BACKGROUND: Identifying sets of related genes (gene sets) that are empirically associated with a treatment or phenotype often yields valuable biological insights. Several methods effectively identify gene sets in which individual genes have simple monotonic relationships with categorical, quantitative, or censored event-time variables. Some distance-based methods, such as distance correlations, may detect complex non-monotone associations of a gene-set with a quantitative variable that elude other methods. However, the distance correlations have yet to be generalized to associate gene-sets with categorical and censored event-time endpoints. Also, there is a need to determine which genes empirically drive the significance of an association of a gene set with an endpoint. RESULTS: We develop gene-set distance analysis (GSDA) by generalizing distance correlations to evaluate the association of a gene set with categorical and censored event-time variables. We also develop a backward elimination procedure to identify a subset of genes that empirically drive significant associations. In simulation studies, GSDA more effectively identified complex non-monotone gene-set associations than did six other published methods. In the analysis of a pediatric acute myeloid leukemia (AML) data set, GSDA was the only method to discover that event-free survival (EFS) was associated with the 56-gene AML pathway gene-set, narrow that result down to 5 genes, and confirm the association of those 5 genes with EFS in a separate validation cohort. These results indicate that GSDA effectively identifies and characterizes complex non-monotonic gene-set associations that are missed by other methods. CONCLUSION: GSDA is a powerful and flexible method to detect gene-set association with categorical, quantitative, or censored event-time variables, especially to detect complex non-monotonic gene-set associations. Available at https://CRAN.R-project.org/package=GSDA .

Genome-wide association studies have discovered many biologically important associations of genes with phenotypes. Typically, genome-wide association analyses formally test the association of each genetic feature (SNP, CNV, etc) with the phenotype of interest and summarize the results with multiplicity-adjusted p-values. However, very small p-values only provide evidence against the null hypothesis of no association without indicating which biological model best explains the observed data. Correctly identifying a specific biological model may improve the scientific interpretation and can be used to more effectively select and design a follow-up validation study. Thus, statistical methodology to identify the correct biological model for a particular genotype-phenotype association can be very useful to investigators. Here, we propose a general statistical method to summarize how accurately each of five biological models (null, additive, dominant, recessive, co-dominant) represents the data observed for each variant in a GWAS study. We show that the new method stringently controls the false discovery rate and asymptotically selects the correct biological model. Simulations of two-stage discovery-validation studies show that the new method has these properties and that its validation power is similar to or exceeds that of simple methods that use the same statistical model for all SNPs. Example analyses of three data sets also highlight these advantages of the new method. An R package is freely available at www.stjuderesearch.org/site/depts/biostats/software.

Evaluating the differential expression of a set of genes belonging to a common biological process or ontology has proven to be a very useful tool for biological discovery. However, existing gene-set association methods are limited to applications that evaluate differential expression across k ≥ 2 treatment groups or biological categories. This limitation precludes researchers from most effectively evaluating the association with other phenotypes that may be more clinically meaningful, such as quantitative variables or censored survival time variables. Projection onto the Orthogonal Space Testing (POST) is proposed as a general procedure that can robustly evaluate the association of a gene-set with several different types of phenotypic data (categorical, ordinal, continuous, or censored). For each gene-set, POST transforms the gene profiles into a set of eigenvectors and then uses statistical modeling to compute a set of z-statistics that measure the association of each eigenvector with the phenotype. The overall gene-set statistic is the sum of squared z-statistics weighted by the corresponding eigenvalues. Finally, bootstrapping is used to compute a p-value. POST may evaluate associations with or without adjustment for covariates. In simulation studies, it is shown that the performance of POST in evaluating the association with a categorical phenotype is similar to or exceeds that of existing methods. In evaluating the association of 875 biological processes with the time to relapse of pediatric acute myeloid leukemia, POST identified the well-known oncogenic WNT signaling pathway as its top hit. These results indicate that POST can be a very useful tool for evaluating the association of a gene-set with a variety of different phenotypes.

BACKGROUND: As new technologies allow investigators to collect multiple forms of molecular data (genomic, epigenomic, transcriptomic, etc) and multiple endpoints on a clinical trial cohort, it will become necessary to effectively integrate all these data in a way that reliably identifies biologically important genes. METHODS: We introduce CC-PROMISE as an integrated data analysis method that combines components of canonical correlation (CC) and projection onto the most interesting evidence (PROMISE). For each gene, CC-PROMISE first uses CC to compute scores that represent the association of two forms of molecular data with each other. Next, these scores are substituted into PROMISE to evaluate the statistical evidence that the molecular data show a biologically meaningful relationship with the endpoints. RESULTS: CC-PROMISE shows outstanding performance in simulation studies and an example application involving pediatric leukemia. In simulation studies, CC-PROMISE controls the type I error (misleading significance) rate very near the nominal level across 100 distinct null settings in which no molecular-endpoint association exists. Also, CC-PROMISE has better statistical power than three other methods that control type I error in 396 of 400 (99 %) alternative settings for which a molecular-endpoint association is present; the power advantage of CC-PROMISE exceeds 30 % in 127 of the 400 (32 %) alternative settings. These advantages of CC-PROMISE are also observed in an example application. CONCLUSION: CC-PROMISE very effectively identifies genes for which some form of molecular data shows a biologically meaningful association with multiple related endpoints. AVAILABILITY: The R package CCPROMISE is currently available from www.stjuderesearch.org/site/depts/biostats/software .

I1 Proceedings of the Fifteenth Annual UT- KBRIN Bioinformatics Summit 2016 Eric C. Rouchka, Julia H. Chariker, Benjamin J. Harrison, Juw Won Park P1 CC-PROMISE: Projection onto the Most Interesting Statistical Evidence (PROMISE) with Canonical Correlation to integrate gene expression and methylation data with multiple pharmacologic and clinical endpoints Xueyuan Cao, Stanley Pounds, Susana Raimondi, James Downing, Raul Ribeiro, Jeffery Rubnitz, Jatinder Lamba P2 Integration of microRNA-mRNA interaction networks with gene expression data to increase experimental power Bernie J Daigle, Jr. P3 Designing and writing software for in silico subtractive hybridization of large eukaryotic genomes Deborah Burgess, Stephanie Gehrlich, John C Carmen P4 Tracking the molecular evolution of Pax gene Nicholas Johnson; Chandrakanth Emani P5 Identifying genetic differences in thermally dimorphic and state specific fungi using in silico genomic comparison Stephanie Gehrlich, Deborah Burgess, John C Carmen P6 Identification of conserved genomic regions and variation therein amongst Cetartiodactyla species using next generation sequencing Kalpani De Silva, Michael P Heaton, Theodore S Kalbfleisch P7 Mining physiological data to identify patients with similar medical events and phenotypes Teeradache Viangteeravat, Rahul Mudunuri, Oluwaseun Ajayi, Fatih Şen, Eunice Y Huang P8 Smart brief for home health monitoring Mohammad Mohebbi, Luaire Florian, Douglas J Jackson, John F Naber P9 Side-effect term matching for computational adverse drug reaction predictions AKM Sabbir, Sally R Ellingson P10 Enrichment vs robustness: A comparison of transcriptomic data clustering metrics Yuping Lu, Charles A Phillips, Michael A Langston P11 Deep neural networks for transcriptome-based cancer classification Rahul K Sevakula, Raghuveer Thirukovalluru, Nishchal K. Verma, Yan Cui P12 Motif discovery using K-means clustering Mohammed Sayed, Juw Won Park P13 Large scale discovery of active enhancers from nascent RNA sequencing Jing Wang, Qi Liu, Yu Shyr P14 Computationally characterizing genomic pipelines and benchmarking results using GATK best practices on the high performance computing cluster at the University of Kentucky Xiaofei Zhang, Sally R Ellingson P15 Development of approaches enabling the identification of abnormal gene expression from RNA-Seq in personalized oncology Naresh Prodduturi, Gavin R Oliver, Diane Grill, Jie Na, Jeanette Eckel-Passow, Eric W Klee P16 Processing RNA-Seq data of plants infected with coffee ringspot virus Michael M Goodin, Mark Farman, Harrison Inocencio, Chanyong Jang, Jerzy W Jaromczyk, Neil Moore, Kelly Sovacool P17 Comparative transcriptomics of three Acinetobacter baumanii clinical isolates with different antibiotic resistance patterns Leon Dent, Mike Izban, Sammed Mandape, Shruti Sakhare, Siddharth Pratap, Dana Marshall P18 Metagenomic assessment of possible microbial contamination in the equine reference genome assembly M Scotty DePriest, James N MacLeod, Theodore S Kalbfleisch P19 Molecular evolution of cancer driver genes Chandrakanth Emani, Hanady Adam, Ethan Blandford, Joel Campbell, Joshua Castlen, Brittany Dixon, Ginger Gilbert, Aaron Hall, Philip Kreisle, Jessica Lasher, Bethany Oakes, Allison Speer, Maximilian Valentine P20 Biorepository Laboratory Information Management System Naga Satya V Rao Nagisetty, Rony Jose, Teeradache Viangteeravat, Robert Rooney, David Hains

SUMMARY: Several outlier and subgroup identification statistics (OASIS) have been proposed to discover transcriptomic features with outliers or multiple modes in expression that are indicative of distinct biological processes or subgroups. Here, we borrow ideas from the OASIS methods in the bioinformatics and statistics literature to develop the 'most informative spacing test' (MIST) for unsupervised detection of such transcriptomic features. In an example application involving 14 cases of pediatric acute megakaryoblastic leukemia, MIST more robustly identified features that perfectly discriminate subjects according to gender or the presence of a prognostically relevant fusion-gene than did seven other OASIS methods in the analysis of RNA-seq exon expression, RNA-seq exon junction expression and micorarray exon expression data. MIST was also effective at identifying features related to gender or molecular subtype in an example application involving 157 adult cases of acute myeloid leukemia. AVAILABILITY: MIST will be freely available in the OASIS R package at http://www.stjuderesearch.org/site/depts/biostats CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: It is scientifically and ethically imperative that the results of statistical analysis of biomedical research data be computationally reproducible in the sense that the reported results can be easily recapitulated from the study data. Some statistical analyses are computationally a function of many data files, program files, and other details that are updated or corrected over time. In many applications, it is infeasible to manually maintain an accurate and complete record of all these details about a particular analysis. RESULTS: Therefore, we developed the rctrack package that automatically collects and archives read only copies of program files, data files, and other details needed to computationally reproduce an analysis. CONCLUSIONS: The rctrack package uses the trace function to temporarily embed detail collection procedures into functions that read files, write files, or generate random numbers so that no special modifications of the primary R program are necessary. At the conclusion of the analysis, rctrack uses these details to automatically generate a read only archive of data files, program files, result files, and other details needed to recapitulate the analysis results. Information about this archive may be included as an appendix of a report generated by Sweave or knitR. Here, we describe the usage, implementation, and other features of the rctrack package. The rctrack package is freely available from http://www.stjuderesearch.org/site/depts/biostats/rctrack under the GPL license.

MOTIVATION: There is a substantial body of works in the biology literature that seeks to characterize the cyclic behavior of genes during cell division. Gene expression microarrays made it possible to measure the expression profiles of thousands of genes simultaneously in time-course experiments to assess changes in the expression levels of genes over time. In this context, the commonly used procedures for testing include the permutation test by de Lichtenberg et al. and the Fisher's G-test, both of which are designed to evaluate periodicity against noise. However, it is possible that a gene of interest may have expression that is neither cyclic nor just noise. Thus, there is a need for a new test for periodicity that can identify cyclic patterns against not only noise but also other non-cyclic patterns such as linear, quadratic or higher order polynomial patterns. RESULTS: To address this weakness, we have introduced an empirical Bayes approach to test for periodicity and compare its performance in terms of sensitivity and specificity with that of the permutation test and Fisher's G-test through extensive simulations and by application to a set of time-course experiments on the Schizosaccharomyces pombe cell-cycle gene expression. We use 'conserved' and 'cycling' genes by Lu et al. to assess the sensitivity and CESR genes by Chenet al. to assess the specificity of our new empirical Bayes method. AVAILABILITY AND IMPLEMENTATION: The SAS Macro for our empirical Bayes test for periodicity is included in the supplementary materials along with a sample run of the MACRO program. CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Tumors exhibit numerous genomic lesions such as copy number variations, structural variations and sequence variations. It is difficult to determine whether a specific constellation of lesions observed across a cohort of multiple tumors provides statistically significant evidence that the lesions target a set of genes that may be located across different chromosomes but yet are all involved in a single specific biological process or function. RESULTS: We introduce the genomic random interval (GRIN) statistical model and analysis method that evaluates the statistical significance of the abundance of genomic lesions that overlap a specific locus or a pre-defined set of biologically related loci. The GRIN model retains certain biologically important properties of genomic lesions that are ignored by other methods. In a simulation study and two example analyses of leukemia genomic lesion data, GRIN more effectively identified important loci as significant than did three methods based on a permutation-of-markers model. GRIN also identified biologically relevant pathways with a significant abundance of lesions in both examples. AVAILABILITY: An R package will be freely available at CRAN and www.stjuderesearch.org/site/depts/biostats/software.

MOTIVATION: Normalization is critical in DNA copy number analysis. We propose a new method to correctly identify two-copy probes from the genome to obtain representative references for normalization in single nucleotide polymorphism arrays. The method is based on a two-state Hidden Markov Model. Unlike most currently available methods in the literature, the proposed method does not need to assume that the percentage of two-copy state probes is dominant in the genome, as long as there do exist two-copy probes. RESULTS: The real data analysis and simulation study show that the proposed algorithm is successful in that (i) it performs as well as the current methods (e.g. CGHnormaliter and popLowess) for samples with dominant two-copy states and outperforms these methods for samples with less dominant two-copy states; (ii) it can identify the copy-neutral loss of heterozygosity; and (iii) it is efficient in terms of the computational time used. AVAILABILITY: R scripts are available at http://publichealth.lsuhsc.edu/PAIR.html.

Materials and methods Here, we describe our three-component strategy to achieve and document permanent reproducible computing in our research environment. First, we use the Sweave literate programming infrastructure to embed R code and report text in the same file. Sweave performs the specified calculations in R, inserts those results directly into a LaTeX typesetting command file, and finally compiles the LaTeX typesetting file into a PDF file. Thus, a Sweave file internally documents the top-level R code that produces the reported results. However, a Sweave report does not retain its reproducibility if the input data files and lowerlevel R code are modified later. Therefore, as the second component of our strategy, we developed the Igloo system to archive and freeze files for permanent reproducibility. The Igloo system requests that the user document every file that is transferred to a frozen archive. The Igloo system freezes the files in an archive with a directory structure that annotates the files by research team (leukemia, brain tumor, etc) and category (code file, type of data file, etc). The archive directory is visible in our Windows and Linux high-performance computing environments and has permission controls to ensure appropriate access to the files. However, neither Sweave nor Igloo assists with the cumbersome task of identifying specific input files that should be frozen to ensure permanent reproducibility. As the third component of our strategy, we developed the R package rctrack that computationally tracks the accession and generation of files by an R analysis program. The rctrack package defines a function that identifies files which need to be frozen in order to ensure permanent reproducibility. Additionally, rctrack provides mechanisms to track and document the usage of other software for some calculations. Finally, the rctrack package defines a function that generates a Sweave appendix with details regarding the input data and code files and their impact on the reproducibility of the report.

Background Differential expression analysis of digital gene expression data involves performing a large number of hypothesis tests that compare the expression count data of each gene or transcript across two or more biological conditions. The assumptions of any specific hypothesis-testing method will probably not be valid for each of a very large number of genes. Thus, computational evaluation of assumptions should be incorporated into the analysis to select an appropriate hypothesis-testing method for each gene.

High-dimensional biological data presents the opportunity to discover novel forms of biological heterogeneity, such as overexpression or suppression of expression of a particular gene in a subset of a cohort. This novel biological heterogeneity appears in the data as outliers or distinct subgroups. Here, we describe and evaluate three procedures for subgroup and outlier detection analysis (SODA): a leave-one-out (LOO) procedure that is widely used for outlier detection in the bioinformatics literature, the least median squares (LMS) procedure from the statistics literature, and the dip test (DT) from the statistics literature. We also propose and evaluate the max spacing test (MST) as a novel SODA method. In simulation studies, we found that LMS, DT, and MST are each the best method in specific settings. In an example analysis, we found that LMS and MST effectively identified confirmed fusion genes as outliers and DT and MST effectively identified genes that distinguish between two confirmed subtypes of pediatric acute megakaryoblastic leukemia. We conclude that LMS, DT, and MST are robust and complimentary methods for SODA.

MOTIVATION: Animal models play a pivotal role in translation biomedical research. The scientific value of an animal model depends on how accurately it mimics the human disease. In principle, microarrays collect the necessary data to evaluate the transcriptomic fidelity of an animal model in terms of the similarity of expression with the human disease. However, statistical methods for this purpose are lacking. RESULTS: We develop the agreement of differential expression (AGDEX) procedure to measure and determine the statistical significance of the similarity of the results of two experiments that measure differential expression across two groups. AGDEX defines a metric of agreement and determines statistical significance by permutation of each experiment's group labels. Additionally, AGDEX performs a comprehensive permutation-based analysis of differential expression for each experiment, including gene-set analyses and meta-analytic integration of results across studies. As an example, we show how AGDEX was recently used to evaluate the similarity of the transcriptome of a novel model of the brain tumor ependymoma in mice to that of a subtype of the human disease. This result, combined with other observations, helped us to infer the cell of origin of this devastating human cancer. AVAILABILITY: An R package is currently available from www.stjuderesearch.org/site/depts/biostats/agdex and will shortly be available from www.bioconductor.org.

Background Recent technical developments have made it possible to collect high-resolution genomics data using single nucleotide polymorphism (SNP) arrays. These arrays can be used in a paired data context to compare cancer tissue to normal samples in an effort to identify regions of genomic amplification or deletion. Such regions potentially contain oncogenes or tumor suppressor genes and are therefore of particular interest. However, using SNP array signals to identifying regions of copy number alteration is a challenging task due to the properties of the derived measurements.

Powerful methods for integrated analysis of multiple biological data sets must be developed to maximize researchers' ability to interpret them and acquire meaningful knowledge. Projection onto the most interesting statistical evidence (PROMISE) is a powerful statistical procedure that has been recently developed to incorporate a biological paradigm for the relationships among endpoint variables into an integrated analysis of microarray gene expression data with multiple biological and clinical endpoints. Here, PROMISE is adapted to perform an integrated analysis of pharmacokinetic, clinical, and genome-wide genotype data that incorporates a clinically relevant biological paradigm for pharmacokinetic and clinical response data. An efficient permutation-testing algorithm is introduced so that statistical calculations are computationally feasible in this higher-dimension setting. The new method is applied to a pediatric leukemia data set. The results clearly indicate that PROMISE can be a very powerful statistical tool for identifying genomic features that exhibit a pattern of association with multiple endpoint variables which is concordant with a practically useful biological paradigm.

Abstract Contact: [email protected]

MOTIVATION: In some applications, prior biological knowledge can be used to define a specific pattern of association of multiple endpoint variables with a genomic variable that is biologically most interesting. However, to our knowledge, there is no statistical procedure designed to detect specific patterns of association with multiple endpoint variables. RESULTS: Projection onto the most interesting statistical evidence (PROMISE) is proposed as a general procedure to identify genomic variables that exhibit a specific biologically interesting pattern of association with multiple endpoint variables. Biological knowledge of the endpoint variables is used to define a vector that represents the biologically most interesting values for statistics that characterize the associations of the endpoint variables with a genomic variable. A test statistic is defined as the dot-product of the vector of the observed association statistics and the vector of the most interesting values of the association statistics. By definition, this test statistic is proportional to the length of the projection of the observed vector of correlations onto the vector of most interesting associations. Statistical significance is determined via permutation. In simulation studies and an example application, PROMISE shows greater statistical power to identify genes with the interesting pattern of associations than classical multivariate procedures, individual endpoint analyses or listing genes that have the pattern of interest and are significant in more than one individual endpoint analysis. AVAILABILITY: Documented R routines are freely available from www.stjuderesearch.org/depts/biostats and will soon be available as a Bioconductor package from www.bioconductor.org.

UNLABELLED: A new procedure to align single nucleotide polymorphism (SNP) microarray signals for copy number analysis is proposed. For each individual array, this reference alignment procedure (RAP) uses a set of selected markers as internal references to direct the signal alignment. RAP aligns the signals so that each array has a similar signal distribution among its reference markers. An accompanying reference selection algorithm (RSA) uses genotype calls and initial signal intensities to choose two-copy markers as the internal references for each array. After RSA and RAP are applied, each array has a similar distribution of signals of two-copy markers so that across-array signal comparisons are biologically meaningful. An upper bound for a statistical metric of signal misalignment is derived and provides a theoretical basis to choose RSA-RAP over other alignment procedures for copy number analysis of cancers. In our study of acute lymphoblastic leukemia, RSA-RAP gives copy number analysis results that show substantially better concordance with cytogenetics than do two other alignment procedures. AVAILABILITY: Documented R code is freely available from www.stjuderesearch.org/depts/biostats/refnorm.

The analysis of microarray data often involves performing a large number of statistical tests, usually at least one test per queried gene. Each test has a certain probability of reaching an incorrect inference; therefore, it is crucial to estimate or control error rates that measure the occurrence of erroneous conclusions in reporting and interpreting the results of a microarray study. In recent years, many innovative statistical methods have been developed to estimate or control various error rates for microarray studies. Researchers need guidance choosing the appropriate statistical methods for analysing these types of data sets. This review describes a family of methods that use a set of P-values to estimate or control the false discovery rate and similar error rates. Finally, these methods are classified in a manner that suggests the appropriate method for specific applications and diagnostic procedures that can identify problems in the analysis are described.

MOTIVATION: Presently available methods that use p-values to estimate or control the false discovery rate (FDR) implicitly assume that p-values are continuously distributed and based on two-sided tests. Therefore, it is difficult to reliably estimate the FDR when p-values are discrete or based on one-sided tests. RESULTS: A simple and robust method to estimate the FDR is proposed. The proposed method does not rely on implicit assumptions that tests are two-sided or yield continuously distributed p-values. The proposed method is proven to be conservative and have desirable large-sample properties. In addition, the proposed method was among the best performers across a series of 'real data simulations' comparing the performance of five currently available methods. AVAILABILITY: Libraries of S-plus and R routines to implement the method are freely available from www.stjuderesearch.org/depts/biostats.

MOTIVATION: There is not a widely applicable method to determine the sample size for experiments basing statistical significance on the false discovery rate (FDR). RESULTS: We propose and develop the anticipated FDR (aFDR) as a conceptual tool for determining sample size. We derive mathematical expressions for the aFDR and anticipated average statistical power. These expressions are used to develop a general algorithm to determine sample size. We provide specific details on how to implement the algorithm for a k-group (k > or = 2) comparisons. The algorithm performs well for k-group comparisons in a series of traditional simulations and in a real-data simulation conducted by resampling from a large, publicly available dataset. AVAILABILITY: Documented S-plus and R code libraries are freely available from www.stjuderesearch.org/depts/biostats.

MOTIVATION: Recent attempts to account for multiple testing in the analysis of microarray data have focused on controlling the false discovery rate (FDR). However, rigorous control of the FDR at a preselected level is often impractical. Consequently, it has been suggested to use the q-value as an estimate of the proportion of false discoveries among a set of significant findings. However, such an interpretation of the q-value may be unwarranted considering that the q-value is based on an unstable estimator of the positive FDR (pFDR). Another method proposes estimating the FDR by modeling p-values as arising from a beta-uniform mixture (BUM) distribution. Unfortunately, the BUM approach is reliable only in settings where the assumed model accurately represents the actual distribution of p-values. METHODS: A method called the spacings LOESS histogram (SPLOSH) is proposed for estimating the conditional FDR (cFDR), the expected proportion of false positives conditioned on having k 'significant' findings. SPLOSH is designed to be more stable than the q-value and applicable in a wider variety of settings than BUM. RESULTS: In a simulation study and data analysis example, SPLOSH exhibits the desired characteristics relative to the q-value and BUM. AVAILABILITY: The Web site www.stjuderesearch.org/statistics/splosh.html has links to freely available S-plus code to implement the proposed procedure.

MOTIVATION: The occurrence of false positives and false negatives in a microarray analysis could be easily estimated if the distribution of p-values were approximated and then expressed as a mixture of null and alternative densities. Essentially any distribution of p-values can be expressed as such a mixture by extracting a uniform density from it. RESULTS: The occurrence of false positives and false negatives in a microarray analysis could be easily estimated if the distribution of p-values were approximated and then expressed as a mixture of null and alternative densities. Essentially any distribution of p-values can be expressed as such a mixture by extracting a uniform density from it. AVAILABILITY: An S-plus function library is available from http://www.stjuderesearch.org/statistics.