Simultaneous rank tests for detecting differentially expressed genes

Rank tests are known to be robust to outliers and violation of distributional assumptions. Two major issues besetting microarray data are violation of the normality assumption and contamination by outliers. In this article, we formulate the normal theory simultaneous tests and their aligned rank transformation (ART) analog for detecting differentially expressed genes. These tests are based on the least-squares estimates of the effects when data follow a linear model. Application of the two methods are then demonstrated on a real data set. To evaluate the performance of the aligned rank transform method with the corresponding normal theory method, data were simulated according to the characteristics of a real gene expression data. These simulated data are then used to compare the two methods with respect to their sensitivity to the distributional assumption and to outliers for controlling the family-wise Type I error rate, power, and false discovery rate. It is demonstrated that the ART generally possesses the robustness of validity property even for microarray data with small number of replications. Although these methods can be applied to more general designs, in this article the simulation study is carried out for a dye-swap design since this design is broadly used in cDNA microarray experiments.     

Evaluation of false discovery rate and power via sample size in microarray studies

Microarray studies are now common for human, agricultural plant and animal studies. False discovery rate (FDR) is widely used in the analysis of large-scale microarray data to account for problems associated with multiple testing. A well-designed microarray study should have adequate statistical power to detect the differentially expressed (DE) genes, while keeping the FDR acceptably low. In this paper, we used a mixture model of expression responses involving DE genes and non-DE genes to analyse theoretical FDR and power for simple scenarios where it is assumed that each gene has equal error variance and the gene effects are independent. A simulation study was used to evaluate the empirical FDR and power for more complex scenarios with unequal error variance and gene dependence. Based on this approach, we present a general guide for sample size requirement at the experimental design stage for prospective microarray studies. This paper presented an approach to explicitly connect the sample size with FDR and power. While the methods have been developed in the context of one-sample microarray studies, they are readily applicable to two-sample, and could be adapted to multiple-sample studies.     

Clustering and classification problems in genetics through U-statistics

ABSTRACT

Genetic data are frequently categorical and have complex dependence structures that are not always well understood. For this reason, clustering and classification based on genetic data, while highly relevant, are challenging statistical problems. Here we consider a versatile U-statistics-based approach for non-parametric clustering that allows for an unconventional way of solving these problems. In this paper we propose a statistical test to assess group homogeneity taking into account multiple testing issues and a clustering algorithm based on dissimilarities within and between groups that highly speeds up the homogeneity test. We also propose a test to verify classification significance of a sample in one of two groups. We present Monte Carlo simulations that evaluate size and power of the proposed tests under different scenarios. Finally, the methodology is applied to three different genetic data sets: global human genetic diversity, breast tumour gene expression and Dengue virus serotypes. These applications showcase this statistical framework's ability to answer diverse biological questions in the high dimension low sample size scenario while adapting to the specificities of the different datatypes.     

On simultaneous confidence intervals based on rank-estimates with application to analysis of gene expression data

Abstract

Inferential methods based on ranks present robust and powerful alternative methodology for testing and estimation. In this article, two objectives are followed. First, develop a general method of simultaneous confidence intervals based on the rank estimates of the parameters of a general linear model and derive the asymptotic distribution of the pivotal quantity. Second, extend the method to high dimensional data such as gene expression data for which the usual large sample approximation does not apply. It is common in practice to use the asymptotic distribution to make inference for small samples. The empirical investigation in this article shows that for methods based on the rank-estimates, this approach does not produce a viable inference and should be avoided. A method based on the bootstrap is outlined and it is shown to provide a reliable and accurate method of constructing simultaneous confidence intervals based on rank estimates. In particular it is shown that commonly applied methods of normal or t-approximation are not satisfactory, particularly for large-scale inferences. Methods based on ranks are uniquely suitable for analysis of microarray gene expression data since they often involve large scale inferences based on small samples containing a large number of outliers and violate the assumption of normality. A real microarray data is analyzed using the rank-estimate simultaneous confidence intervals. Viability of the proposed method is assessed through a Monte Carlo simulation study under varied assumptions.     

Minimum breakdown designs in blocks of size two

In scientific investigations, there are many situations where each two experimental units have to be grouped into a block of size two. For planning such experiments, the variance-based optimality criteria like A-, D- and E-criterion are typically employed to choose efficient designs, if the estimation efficiency of treatment contrasts is primarily concerned. Alternatively, if there are observations which tend to become lost during the experimental period, the robustness criteria against the unavailability of data should be strongly recommended for selecting the planning scheme. In this study, a new criterion, called minimum breakdown criterion, is proposed to quantify the robustness of designs in blocks of size two. Based on the proposed criterion, a new class of robust designs, called minimum breakdown designs, is defined. When various numbers of blocks are missing, the minimum breakdown designs provide the highest probabilities that all the treatment contrasts are estimable. An exhaustive search procedure is proposed to generate such designs. In addition, two classes of uniformly minimum breakdown designs are theoretically verified.     

Clustering time-course microarray data using functional Bayesian infinite mixture model

This paper presents a new Bayesian, infinite mixture model based, clustering approach, specifically designed for time-course microarray data. The problem is to group together genes which have “similar” expression profiles, given the set of noisy measurements of their expression levels over a specific time interval. In order to capture temporal variations of each curve, a non-parametric regression approach is used. Each expression profile is expanded over a set of basis functions and the sets of coefficients of each curve are subsequently modeled through a Bayesian infinite mixture of Gaussian distributions. Therefore, the task of finding clusters of genes with similar expression profiles is then reduced to the problem of grouping together genes whose coefficients are sampled from the same distribution in the mixture. Dirichlet processes prior is naturally employed in such kinds of models, since it allows one to deal automatically with the uncertainty about the number of clusters. The posterior inference is carried out by a split and merge MCMC sampling scheme which integrates out parameters of the component distributions and updates only the latent vector of the cluster membership. The final configuration is obtained via the maximum a posteriori estimator. The performance of the method is studied using synthetic and real microarray data and is compared with the performances of competitive techniques.     

Estimation of a residual distribution with small numbers of repeated measurements

The authors consider the estimation of a residual distribution for different measurement problems with a common measurement error process. The problem is motivated by issues arising in the analysis of gene expression data but should have application in other similar settings. It is implicitly assumed throughout that there are large numbers of measurements but small numbers of repeated measurements. As a consequence, the distribution of the estimated residuals is a biased estimate of the residual distribution. The authors present two methods for the estimation of the residual distribution with some restriction on the form of the distribution. They give an upper bound for the rate of convergence for an estimator based on the characteristic function and compare its performance with that of another estimator with simulations.     

Multiphase experiments with at least one later laboratory phase. II. Nonorthogonal designs

Principles and laws that apply to nonorthogonal multiphase experiments are developed and illustrated using examples that are nonorthogonal but structure‐balanced, not structure, but first‐order, balanced or unbalanced, thus exposing the differences between the different design types. The design of such experiments using standard designs, a catalogue of designs and computer searches is exemplified. Factor–allocation diagrams are employed to depict the allocations in the examples, and used in producing the anatomies of designs or, when possible, the related skeleton‐analysis‐of‐variance tables, to assess the properties of designs. The formulation of mixed models based on them is also described. Tools used for structure‐balanced experiments are also shown to be applicable to those experiments that are not.     

An extended association rule mining strategy for gene relationship discovery from microarray data

DNA microarrays allow for measuring expression levels of a large number of genes between different experimental conditions and/or samples. Association rule mining (ARM) methods are helpful in finding associational relationships between genes. However, classical association rule mining (CARM) algorithms extract only a subset of the associations that exist among different binary states; therefore can only infer part of the relationships on gene regulations. To resolve this problem, we developed an extended association rule mining (EARM) strategy along with a new way of the association rule definition. Compared with the CARM method, our new approach extracted more frequent genesets from a public microarray data set. The EARM method discovered some biologically interesting association rules that were not detected by CARM. Therefore, EARM provides an effective tool for exploring relationships among genes.