In a genome-wide association study (GWAS), the probability that a single nucleotide polymorphism (SNP) is not associated with a disease is its local false discovery rate (LFDR). The LFDR for each SNP is relative to a reference class of SNPs. For example, the LFDR of an exonic SNP can vary widely depending on whether it is considered relative to the separate reference class of other exonic SNPs or relative to the combined reference class of all SNPs in the data set. As a result, the analysis of the data based on the combined reference class might indicate that a specific exonic SNP is associated with the disease, while using the separate reference class indicates that it is not associated, or vice versa. To address that, we introduce empirical Bayes methods that simultaneously consider a combined reference class and a separate reference class. Our simulation studies indicate that the proposed methods lead to improved performance. The new maximum entropy method achieves that by depending on the separate class when it has enough SNPs for reliable LFDR estimation and depending solely on the combined class otherwise. We used the new methods to analyze data from a GWAS of 2,000 cases and 3,000 controls. R functions implementing the proposed methods are available on CRAN and Shiny .

BACKGROUND: In investigating differentially expressed genes or other selected features, researchers conduct hypothesis tests to determine which biological categories, such as those of the Gene Ontology (GO), are enriched for the selected features. Multiple comparison procedures (MCPs) are commonly used to prevent excessive false positive rates. Traditional MCPs, e.g., the Bonferroni method, go to the opposite extreme: strictly controlling a family-wise error rate, resulting in excessive false negative rates. Researchers generally prefer the more balanced approach of instead controlling the false discovery rate (FDR). However, the q-values that methods of FDR control assign to biological categories tend to be too low to reliably estimate the probability that a biological category is not enriched for the preselected features. Thus, we study an application of the other estimators of that probability, which is called the local FDR (LFDR). RESULTS: We considered five LFDR estimators for detecting enriched GO terms: a binomial-based estimator (BBE), a maximum likelihood estimator (MLE), a normalized MLE (NMLE), a histogram-based estimator assuming a theoretical null hypothesis (HBE), and a histogram-based estimator assuming an empirical null hypothesis (HBE-EN). Since NMLE depends not only on the data but also on the specified value of π0, the proportion of non-enriched GO terms, it is only advantageous when either π0 is already known with sufficient accuracy or there are data for only 1 GO term. By contrast, the other estimators work without specifying π0 but require data for at least 2 GO terms. Our simulation studies yielded the following summaries of the relative performance of each of those four estimators. HBE and HBE-EN produced larger biases for 2, 4, 8, 32, and 100 GO terms than BBE and MLE. BBE has the lowest bias if π0 is 1 and if the number of GO terms is between 2 and 32. The bias of MLE is no worse than that of BBE for 100 GO terms even when the ideal number of components in its underlying mixture model is unknown, but has high bias when the number of GO terms is small compared to the number of estimated parameters. For unknown values of π0, BBE has the lowest bias for a small number of GO terms (2-32 GO terms), and MLE has the lowest bias for a medium number of GO terms (100 GO terms). CONCLUSIONS: For enrichment detection, we recommend estimating the LFDR by MLE given at least a medium number of GO terms, by BBE given a small number of GO terms, and by NMLE given either only 1 GO term or precise knowledge of π0.

Abstract—Many genome-wide association studies have been conducted to identify single nucleotide polymorphisms (SNPs) that are associated with particular diseases or other traits. The local false discovery rate (LFDR) estimated using semiparametric models has enjoyed success in simultaneous inference. However, semiparametric LFDR estimators can be biased because they tend to overestimate the proportion of the nonassociated SNPs. We address the problem by adapting a simple parametric mixture model (PMM) and by comparing this model to the semiparametric mixture model (SMM) behind an LFDR estimator that is known to be conservatively biased. Then, we also compare the PMM with a parametric nonmixture model (PNM). In our simulation studies, we thoroughly analyze the performances of the three models under different values of p1, a prior probability that is approximately equal to the proportion of SNPs that are associated with the disease. When p₁ > 10%, the PMM generally performs better than the SMM. When p₁ < 0.1%, the SMM outperforms PMM. When p₁ lies between 0.1 and 10 percent, both methods have about the same performance. In that setting, the PMM may be preferred since it has the advantage of supplying an estimate of the detectability level of the nonassociated SNPs.

BACKGROUND: Sustained research on the problem of determining which genes are differentially expressed on the basis of microarray data has yielded a plethora of statistical algorithms, each justified by theory, simulation, or ad hoc validation and yet differing in practical results from equally justified algorithms. Recently, a concordance method that measures agreement among gene lists have been introduced to assess various aspects of differential gene expression detection. This method has the advantage of basing its assessment solely on the results of real data analyses, but as it requires examining gene lists of given sizes, it may be unstable. RESULTS: Two methodologies for assessing predictive error are described: a cross-validation method and a posterior predictive method. As a nonparametric method of estimating prediction error from observed expression levels, cross validation provides an empirical approach to assessing algorithms for detecting differential gene expression that is fully justified for large numbers of biological replicates. Because it leverages the knowledge that only a small portion of genes are differentially expressed, the posterior predictive method is expected to provide more reliable estimates of algorithm performance, allaying concerns about limited biological replication. In practice, the posterior predictive method can assess when its approximations are valid and when they are inaccurate. Under conditions in which its approximations are valid, it corroborates the results of cross validation. Both comparison methodologies are applicable to both single-channel and dual-channel microarrays. For the data sets considered, estimating prediction error by cross validation demonstrates that empirical Bayes methods based on hierarchical models tend to outperform algorithms based on selecting genes by their fold changes or by non-hierarchical model-selection criteria. (The latter two approaches have comparable performance.) The posterior predictive assessment corroborates these findings. CONCLUSIONS: Algorithms for detecting differential gene expression may be compared by estimating each algorithm's error in predicting expression ratios, whether such ratios are defined across microarray channels or between two independent groups.According to two distinct estimators of prediction error, algorithms using hierarchical models outperform the other algorithms of the study. The fact that fold-change shrinkage performed as well as conventional model selection criteria calls for investigating algorithms that combine the strengths of significance testing and fold-change estimation.

MOTIVATION: Measurements of gene expression over time enable the reconstruction of transcriptional networks. However, Bayesian networks and many other current reconstruction methods rely on assumptions that conflict with the differential equations that describe transcriptional kinetics. Practical approximations of kinetic models would enable inferring causal relationships between genes from expression data of microarray, tag-based and conventional platforms, but conclusions are sensitive to the assumptions made. RESULTS: The representation of a sufficiently large portion of genome enables computation of an upper bound on how much confidence one may place in influences between genes on the basis of expression data. Information about which genes encode transcription factors is not necessary but may be incorporated if available. The methodology is generalized to cover cases in which expression measurements are missing for many of the genes that might control the transcription of the genes of interest. The assumption that the gene expression level is roughly proportional to the rate of translation led to better empirical performance than did either the assumption that the gene expression level is roughly proportional to the protein level or the Bayesian model average of both assumptions. AVAILABILITY: http://www.oisb.ca points to R code implementing the methods (R Development Core Team 2004). SUPPLEMENTARY INFORMATION: http://www.davidbickel.com.

Abstract Motivation: The reconstruction of gene networks from gene-expression microarrays is gaining popularity as methods improve and as more data become available. The reliability of such networks could be judged by the probability that a connection between genes is spurious, resulting from chance fluctuations rather than from a true biological relationship. Results: Unlike the false discovery rate and positive false discovery rate, the decisive false discovery rate (dFDR) is exactly equal to a conditional probability without assuming independence or the randomness of hypothesis truth values. This property is useful not only in the common application to the detection of differential gene expression, but also in determining the probability of a spurious connection in a reconstructed gene network. Estimators of the dFDR can estimate each of three probabilities: (1) The probability that two genes that appear to be associated with each other lack such association. (2) The probability that a time ordering observed for two associated genes is misleading. (3) The probability that a time ordering observed for two genes is misleading, either because they are not associated or because they are associated without a lag in time. The first probability applies to both static and dynamic gene networks, and the other two only apply to dynamic gene networks. Availability: Cross-platform software for network reconstruction, probability estimation, and plotting is free from http://www.davidbickel.com in Statomics, a suite of R functions with a Java application. Contact: [email protected] Supplementary information: Color figures are available from http://www.davidbickel.com

MOTIVATION: Many methods of identifying differential expression in genes depend on testing the null hypotheses of exactly equal means or distributions of expression levels for each gene across groups, even though a statistically significant difference in the expression level does not imply the occurrence of any difference of biological or clinical significance. This is because a mathematical definition of 'differential expression' as any non-zero difference does not correspond to the differential expression biologists seek. Furthermore, while some current methods account for multiple comparisons in hypothesis tests, they do not accordingly adjust estimates of the degrees to which genes are differentially expressed. Both problems lead to overstating the relevance of findings. RESULTS: Testing whether genes have relevant differential expression can be accomplished with customized null hypotheses, thereby redefining 'differential expression' in a way that is more biologically meaningful. When such tests control the false discovery rate, they effectively discover genes based on a desired quantile of differential gene expression. Estimation of the degree to which genes are differentially expressed has been corrected for multiple comparisons. AVAILABILITY: R code is freely available from http://www.davidbickel.com and may become available from www.r-project.org or www.bioconductor.org SUPPLEMENTARY INFORMATION: Applications to cancer microarrays, an application in the absence of differential expression, pseudocode, and a guide to customizing the methods may be found at www.davidbickel.com and www.mathpreprints.com

Abstract Motivation: The success of each method of cluster analysis depends on how well its underlying model describes the patterns of expression. Outlier-resistant and distribution-insensitive clustering of genes are robust against violations of model assumptions. Results: A measure of dissimilarity that combines advantages of the Euclidean distance and the correlation coefficient is introduced. The measure can be made robust using a rank order correlation coefficient. A robust graphical method of summarizing the results of cluster analysis and a biological method of determining the number of clusters are also presented. These methods are applied to a public data set, showing that rank-based methods perform better than log-based methods. Availability: Software is available from http://www.davidbickel.com. Contact: [email protected] Supplementary Information: http://www.davidbickel.com will have updates and related articles * To whom correspondence should be addressed.