多重检验法在基因芯片研究中鉴定差异表达基因的统计功效

鉴于基因芯片实验的造价,在基因芯片实验设计中,首要考虑的因素是需要多少重复才能检测出一个具有显著差异表达的基因。计算多重检验法要求的重复数（样本大小）或功效可为基因芯片实验设计提供重要的参考。为此,本文基于置换重抽样法构建了一种基因表达噪声混合分布模型。该方法适用各类基因表达数据,即无论是基因表达单噪声源或是多噪声源都可行。应用混合模型和多重检验法并给定统计功效。研究者能在基因芯片实验中获得所需要的最少生物学重复数：或者根据样本大小来确定测定一个显著差异表达的基因所具有的检验功效;或者根据样本大小和统计检验功效,选择最好的统计测验方法。本文以一组在老鼠中与中风有关的3000个基因的基因芯片实验所获得的数据为例,应用该方法拟和后组建了一个单分布模型（即表达单噪声源的分布模型）。根据该模型,我们计算了4种多重检验法在鉴定一个具有表达差异（D）值的基因中所需要的统计功效。结果表明。检测一个小的差异D值,4种多重检验法中B方法的统计功效最低,而BH方法最高。但是,对于鉴定一个具有最大表达差异的基因时,4种方法有相同的鉴定功效。与传统的单个检验法一样,BH方法检测一个小的变化所需要的效率不会随基因数目增加而改变,其他3种多重检验法的检测功效则随基因数目增加而降低。     

How many replicates of arrays are required to detect gene expression changes in microarray experiments? A mixture model approach

Background  

It has been recognized that replicates of arrays (or spots) may be necessary for reliably detecting differentially expressed genes in microarray experiments. However, the often-asked question of how many replicates are required has barely been addressed in the literature. In general, the answer depends on several factors: a given magnitude of expression change, a desired statistical power (that is, probability) to detect it, a specified Type I error rate, and the statistical method being used to detect the change. Here, we discuss how to calculate the number of replicates in the context of applying a nonparametric statistical method, the normal mixture model approach, to detect changes in gene expression.     

Calculation of the minimum number of replicate spots required for detection of significant gene expression fold change in microarray experiments

MOTIVATION: We present statistical methods for determining the number of per gene replicate spots required in microarray experiments. The purpose of these methods is to obtain an estimate of the sampling variability present in microarray data, and to determine the number of replicate spots required to achieve a high probability of detecting a significant fold change in gene expression, while maintaining a low error rate. Our approach is based on data from control microarrays, and involves the use of standard statistical estimation techniques. RESULTS: After analyzing two experimental data sets containing control array data, we were able to determine the statistical power available for the detection of significant differential expression given differing levels of replication. The inclusion of replicate spots on microarrays not only allows more accurate estimation of the variability present in an experiment, but more importantly increases the probability of detecting genes undergoing significant fold changes in expression, while substantially decreasing the probability of observing fold changes due to chance rather than true differential expression.     

Microarray segmentation methods significantly influence data precision

Little consideration has been given to the effect of different segmentation methods on the variability of data derived from microarray images. Previous work has suggested that the significant source of variability from microarray image analysis is from estimation of local background. In this study, we used Analysis of Variance (ANOVA) models to investigate the effect of methods of segmentation on the precision of measurements obtained from replicate microarray experiments. We used four different methods of spot segmentation (adaptive, fixed circle, histogram and GenePix) to analyse a total number of 156 172 spots from 12 microarray experiments. Using a two-way ANOVA model and the coefficient of repeatability, we show that the method of segmentation significantly affects the precision of the microarray data. The histogram method gave the lowest variability across replicate spots compared to other methods, and had the lowest pixel-to-pixel variability within spots. This effect on precision was independent of background subtraction. We show that these findings have direct, practical implications as the variability in precision between the four methods resulted in different numbers of genes being identified as differentially expressed. Segmentation method is an important source of variability in microarray data that directly affects precision and the identification of differentially expressed genes.     

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.     

Noise sampling method: an ANOVA approach allowing robust selection of differentially regulated genes measured by DNA microarrays

MOTIVATION: A crucial step in microarray data analysis is the selection of subsets of interesting genes from the initial set of genes. In many cases, especially when comparing a specific condition to a reference, the genes of interest are those which are differentially expressed. Two common methods for gene selection are: (a) selection by fold difference (at least n fold variation) and (b) selection by altered ratio (at least n standard deviations away from the mean ratio). RESULTS: The novel method proposed here is based on ANOVA and uses replicate spots to estimate an empirical distribution of the noise. The measured intensity range is divided in a number of intervals. A noise distribution is constructed for each such interval. Bootstrapping is used to map the desired confidence levels from the noise distribution corresponding to a given interval to the measured log ratios in that interval. If the method is applied on individual arrays having replicate spots, the method can calculate an overall width of the noise distribution which can be used as an indicator of the array quality. We compared this method with the fold change and unusual ratio method. We also discuss the relationship with an ANOVA model proposed by Churchill et al. In silico experiments were performed while controlling the degree of regulation as well as the amount of noise. Such experiments show the performance of the classical methods can be very unsatisfactory. We also compared the results of the 2-fold method with the results of the noise sampling method using pre and post immortalization cell lines derived from the MDAH041 fibroblasts hybridized on Affymetrix GeneChip arrays. The 2-fold method reported 198 genes as upregulated and 493 genes as downregulated. The noise sampling method reported 98 gene upregulated and 240 genes downregulated at the 99.99% confidence level. The methods agreed on 221 genes downregulated and 66 genes upregulated. Fourteen genes from the subset of genes reported by both methods were all confirmed by Q-RT-PCR. Alternative assays on various subsets of genes on which the two methods disagreed suggested that the noise sampling method is likely to provide fewer false positives.     

Identification of differentially expressed gene categories in microarray studies using nonparametric multivariate analysis

MOTIVATION: The field of microarray data analysis is shifting emphasis from methods for identifying differentially expressed genes to methods for identifying differentially expressed gene categories. The latter approaches utilize a priori information about genes to group genes into categories and enhance the interpretation of experiments aimed at identifying expression differences across treatments. While almost all of the existing approaches for identifying differentially expressed gene categories are practically useful, they suffer from a variety of drawbacks. Perhaps most notably, many popular tools are based exclusively on gene-specific statistics that cannot detect many types of multivariate expression change. RESULTS: We have developed a nonparametric multivariate method for identifying gene categories whose multivariate expression distribution differs across two or more conditions. We illustrate our approach and compare its performance to several existing procedures via the analysis of a real data set and a unique data-based simulation study designed to capture the challenges and complexities of practical data analysis. We show that our method has good power for differentiating between differentially expressed and non-differentially expressed gene categories, and we utilize a resampling based strategy for controlling the false discovery rate when testing multiple categories. AVAILABILITY: R code (www.r-project.org) for implementing our approach is available from the first author by request.     

Rank products: a simple, yet powerful, new method to detect differentially regulated genes in replicated microarray experiments

One of the main objectives in the analysis of microarray experiments is the identification of genes that are differentially expressed under two experimental conditions. This task is complicated by the noisiness of the data and the large number of genes that are examined simultaneously. Here, we present a novel technique for identifying differentially expressed genes that does not originate from a sophisticated statistical model but rather from an analysis of biological reasoning. The new technique, which is based on calculating rank products (RP) from replicate experiments, is fast and simple. At the same time, it provides a straightforward and statistically stringent way to determine the significance level for each gene and allows for the flexible control of the false-detection rate and familywise error rate in the multiple testing situation of a microarray experiment. We use the RP technique on three biological data sets and show that in each case it performs more reliably and consistently than the non-parametric t-test variant implemented in Tusher et al.'s significance analysis of microarrays (SAM). We also show that the RP results are reliable in highly noisy data. An analysis of the physiological function of the identified genes indicates that the RP approach is powerful for identifying biologically relevant expression changes. In addition, using RP can lead to a sharp reduction in the number of replicate experiments needed to obtain reproducible results.     

A Regression-Based Differential Expression Detection Algorithm for Microarray Studies with Ultra-Low Sample Size

Global gene expression analysis using microarrays and, more recently, RNA-seq, has allowed investigators to understand biological processes at a system level. However, the identification of differentially expressed genes in experiments with small sample size, high dimensionality, and high variance remains challenging, limiting the usability of these tens of thousands of publicly available, and possibly many more unpublished, gene expression datasets. We propose a novel variable selection algorithm for ultra-low-n microarray studies using generalized linear model-based variable selection with a penalized binomial regression algorithm called penalized Euclidean distance (PED). Our method uses PED to build a classifier on the experimental data to rank genes by importance. In place of cross-validation, which is required by most similar methods but not reliable for experiments with small sample size, we use a simulation-based approach to additively build a list of differentially expressed genes from the rank-ordered list. Our simulation-based approach maintains a low false discovery rate while maximizing the number of differentially expressed genes identified, a feature critical for downstream pathway analysis. We apply our method to microarray data from an experiment perturbing the Notch signaling pathway in Xenopus laevis embryos. This dataset was chosen because it showed very little differential expression according to limma, a powerful and widely-used method for microarray analysis. Our method was able to detect a significant number of differentially expressed genes in this dataset and suggest future directions for investigation. Our method is easily adaptable for analysis of data from RNA-seq and other global expression experiments with low sample size and high dimensionality.     

Internal standard-based analysis of microarray data. Part 1: analysis of differential gene expressions

Genome-scale microarray experiments for comparative analysis of gene expressions produce massive amounts of information. Traditional statistical approaches fail to achieve the required accuracy in sensitivity and specificity of the analysis. Since the problem can be resolved neither by increasing the number of replicates nor by manipulating thresholds, one needs a novel approach to the analysis. This article describes methods to improve the power of microarray analyses by defining internal standards to characterize features of the biological system being studied and the technological processes underlying the microarray experiments. Applying these methods, internal standards are identified and then the obtained parameters are used to define (i) genes that are distinct in their expression from background; (ii) genes that are differentially expressed; and finally (iii) genes that have similar dynamical behavior.     

Microarray results improve significantly as hybridization approaches equilibrium

Dual-channel long oligonucleotide microarrays are in widespread use. Although much attention has been given to proper experimental design and analysis regarding long oligonucleotide microarrays, relatively little information is available concerning the optimization of protocols. We carried out a series of microarray experiments designed to investigate the effects of different levels of target concentration and hybridization times using a long oligonucleotide library. Based on principles developed from nucleic acid renaturation kinetics studies, we show that increasing the time of hybridization from 18 h to 42 h and 66 h, especially when lower than optimal concentrations of target were used, significantly improved the quality of the microarray results. Longer hybridization times significantly increased the number of spots detected, signal-to-noise ratios, and the number of differentially expressed genes and correlations among replicate arrays. We conclude that at 18 h of incubation, target-to-probe hybridization has not reached equilibrium and that a relatively high proportion of nonspecific hybridization occurs. This result is striking, given that most, if not all, published microarray protocols stipulate 8-24 h for hybridization. Using shorter than optimal hybridization times (i.e., not allowing hybridization to reach equilibrium) has the consequence of underestimating the fold change of differentially expressed genes and of missing less represented sequences.     

Effect of pooling samples on the efficiency of comparative studies using microarrays

MOTIVATION: Many biomedical experiments are carried out by pooling individual biological samples. However, pooling samples can potentially hide biological variance and give false confidence concerning the data significance. In the context of microarray experiments for detecting differentially expressed genes, recent publications have addressed the problem of the efficiency of sample pooling, and some approximate formulas were provided for the power and sample size calculations. It is desirable to have exact formulas for these calculations and have the approximate results checked against the exact ones. We show that the difference between the approximate and the exact results can be large. RESULTS: In this study, we have characterized quantitatively the effect of pooling samples on the efficiency of microarray experiments for the detection of differential gene expression between two classes. We present exact formulas for calculating the power of microarray experimental designs involving sample pooling and technical replications. The formulas can be used to determine the total number of arrays and biological subjects required in an experiment to achieve the desired power at a given significance level. The conditions under which pooled design becomes preferable to non-pooled design can then be derived given the unit cost associated with a microarray and that with a biological subject. This paper thus serves to provide guidance on sample pooling and cost-effectiveness. The formulation in this paper is outlined in the context of performing microarray comparative studies, but its applicability is not limited to microarray experiments. It is also applicable to a wide range of biomedical comparative studies where sample pooling may be involved.     

A global approach to identify differentially expressed genes in cDNA (two-color) microarray experiments

MOTIVATION: Currently most of the methods for identifying differentially expressed genes fall into the category of so called single-gene-analysis, performing hypothesis testing on a gene-by-gene basis. In a single-gene-analysis approach, estimating the variability of each gene is required to determine whether a gene is differentially expressed or not. Poor accuracy of variability estimation makes it difficult to identify genes with small fold-changes unless a very large number of replicate experiments are performed. RESULTS: We propose a method that can avoid the difficult task of estimating variability for each gene, while reliably identifying a group of differentially expressed genes with low false discovery rates, even when the fold-changes are very small. In this article, a new characterization of differentially expressed genes is established based on a theorem about the distribution of ranks of genes sorted by (log) ratios within each array. This characterization of differentially expressed genes based on rank is an example of all-gene-analysis instead of single gene analysis. We apply the method to a cDNA microarray dataset and many low fold-changed genes (as low as 1.3 fold-changes) are reliably identified without carrying out hypothesis testing on a gene-by-gene basis. The false discovery rate is estimated in two different ways reflecting the variability from all the genes without the complications related to multiple hypothesis testing. We also provide some comparisons between our approach and single-gene-analysis based methods. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Comparison of methods for identifying differentially expressed genes across multiple conditions from microarray data

Identification of genes differentially expressed across multiple conditions has become an important statistical problem in analyzing large-scale microarray data. Many statistical methods have been developed to address the challenging problem. Therefore, an extensive comparison among these statistical methods is extremely important for experimental scientists to choose a valid method for their data analysis. In this study, we conducted simulation studies to compare six statistical methods: the Bonferroni (B-) procedure, the Benjamini and Hochberg (BH-) procedure, the Local false discovery rate (Localfdr) method, the Optimal Discovery Procedure (ODP), the Ranking Analysis of F-statistics (RAF), and the Significant Analysis of Microarray data (SAM) in identifying differentially expressed genes. We demonstrated that the strength of treatment effect, the sample size, proportion of differentially expressed genes and variance of gene expression will significantly affect the performance of different methods. The simulated results show that ODP exhibits an extremely high power in indentifying differentially expressed genes, but significantly underestimates the False Discovery Rate (FDR) in all different data scenarios. The SAM has poor performance when the sample size is small, but is among the best-performing methods when the sample size is large. The B-procedure is stringent and thus has a low power in all data scenarios. Localfdr and RAF show comparable statistical behaviors with the BH-procedure with favorable power and conservativeness of FDR estimation. RAF performs the best when proportion of differentially expressed genes is small and treatment effect is weak, but Localfdr is better than RAF when proportion of differentially expressed genes is large.     

Comparison of algorithms for the analysis of Affymetrix microarray data as evaluated by co-expression of genes in known operons

Oligonucleotide microarrays are an informative tool to elucidate gene regulatory networks. In order for gene expression levels to be comparable across microarrays, normalization procedures have to be invoked. A large number of methods have been described to correct for systematic biases in microarray experiments. The performance of these methods has been tested only to a limited extend. Here, we evaluate two different types of microarray analyses: (i) the same gene in replicate samples and (ii) different, but co-expressed genes in the same sample. The reliability of the latter analysis needs to be determined for the analysis of regulatory networks and our report is the first attempt to evaluate for the accuracy of different microarray normalization methods in this respect. Consistent with previous results we observed a large effect of the normalization method on the outcome of the expression analyses. Our analyses indicate that different normalization methods should be performed depending on whether a study is aiming to detect differential gene expression between independent samples or whether co-expressed genes should be identified. We make recommendations about the most appropriate method to use.     

The PowerAtlas: a power and sample size atlas for microarray experimental design and research

Background  

Microarrays permit biologists to simultaneously measure the mRNA abundance of thousands of genes. An important issue facing investigators planning microarray experiments is how to estimate the sample size required for good statistical power. What is the projected sample size or number of replicate chips needed to address the multiple hypotheses with acceptable accuracy? Statistical methods exist for calculating power based upon a single hypothesis, using estimates of the variability in data from pilot studies. There is, however, a need for methods to estimate power and/or required sample sizes in situations where multiple hypotheses are being tested, such as in microarray experiments. In addition, investigators frequently do not have pilot data to estimate the sample sizes required for microarray studies.     

Intensity dependent estimation of noise in microarrays improves detection of differentially expressed genes

Background  

In many microarray experiments, analysis is severely hindered by a major difficulty: the small number of samples for which expression data has been measured. When one searches for differentially expressed genes, the small number of samples gives rise to an inaccurate estimation of the experimental noise. This, in turn, leads to loss of statistical power.     

Sample size for identifying differentially expressed genes in microarray experiments.

Microarray technology allows simultaneous comparison of expression levels of thousands of genes under each condition. This paper concerns sample size calculation in the identification of differentially expressed genes between a control and a treated sample. In a typical experiment, only a fraction of genes (altered genes) is expected to be differentially expressed between two samples. Sample size determination depends on a number of factors including the specified significance level (alpha), the desired statistical power (1-beta), the fraction (eta) of truly altered genes out of the total g genes studied, and the effect sizes (Delta) for the altered genes. This paper proposes a method to calculate the number of arrays required to detect at least 100lambda % (where 0 < lambda < or = 1) of the truly altered genes under the model of an equal effect size for all altered genes. The required numbers of arrays are tabulated for various values of alpha, beta, Delta, eta, and lambda for the one-sample and two-sample t-tests for g = 10,000. Based on the proposed approach, to identify up to 90% of truly altered genes among the unknown number of truly altered genes, the estimated numbers of arrays needed appear to be manageable. For instance, when the standardized effect size is at least 2.0, the number of arrays needed is less than or equal to 14 for the two-sample t-test and is less than or equal to 10 for the one-sample t-test. As the cost per array declines, such array numbers become practical. The proposed method offers a simple, intuitive, and practical way to determine the number of arrays needed in microarray experiments in which the true correlation structure among the genes under investigation cannot be reasonably assumed. An example dataset is used to illustrate the use of the proposed approach to plan microarray experiments.     

Parallel multiplicity and error discovery rate (EDR) in microarray experiments

Background  

In microarray gene expression profiling experiments, differentially expressed genes (DEGs) are detected from among tens of thousands of genes on an array using statistical tests. It is important to control the number of false positives or errors that are present in the resultant DEG list. To date, more than 20 different multiple test methods have been reported that compute overall Type I error rates in microarray experiments. However, these methods share the following dilemma: they have low power in cases where only a small number of DEGs exist among a large number of total genes on the array.