters” (TCs). To identify TCs, we propose a method based on the normal mixture model (NMM) restricted by a linear combination of means. We provide an expectation–maximization (EM) algorithm to fit the restricted NMM by using the maximum-likelihood method. The convergence property of the EM algorithm and a reasonable set of initial estimates are presented. We demonstrate the method's usefulness and validity through a simulation study and two well-known data sets. The proposed method provides several types of useful clusters, which would be difficult to achieve with conventional clustering or exploratory data analysis methods based on the ordinary NMM. A simple comparison with another target clustering approach shows that the proposed method is promising in the identification.

Distribution mixtures are used as models to analyze grouped data. The estimation of parameters is an important step for mixture distributions. The latent class model is generally used as the analysis of mixture distributions for discrete data. In this paper, we consider the parameter estimation for a mixture of logistic regression models. We know that the expectation maximization (EM) algorithm was most used for estimating the parameters of logistic regression mixture models. In this paper, we propose a new type of fuzzy class model and then derive an algorithm for the parameter estimation of a fuzzy class logistic regression model. The effects of the explanatory variables on the response variables are described. The focus is on binary responses for the logistic regression mixture analysis with a fuzzy class model. An algorithm, called a fuzzy classification maximum likelihood (FCML), is then created. The mean squared error (MSE) based accuracy criterion for the FCML and EM algorithms to the parameter estimation of logistic regression mixture models are compared using the samples drawn from logistic regression mixtures of two classes. Numerical results show that the proposed FCML algorithm presents good accuracy and is recommended as a new tool for the parameter estimation of the logistic regression mixture models.

Mixture modeling provides an effective approach to the differential expression problem in microarray data analysis. Methods based on fully parametric mixture models are available, but lack of fit in some examples indicates that more flexible models may be beneficial. Existing, more flexible, mixture models work at the level of one-dimensional gene-specific summary statistics, and so when there are relatively few measurements per gene these methods may not provide sensitive detectors of differential expression. We propose a hierarchical mixture model to provide methodology that is both sensitive in detecting differential expression and sufficiently flexible to account for the complex variability of normalized microarray data. EM-based algorithms are used to fit both parametric and semiparametric versions of the model. We restrict attention to the two-sample comparison problem; an experiment involving Affymetrix microarrays and yeast translation provides the motivating case study. Gene-specific posterior probabilities of differential expression form the basis of statistical inference; they define short gene lists and false discovery rates. Compared to several competing methodologies, the proposed methodology exhibits good operating characteristics in a simulation study, on the analysis of spike-in data, and in a cross-validation calculation.

Marker-assisted genetic evaluation needs to infer genotypes at quantitative trait loci (QTL) based on the information of linked markers. As the inference usually provides the probability distribution of QTL genotypes rather than a specific genotype, marker-assisted genetic evaluation is characterized by the mixture model because of the uncertainty of QTL genotypes. It is, therefore, necessary to develop a statistical procedure useful for mixture model analyses. In this study, a set of mixture model equations was derived based on the normal mixture model and the EM algorithm for evaluating linear models with uncertain independent variables. The derived equations can be seen as an extension of Henderson's mixed model equations to mixture models and provide a general framework to deal with the issues of uncertain incidence matrices in linear models. The mixture model equations were applied to marker-assisted genetic evaluation with different parameterizations of QTL effects. A sire-QTL-effect model and a founder-QTL-effect model were used to illustrate the application of the mixture model equations. The potential advantages of the mixture model equations for marker-assisted genetic evaluation were discussed. The mixed-effect mixture model equations are flexible in modelling QTL effects and show desirable properties in estimating QTL effects, compared with Henderson's mixed model equations.

In this paper we address the problem of estimating the parameters of a Gaussian mixture model. Although the EM (Expectation-Maximization) algorithm yields the maximum-likelihood solution it requires a careful initialization of the parameters and the optimal number of kernels in the mixture may be unknown beforehand. We propose a criterion based on the entropy of the pdf (probability density function) associated to each kernel to measure the quality of a given mixture model. Two different methods for estimating Shannon entropy are proposed and a modification of the classical EM algorithm to find the optimal number of kernels in the mixture is presented. We test our algorithm in probability density estimation, pattern recognition and color image segmentation.

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.

Mixture modeling is a popular technique for identifying unobserved subpopulations (e.g., components) within a data set, with Gaussian (normal) mixture modeling being the form most widely used. Generally, the parameters of these Gaussian mixtures cannot be estimated in closed form, so estimates are typically obtained via an iterative process. The most common estimation procedure is maximum likelihood via the expectation-maximization (EM) algorithm. Like many approaches for identifying subpopulations, finite mixture modeling can suffer from locally optimal solutions, and the final parameter estimates are dependent on the initial starting values of the EM algorithm. Initial values have been shown to significantly impact the quality of the solution, and researchers have proposed several approaches for selecting the set of starting values. Five techniques for obtaining starting values that are implemented in popular software packages are compared. Their performances are assessed in terms of the following four measures: (1) the ability to find the best observed solution, (2) settling on a solution that classifies observations correctly, (3) the number of local solutions found by each technique, and (4) the speed at which the start values are obtained. On the basis of these results, a set of recommendations is provided to the user.