Split-Plot Microarray Experiments

This article focuses on microarray experiments with two or more factors in which treatment combinations of the factors corresponding to the samples paired together onto arrays are not completely random. A main effect of one (or more) factor(s) is confounded with arrays (the experimental blocks). This is called a split-plot microarray experiment. We utilise an analysis of variance (ANOVA) model to assess differentially expressed genes for between-array and within-array comparisons that are generic under a split-plot microarray experiment. Instead of standard t- or F-test statistics that rely on mean square errors of the ANOVA model, we use a robust method, referred to as 'a pooled percentile estimator', to identify genes that are differentially expressed across different treatment conditions. We illustrate the design and analysis of split-plot microarray experiments based on a case application described by Jin et al. A brief discussion of power and sample size for split-plot microarray experiments is also presented.     

Non-linear normalization and background correction in one-channel cDNA microarray studies

MOTIVATION: Data from one-channel cDNA microarray studies may exhibit poor reproducibility due to spatial heterogeneity, non-linear array-to-array variation and problems in correcting for background. Uncorrected, these phenomena can give rise to misleading conclusions. RESULTS: Spatial heterogeneity may be corrected using two-dimensional loess smoothing (Colantuoni et al., 2002). Non-linear between-array variation may be corrected using an iterative application of one-dimensional loess smoothing. A method for background correction using a smoothing function rather than simple subtraction is described. These techniques promote within-array spatial uniformity and between-array reproducibility. Their application is illustrated using data from a study of the effects of an insulin sensitizer, rosiglitazone, on gene expression in white adipose tissue in diabetic db/db mice. They may also be useful with data from two-channel cDNA microarrays and from oligonucleotide arrays. AVAILABILITY: R functions for the methods described are available on request from the author.     

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.     

Data extraction from composite oligonucleotide microarrays

Microarray or DNA chip technology is revolutionizing biology by empowering researchers in the collection of broad-scope gene information. It is well known that microarray-based measurements exhibit a substantial amount of variability due to a number of possible sources, ranging from hybridization conditions to image capture and analysis. In order to make reliable inferences and carry out quantitative analysis with microarray data, it is generally advisable to have more than one measurement of each gene. The availability of both between-array and within-array replicate measurements is essential for this purpose. Although statistical considerations call for increasing the number of replicates of both types, the latter is particularly challenging in practice due to a number of limiting factors, especially for in-house spotting facilities. We propose a novel approach to design so-called composite microarrays, which allow more replicates to be obtained without increasing the number of printed spots.     

limmaGUI: a graphical user interface for linear modeling of microarray data

SUMMARY: limmaGUI is a graphical user interface (GUI) based on R-Tcl/Tk for the exploration and linear modeling of data from two-color spotted microarray experiments, especially the assessment of differential expression in complex experiments. limmaGUI provides an interface to the statistical methods of the limma package for R, and is itself implemented as an R package. The software provides point and click access to a range of methods for background correction, graphical display, normalization, and analysis of microarray data. Arbitrarily complex microarray experiments involving multiple RNA sources can be accomodated using linear models and contrasts. Empirical Bayes shrinkage of the gene-wise residual variances is provided to ensure stable results even when the number of arrays is small. Integrated support is provided for quantitative spot quality weights, control spots, within-array replicate spots and multiple testing. limmaGUI is available for most platforms on the which R runs including Windows, Mac and most flavors of Unix. AVAILABILITY: http://bioinf.wehi.edu.au/limmaGUI.     

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.     

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.     

A structural mixed model for variances in differential gene expression studies

The importance of variance modelling is now widely known for the analysis of microarray data. In particular the power and accuracy of statistical tests for differential gene expressions are highly dependent on variance modelling. The aim of this paper is to use a structural model on the variances, which includes a condition effect and a random gene effect, and to propose a simple estimation procedure for these parameters by working on the empirical variances. The proposed variance model was compared with various methods on both real and simulated data. It proved to be more powerful than the gene-by-gene analysis and more robust to the number of false positives than the homogeneous variance model. It performed well compared with recently proposed approaches such as SAM and VarMixt even for a small number of replicates, and performed similarly to Limma. The main advantage of the structural model is that, thanks to the use of a linear mixed model on the logarithm of the variances, various factors of variation can easily be incorporated in the model, which is not the case for previously proposed empirical Bayes methods. It is also very fast to compute and is adapted to the comparison of more than two conditions.     

Improved statistical tests for differential gene expression by shrinking variance components estimates

Combining information across genes in the statistical analysis of microarray data is desirable because of the relatively small number of data points obtained for each individual gene. Here we develop an estimator of the error variance that can borrow information across genes using the James-Stein shrinkage concept. A new test statistic (FS) is constructed using this estimator. The new statistic is compared with other statistics used to test for differential expression: the gene-specific F test (F1), the pooled-variance F statistic (F3), a hybrid statistic (F2) that uses the average of the individual and pooled variances, the regularized t-statistic, the posterior odds statistic B, and the SAM t-test. The FS-test shows best or nearly best power for detecting differentially expressed genes over a wide range of simulated data in which the variance components associated with individual genes are either homogeneous or heterogeneous. Thus FS provides a powerful and robust approach to test differential expression of genes that utilizes information not available in individual gene testing approaches and does not suffer from biases of the pooled variance approach.     

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.     

Use of within-array replicate spots for assessing differential expression in microarray experiments

MOTIVATION: Spotted arrays are often printed with probes in duplicate or triplicate, but current methods for assessing differential expression are not able to make full use of the resulting information. The usual practice is to average the duplicate or triplicate results for each probe before assessing differential expression. This results in the loss of valuable information about genewise variability. RESULTS: A method is proposed for extracting more information from within-array replicate spots in microarray experiments by estimating the strength of the correlation between them. The method involves fitting separate linear models to the expression data for each gene but with a common value for the between-replicate correlation. The method greatly improves the precision with which the genewise variances are estimated and thereby improves inference methods designed to identify differentially expressed genes. The method may be combined with empirical Bayes methods for moderating the genewise variances between genes. The method is validated using data from a microarray experiment involving calibration and ratio control spots in conjunction with spiked-in RNA. Comparing results for calibration and ratio control spots shows that the common correlation method results in substantially better discrimination of differentially expressed genes from those which are not. The spike-in experiment also confirms that the results may be further improved by empirical Bayes smoothing of the variances when the sample size is small. AVAILABILITY: The methodology is implemented in the limma software package for R, available from the CRAN repository http://www.r-project.org     

Bivariate microarray analysis: statistical interpretation of two-channel functional genomics data

Conventional statistical methods for interpreting microarray data require large numbers of replicates in order to provide sufficient levels of sensitivity. We recently described a method for identifying differentially-expressed genes in one-channel microarray data 1. Based on the idea that the variance structure of microarray data can itself be a reliable measure of noise, this method allows statistically sound interpretation of as few as two replicates per treatment condition. Unlike the one-channel array, the two-channel platform simultaneously compares gene expression in two RNA samples. This leads to covariation of the measured signals. Hence, by accounting for covariation in the variance model, we can significantly increase the power of the statistical test. We believe that this approach has the potential to overcome limitations of existing methods. We present here a novel approach for the analysis of microarray data that involves modeling the variance structure of paired expression data in the context of a Bayesian framework. We also describe a novel statistical test that can be used to identify differentially-expressed genes. This method, bivariate microarray analysis (BMA), demonstrates dramatically improved sensitivity over existing approaches. We show that with only two array replicates, it is possible to detect gene expression changes that are at best detected with six array replicates by other methods. Further, we show that combining results from BMA with Gene Ontology annotation yields biologically significant results in a ligand-treated macrophage cell system.     

Comparison of microarray designs for class comparison and class discovery

MOTIVATION: Two-color microarray experiments in which an aliquot derived from a common RNA sample is placed on each array are called reference designs. Traditionally, microarray experiments have used reference designs, but designs without a reference have recently been proposed as alternatives. RESULTS: We develop a statistical model that distinguishes the different levels of variation typically present in cancer data, including biological variation among RNA samples, experimental error and variation attributable to phenotype. Within the context of this model, we examine the reference design and two designs which do not use a reference, the balanced block design and the loop design, focusing particularly on efficiency of estimates and the performance of cluster analysis. We calculate the relative efficiency of designs when there are a fixed number of arrays available, and when there are a fixed number of samples available. Monte Carlo simulation is used to compare the designs when the objective is class discovery based on cluster analysis of the samples. The number of discrepancies between the estimated clusters and the true clusters were significantly smaller for the reference design than for the loop design. The efficiency of the reference design relative to the loop and block designs depends on the relation between inter- and intra-sample variance. These results suggest that if cluster analysis is a major goal of the experiment, then a reference design is preferable. If identification of differentially expressed genes is the main concern, then design selection may involve a consideration of several factors.     

Spatial heterogeneity of periphytic microbial communities in a small pesticide-polluted river

Spatial variability in the microbial community composition of river biofilms was investigated in a small river using two spatial scales: one monitored the upstream–downstream pesticide contamination gradient, referred to as the 'between-section variability', and the other monitored a 100-m longitudinal transect (eight sampling sites per section) within each sampling section, referred to as the 'within-section variability'. Periphyton samples were collected in spring and winter on artificial substrates placed in the main channel of the river. Denaturing gradient gel electrophoresis (DGGE) was used to assess the prokaryotic and eukaryotic community richness and diversity, and HPLC pigment analysis to assess the global taxonomic composition of the photoautotrophic community. In order to try to reduce the biological variability due to differences in flow velocity and in light conditions within each sampling section, and consequently to take into account only the changes due to water chemistry, nine plates (three per sampling section) subjected to similar physical conditions were chosen, and the results for these plates were compared with those obtained for all 24 plates. As shown by DGGE and by HPLC analyses, using these three substrate plates exposed to similar environmental conditions did indeed reduce the within-section variability and maximize the between-section variability. This sampling strategy also improved the evaluation of the impact of pollutants on the periphytic communities, measured using short-term sensitivity testing.     

Statistical tests for differential expression in cDNA microarray experiments

Extracting biological information from microarray data requires appropriate statistical methods. The simplest statistical method for detecting differential expression is the t test, which can be used to compare two conditions when there is replication of samples. With more than two conditions, analysis of variance (ANOVA) can be used, and the mixed ANOVA model is a general and powerful approach for microarray experiments with multiple factors and/or several sources of variation.     

Optimising two-dye microarray designs for estimating associations with a quantitative trait

We discuss how the samples should be arranged in two-dye microarray studies when the objective is to investigate associations between gene expression and quantitative traits measured on each sample. Because there is typically large between array variation, information about the association will come from the differences in traits and expression measurements between the two values hybridised to the two dyes on the same array. It is shown that within-slide correlation of trait values should be minimised. The arrangement of samples for which this occurs will depend on the trait values in question, and is a computationally demanding problem. An alternative is to minimise the rank correlation. We discuss this and related issues for different combinations of numbers of samples and arrays. Data analysis, including estimation of the variance components, is also described.     

Heritabilities and additive genetic variances of the activities of some enzymes in Drosophila melanogaster populations living in different habitats

Drosophila melanogaster samples were collected from a large population in two habitats: farmyards and distilleries. Samples were taken from two villages in each habitat. Three isofemale lines were established from all four samples and full-sib crosses were set in each isofemale line. Activities of four enzymes (ADH, alpha GPDH, IDH and 6PGDH) were measured in the offspring of each cross on starch gel after electrophoresis. Broad sense heritabilities and additive genetic variances were estimated in all four samples. Most of the activity variation was observed within the isofemale lines. The isofemale lines tended to be more different in the distilleries than in the farmyards. There was no significant difference in the average activities between the two habitats for any of the enzymes investigated. The additive genetic variance of the enzyme activities did not exhibit a consistent habitat pattern. In the farmyard habitat, we detected a higher activity variation in Tiszafüred than in the other village. Strong correlation was observed among the activities of the enzymes investigated. Correlation coefficients indicated higher level of correlation in the samples collected in Tiszafüred than in those originating from Tiszaszolos. The heritability values were rather high and they had a considerable variation both between the habitats and across the enzymes.     

Illumina WG-6 BeadChip strips should be normalized separately

Background  

Illumina Sentrix-6 Whole-Genome Expression BeadChips are relatively new microarray platforms which have been used in many microarray studies in the past few years. These Chips have a unique design in which each Chip contains six microarrays and each microarray consists of two separate physical strips, posing special challenges for precise between-array normalization of expression values.