Size, power and false discovery rates

Modern scientific technology has provided a new class of large-scale simultaneous inference problems, with thousands of hypothesis tests to consider at the same time. Microarrays epitomize this type of technology, but similar situations arise in proteomics, spectroscopy, imaging, and social science surveys. This paper uses false discovery rate methods to carry out both size and power calculations on large-scale problems. A simple empirical Bayes approach allows the false discovery rate (fdr) analysis to proceed with a minimum of frequentist or Bayesian modeling assumptions. Closed-form accuracy formulas are derived for estimated false discovery rates, and used to compare different methodologies: local or tail-area fdr's, theoretical, permutation, or empirical null hypothesis estimates. Two microarray data sets as well as simulations are used to evaluate the methodology, the power diagnostics showing why nonnull cases might easily fail to appear on a list of ``significant'' discoveries.Comment: Published in at http://dx.doi.org/10.1214/009053606000001460 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric tests for differential gene expression and interaction effects in multi-factorial microarray experiments

BACKGROUND: Numerous nonparametric approaches have been proposed in literature to detect differential gene expression in the setting of two user-defined groups. However, there is a lack of nonparametric procedures to analyze microarray data with multiple factors attributing to the gene expression. Furthermore, incorporating interaction effects in the analysis of microarray data has long been of great interest to biological scientists, little of which has been investigated in the nonparametric framework. RESULTS: In this paper, we propose a set of nonparametric tests to detect treatment effects, clinical covariate effects, and interaction effects for multifactorial microarray data. When the distribution of expression data is skewed or heavy-tailed, the rank tests are substantially more powerful than the competing parametric F tests. On the other hand, in the case of light or medium-tailed distributions, the rank tests appear to be marginally less powerful than the parametric competitors. CONCLUSION: The proposed rank tests enable us to detect differential gene expression and establish interaction effects for microarray data with various non-normally distributed expression measurements across genome. In the presence of outliers, they are advantageous alternative approaches to the existing parametric F tests due to the robustness feature

OpWise: Operons aid the identification of differentially expressed genes in bacterial microarray experiments

BACKGROUND: Differentially expressed genes are typically identified by analyzing the variation between replicate measurements. These procedures implicitly assume that there are no systematic errors in the data even though several sources of systematic error are known. RESULTS: OpWise estimates the amount of systematic error in bacterial microarray data by assuming that genes in the same operon have matching expression patterns. OpWise then performs a Bayesian analysis of a linear model to estimate significance. In simulations, OpWise corrects for systematic error and is robust to deviations from its assumptions. In several bacterial data sets, significant amounts of systematic error are present, and replicate-based approaches overstate the confidence of the changers dramatically, while OpWise does not. Finally, OpWise can identify additional changers by assigning genes higher confidence if they are consistent with other genes in the same operon. CONCLUSION: Although microarray data can contain large amounts of systematic error, operons provide an external standard and allow for reasonable estimates of significance. OpWise is available at

A bayesian approach to estimation and testing in time-course microarray experiments

The objective of the present paper is to develop a truly functional Bayesian method specifically designed for time series microarray data. The method allows one to identify differentially expressed genes in a time-course microarray experiment, to rank them and to estimate their expression profiles. Each gene expression profile is modeled as an expansion over some orthonormal basis, where the coefficients and the number of basis functions are estimated from the data. The proposed procedure deals successfully with various technical difficulties that arise in typical microarray experiments such as a small number of observations, non-uniform sampling intervals and missing or replicated data. The procedure allows one to account for various types of errors and offers a good compromise between nonparametric techniques and techniques based on normality assumptions. In addition, all evaluations are performed using analytic expressions, so the entire procedure requires very small computational effort. The procedure is studied using both simulated and real data, and is compared with competitive recent approaches. Finally, the procedure is applied to a case study of a human breast cancer cell line stimulated with estrogen. We succeeded in finding new significant genes that were not marked in an earlier work on the same dataset

A Model Based Background Adjustment for Oligonucleotide Expression Arrays

High density oligonucleotide expression arrays are widely used in many areas of biomedical research. Affymetrix GeneChip arrays are the most popular. In the Affymetrix system, a fair amount of further pre-processing and data reduction occurs following the image processing step. Statistical procedures developed by academic groups have been successful at improving the default algorithms provided by the Affymetrix system. In this paper we present a solution to one of the pre-processing steps, background adjustment, based on a formal statistical framework. Our solution greatly improves the performance of the technology in various practical applications. Affymetrix GeneChip arrays use short oligonucleotides to probe for genes in an RNA sample. Typically each gene will be represented by 11-20 pairs of oligonucleotide probes. The first component of these pairs is referred to as a perfect match probe and is designed to hybridize only with transcripts from the intended gene (specific hybridization). However, hybridization by other sequences (non-specific hybridization) is unavoidable. Furthermore, hybridization strengths are measured by a scanner that introduces optical noise. Therefore, the observed intensities need to be adjusted to give accurate measurements of specific hybridization. One approach to adjusting is to pair each perfect match probe with a mismatch probe that is designed with the intention of measuring non-specific hybridization. The default adjustment, provided as part of the Affymetrix system, is based on the difference between perfect match and mismatch probe intensities. We have found that this approach can be improved via the use of estimators derived from a statistical model that use probe sequence information. The model is based on simple hybridization theory from molecular biology and experiments specifically designed to help develop it. A final step in the pre-processing of these arrays is to combine the 11-20 probe pair intensities, after background adjustment and normalization, for a given gene to define a measure of expression that represents the amount of the corresponding mRNA species. In this paper we illustrate the practical consequences of not adjusting appropriately for the presence of nonspecific hybridization and provide a solution based on our background adjustment procedure. Software that computes our adjustment is available as part of the Bioconductor project (http://www.bioconductor

Empirical array quality weights in the analysis of microarray data

BACKGROUND: Assessment of array quality is an essential step in the analysis of data from microarray experiments. Once detected, less reliable arrays are typically excluded or "filtered" from further analysis to avoid misleading results. RESULTS: In this article, a graduated approach to array quality is considered based on empirical reproducibility of the gene expression measures from replicate arrays. Weights are assigned to each microarray by fitting a heteroscedastic linear model with shared array variance terms. A novel gene-by-gene update algorithm is used to efficiently estimate the array variances. The inverse variances are used as weights in the linear model analysis to identify differentially expressed genes. The method successfully assigns lower weights to less reproducible arrays from different experiments. Down-weighting the observations from suspect arrays increases the power to detect differential expression. In smaller experiments, this approach outperforms the usual method of filtering the data. The method is available in the limma software package which is implemented in the R software environment. CONCLUSION: This method complements existing normalisation and spot quality procedures, and allows poorer quality arrays, which would otherwise be discarded, to be included in an analysis. It is applicable to microarray data from experiments with some level of replication

Goodness-of-fit tests in mixed models.

Mixed models, with both random and fixed effects, are most often estimated on the assumption that the random effects are normally distributed. In this paper we propose several formal tests of the hypothesis that the random effects and/or errors are normally distributed. Most of the proposed methods can be extended to generalized linear models where tests for non-normal distributions are of interest. Our tests are nonparametric in the sense that they are designed to detect virtually any alternative to normality. In case of rejection of the null hypothesis, the nonparametric estimation method that is used to construct a test provides an estimator of the alternative distribution.Mixed model; Hypothesis test; Nonparametric test; Minimum distance; Order selection;