Differential analysis of DNA microarray gene expression data

Here, we review briefly the sources of experimental and biological variance that affect the interpretation of high-dimensional DNA microarray experiments. We discuss methods using a regularized t-test based on a Bayesian statistical framework that allow the identification of differentially regulated genes with a higher level of confidence than a simple t-test when only a few experimental replicates are available. We also describe a computational method for calculating the global false-positive and false-negative levels inherent in a DNA microarray data set. This method provides a probability of differential expression for each gene based on experiment-wide false-positive and -negative levels driven by experimental error and biological variance.     

Model-based variance-stabilizing transformation for Illumina microarray data

Variance stabilization is a step in the preprocessing of microarray data that can greatly benefit the performance of subsequent statistical modeling and inference. Due to the often limited number of technical replicates for Affymetrix and cDNA arrays, achieving variance stabilization can be difficult. Although the Illumina microarray platform provides a larger number of technical replicates on each array (usually over 30 randomly distributed beads per probe), these replicates have not been leveraged in the current log2 data transformation process. We devised a variance-stabilizing transformation (VST) method that takes advantage of the technical replicates available on an Illumina microarray. We have compared VST with log2 and Variance-stabilizing normalization (VSN) by using the Kruglyak bead-level data (2006) and Barnes titration data (2005). The results of the Kruglyak data suggest that VST stabilizes variances of bead-replicates within an array. The results of the Barnes data show that VST can improve the detection of differentially expressed genes and reduce false-positive identifications. We conclude that although both VST and VSN are built upon the same model of measurement noise, VST stabilizes the variance better and more efficiently for the Illumina platform by leveraging the availability of a larger number of within-array replicates. The algorithms and Supplementary Data are included in the lumi package of Bioconductor, available at: www.bioconductor.org.     

Improved statistical inference from DNA microarray data using analysis of variance and a Bayesian statistical framework. Analysis of global gene expression in Escherichia coli K12

We describe statistical methods based on the t test that can be conveniently used on high density array data to test for statistically significant differences between treatments. These t tests employ either the observed variance among replicates within treatments or a Bayesian estimate of the variance among replicates within treatments based on a prior estimate obtained from a local estimate of the standard deviation. The Bayesian prior allows statistical inference to be made from microarray data even when experiments are only replicated at nominal levels. We apply these new statistical tests to a data set that examined differential gene expression patterns in IHF(+) and IHF(-) Escherichia coli cells (Arfin, S. M., Long, A. D., Ito, E. T., Tolleri, L., Riehle, M. M., Paegle, E. S., and Hatfield, G. W. (2000) J. Biol. Chem. 275, 29672-29684). These analyses identify a more biologically reasonable set of candidate genes than those identified using statistical tests not incorporating a Bayesian prior. We also show that statistical tests based on analysis of variance and a Bayesian prior identify genes that are up- or down-regulated following an experimental manipulation more reliably than approaches based only on a t test or fold change. All the described tests are implemented in a simple-to-use web interface called Cyber-T that is located on the University of California at Irvine genomics web site.     

Replication, Variation and Normalisation in Microarray Experiments

INTRODUCTION: Microarray experiments often have complex designs that include sample pooling, biological and technical replication, sample pairing and dye-swapping. This article demonstrates how statistical modelling can illuminate issues in the design and analysis of microarray experiments, and this information can then be used to plan effective studies. METHODS: A very detailed statistical model for microarray data is introduced, to show the possible sources of variation that are present in even the simplest microarray experiments. Based on this model, the efficacy of common experimental designs, normalisation methodologies and analyses is determined. RESULTS: When the cost of the arrays is high compared with the cost of samples, sample pooling and spot replication are shown to be efficient variance reduction methods, whereas technical replication of whole arrays is demonstrated to be very inefficient. Dye-swap designs can use biological replicates rather than technical replicates to improve efficiency and simplify analysis. When the cost of samples is high and technical variation is a major portion of the error, technical replication can be cost effective. Normalisation by centreing on a small number of spots may reduce array effects, but can introduce considerable variation in the results. Centreing using the bulk of spots on the array is less variable. Similarly, normalisation methods based on regression methods can introduce variability. Except for normalisation methods based on spiking controls, all normalisation requires that most genes do not differentially express. Methods based on spatial location and/or intensity also require that the nondifferentially expressing genes are at random with respect to location and intensity. Spotting designs should be carefully done so that spot replicates are widely spaced on the array, and genes with similar expression patterns are not clustered. DISCUSSION: The tools for statistical design of experiments can be applied to microarray experiments to improve both efficiency and validity of the studies. Given the high cost of microarray experiments, the benefits of statistical input prior to running the experiment cannot be over-emphasised.     

Rosetta error model for gene expression analysis

MOTIVATION: In microarray gene expression studies, the number of replicated microarrays is usually small because of cost and sample availability, resulting in unreliable variance estimation and thus unreliable statistical hypothesis tests. The unreliable variance estimation is further complicated by the fact that the technology-specific variance is intrinsically intensity-dependent. RESULTS: The Rosetta error model captures the variance-intensity relationship for various types of microarray technologies, such as single-color arrays and two-color arrays. This error model conservatively estimates intensity error and uses this value to stabilize the variance estimation. We present two commonly used error models: the intensity error-model for single-color microarrays and the ratio error model for two-color microarrays or ratios built from two single-color arrays. We present examples to demonstrate the strength of our error models in improving statistical power of microarray data analysis, particularly, in increasing expression detection sensitivity and specificity when the number of replicates is limited.     

Comparing three methods for variance estimation with duplicated high density oligonucleotide arrays

Microarray experiments are being increasingly used in molecular biology. A common task is to detect genes with differential expression across two experimental conditions, such as two different tissues or the same tissue at two time points of biological development. To take proper account of statistical variability, some statistical approaches based on the t-statistic have been proposed. In constructing the t-statistic, one needs to estimate the variance of gene expression levels. With a small number of replicated array experiments, the variance estimation can be challenging. For instance, although the sample variance is unbiased, it may have large variability, leading to a large mean squared error. For duplicated array experiments, a new approach based on simple averaging has recently been proposed in the literature. Here we consider two more general approaches based on nonparametric smoothing. Our goal is to assess the performance of each method empirically. The three methods are applied to a colon cancer data set containing 2,000 genes. Using two arrays, we compare the variance estimates obtained from the three methods. We also consider their impact on the t-statistics. Our results indicate that the three methods give variance estimates close to each other. Due to its simplicity and generality, we recommend the use of the smoothed sample variance for data with a small number of replicates. Electronic Publication     

Improving the statistical detection of regulated genes from microarray data using intensity-based variance estimation

Background  

Gene microarray technology provides the ability to study the regulation of thousands of genes simultaneously, but its potential is limited without an estimate of the statistical significance of the observed changes in gene expression. Due to the large number of genes being tested and the comparatively small number of array replicates (e.g., N = 3), standard statistical methods such as the Student's t-test fail to produce reliable results. Two other statistical approaches commonly used to improve significance estimates are a penalized t-test and a Z-test using intensity-dependent variance estimates.     

Selection and validation of normalization methods for c-DNA microarrays using within-array replications

MOTIVATION: Normalization of microarray data is essential for multiple-array analyses. Several normalization protocols have been proposed based on different biological or statistical assumptions. A fundamental problem arises whether they have effectively normalized arrays. In addition, for a given array, the question arises how to choose a method to most effectively normalize the microarray data. RESULTS: We propose several techniques to compare the effectiveness of different normalization methods. We approach the problem by constructing statistics to test whether there are any systematic biases in the expression profiles among duplicated spots within an array. The test statistics involve estimating the genewise variances. This is accomplished by using several novel methods, including empirical Bayes methods for moderating the genewise variances and the smoothing methods for aggregating variance information. P-values are estimated based on a normal or chi approximation. With estimated P-values, we can choose a most appropriate method to normalize a specific array and assess the extent to which the systematic biases due to the variations of experimental conditions have been removed. The effectiveness and validity of the proposed methods are convincingly illustrated by a carefully designed simulation study. The method is further illustrated by an application to human placenta cDNAs comprising a large number of clones with replications, a customized microarray experiment carrying just a few hundred genes on the study of the molecular roles of Interferons on tumor, and the Agilent microarrays carrying tens of thousands of total RNA samples in the MAQC project on the study of reproducibility, sensitivity and specificity of the data. AVAILABILITY: Code to implement the method in the statistical package R is available from the authors.     

Bayesian model averaging: development of an improved multi-class, gene selection and classification tool for microarray data

MOTIVATION: Selecting a small number of relevant genes for accurate classification of samples is essential for the development of diagnostic tests. We present the Bayesian model averaging (BMA) method for gene selection and classification of microarray data. Typical gene selection and classification procedures ignore model uncertainty and use a single set of relevant genes (model) to predict the class. BMA accounts for the uncertainty about the best set to choose by averaging over multiple models (sets of potentially overlapping relevant genes). RESULTS: We have shown that BMA selects smaller numbers of relevant genes (compared with other methods) and achieves a high prediction accuracy on three microarray datasets. Our BMA algorithm is applicable to microarray datasets with any number of classes, and outputs posterior probabilities for the selected genes and models. Our selected models typically consist of only a few genes. The combination of high accuracy, small numbers of genes and posterior probabilities for the predictions should make BMA a powerful tool for developing diagnostics from expression data. AVAILABILITY: The source codes and datasets used are available from our Supplementary website.     

The effect of replication on gene expression microarray experiments

MOTIVATION: We examine the effect of replication on the detection of apparently differentially expressed genes in gene expression microarray experiments. Our analysis is based on a random sampling approach using real data sets from 16 published studies. We consider both the ability to find genes that meet particular statistical criteria as well as the stability of the results in the face of changing levels of replication. RESULTS: While dependent on the data source, our findings suggest that stable results are typically not obtained until at least five biological replicates have been used. Conversely, for most studies, 10-15 replicates yield results that are quite stable, and there is less improvement in stability as the number of replicates is further increased. Our methods will be of use in evaluating existing data sets and in helping to design new studies.     

Large Scale Comparison of Gene Expression Levels by Microarrays and RNAseq Using TCGA Data

RNAseq and microarray methods are frequently used to measure gene expression level. While similar in purpose, there are fundamental differences between the two technologies. Here, we present the largest comparative study between microarray and RNAseq methods to date using The Cancer Genome Atlas (TCGA) data. We found high correlations between expression data obtained from the Affymetrix one-channel microarray and RNAseq (Spearman correlations coefficients of ∼0.8). We also observed that the low abundance genes had poorer correlations between microarray and RNAseq data than high abundance genes. As expected, due to measurement and normalization differences, Agilent two-channel microarray and RNAseq data were poorly correlated (Spearman correlations coefficients of only ∼0.2). By examining the differentially expressed genes between tumor and normal samples we observed reasonable concordance in directionality between Agilent two-channel microarray and RNAseq data, although a small group of genes were found to have expression changes reported in opposite directions using these two technologies. Overall, RNAseq produces comparable results to microarray technologies in term of expression profiling. The RNAseq normalization methods RPKM and RSEM produce similar results on the gene level and reasonably concordant results on the exon level. Longer exons tended to have better concordance between the two normalization methods than shorter exons.     

Variance stabilization and normalization for one-color microarray data using a data-driven multiscale approach

MOTIVATION: Many standard statistical techniques are effective on data that are normally distributed with constant variance. Microarray data typically violate these assumptions since they come from non-Gaussian distributions with a non-trivial mean-variance relationship. Several methods have been proposed that transform microarray data to stabilize variance and draw its distribution towards the Gaussian. Some methods, such as log or generalized log, rely on an underlying model for the data. Others, such as the spread-versus-level plot, do not. We propose an alternative data-driven multiscale approach, called the Data-Driven Haar-Fisz for microarrays (DDHFm) with replicates. DDHFm has the advantage of being 'distribution-free' in the sense that no parametric model for the underlying microarray data is required to be specified or estimated; hence, DDHFm can be applied very generally, not just to microarray data. RESULTS: DDHFm achieves very good variance stabilization of microarray data with replicates and produces transformed intensities that are approximately normally distributed. Simulation studies show that it performs better than other existing methods. Application of DDHFm to real one-color cDNA data validates these results. AVAILABILITY: The R package of the Data-Driven Haar-Fisz transform (DDHFm) for microarrays is available in Bioconductor and CRAN.     

Leveraging two-way probe-level block design for identifying differential gene expression with high-density oligonucleotide arrays

Background  

To identify differentially expressed genes across experimental conditions in oligonucleotide microarray experiments, existing statistical methods commonly use a summary of probe-level expression data for each probe set and compare replicates of these values across conditions using a form of the t-test or rank sum test. Here we propose the use of a statistical method that takes advantage of the built-in redundancy architecture of high-density oligonucleotide arrays.     

Establishing Informative Prior for Gene Expression Variance from Public Databases

Identifying differential expressed genes across various conditions or genotypes is the most typical approach to studying the regulation of gene expression. An estimate of gene-specific variance is often needed for the assessment of statistical significance in most differential expression (DE) detection methods, including linear models (e.g., for transformed and normalized microarray data) and generalized linear models (e.g., for count data in RNAseq). Due to a common limit in sample size, the variance estimate is often unstable in small experiments. Shrinkage estimates using empirical Bayes methods have proven useful in improving the variance estimate, hence improving the detection of DE. The most widely used empirical Bayes methods borrow information across genes within the same experiments. In these methods, genes are considered exchangeable or exchangeable conditioning on expression level. We propose, with the increasing accumulation of expression data, borrowing information from historical data on the same gene can provide better estimate of gene-specific variance, thus further improve DE detection. Specifically, we show that the variation of gene expression is truly gene-specific and reproducible between different experiments. We present a new method to establish informative gene-specific prior on the variance of expression using existing public data, and illustrate how to shrink the variance estimate and detect DE. We demonstrate improvement in DE detection under our strategy compared to leading DE detection methods.     

EVE (external variance estimation) increases statistical power for detecting differentially expressed genes

Accurately identifying differentially expressed genes from microarray data is not a trivial task, partly because of poor variance estimates of gene expression signals. Here, after analyzing 380 replicated microarray experiments, we found that probesets have typical, distinct variances that can be estimated based on a large number of microarray experiments. These probeset-specific variances depend at least in part on the function of the probed gene: genes for ribosomal or structural proteins often have a small variance, while genes implicated in stress responses often have large variances. We used these variance estimates to develop a statistical test for differentially expressed genes called EVE (external variance estimation). The EVE algorithm performs better than the t-test and LIMMA on some real-world data, where external information from appropriate databases is available. Thus, EVE helps to maximize the information gained from a typical microarray experiment. Nonetheless, only a large number of replicates will guarantee to identify nearly all truly differentially expressed genes. However, our simulation studies suggest that even limited numbers of replicates will usually result in good coverage of strongly differentially expressed genes.     

Microarray Data Analysis

The analysis of differential gene expression in microarray experiments requires the development of adequate statistical tools. This article describes a simple statistical method for detecting differential expression between two conditions with a low number of replicates. When comparing two group means using a traditional t-test, gene-specific variance estimates are unstable and can lead to wrong conclusions. We construct a likelihood ratio test while modelling these variances hierarchically across all genes, and express it as a t-test statistic. By borrowing information across genes we can take advantage of their large numbers, and still yield a gene-specific test statistic. We show that this hierarchical t-test is more powerful than its traditional version and generates less false positives in a simulation study, especially with small sample sizes. This approach can be extended to cases where there are more than two groups.     

An associative analysis of gene expression array data

MOTIVATION: We face the absence of optimized standards to guide normalization, comparative analysis, and interpretation of data sets. One aspect of this is that current methods of statistical analysis do not adequately utilize the information inherent in the large data sets generated in a microarray experiment and require a tradeoff between detection sensitivity and specificity. RESULTS: We present a multistep procedure for analysis of mRNA expression data obtained from cDNA array methods. To identify and classify differentially expressed genes, results from standard paired t-test of normalized data are compared with those from a novel method, denoted an associative analysis. This method associates experimental gene expressions presented as residuals in regression analysis against control averaged expressions to a common standard-the family of similarly computed residuals for low variability genes derived from control experiments. By associating changes in expression of a given gene to a large family of equally expressed genes of the control group, this method utilizes the large data sets inherent in microarray experiments to increase both specificity and sensitivity. The overall procedure is illustrated by tabulation of genes whose expression differs significantly between Snell dwarf mice (dw/dw) and their phenotypically normal littermates (dw/+, +/+). Of the 2,352 genes examined only 450-500 were expressed above the background levels observed in nonexpressed genes and of these 120 were established as differentially expressed in dwarf mice at a significance level that excludes appearance of false positive determinations.     

A new method for nonhomogeneous time course expression analysis

Time course expression analysis constitutes a large portion of applications of microarray experiments. One primary goal of such experiments is to detect genes with the temporal changes over a period of time or at some interested time points. Difficulties arising from data with small number of replicates over only a few unaligned time points in multiple groups pose challenges for efficient statistical analysis. Some known methods are limited by the unverifiable assumptions or by the scope of applications for only two groups. We present a new method for detecting differentially expressed genes under nonhomogeneous time course experiments in multiple groups. The new method first models the time course curve of one gene by a Gaussian process to align the nonhomogeneous time course data and to compute the gradient of the time course curve as well, the latter of which is used as directional information to enhance the sensitivity of detection for temporal changes. Second, we adopt a nonparametric method to test a surrogate hypothesis based on the augmented data from the Gaussian process model. The proposed method is robust in terms of model fitting and testing. It does not require any distributional assumption for the observations or the test statistic and the method works for the case with as few as triplicate samples over four or five time points under multiple groups. We show the effectiveness and superiority of the new method in comparison with some existing methods using simulated models and two real data sets.     

A mixture model approach to detecting differentially expressed genes with microarray data

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.