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Combining p-values to integrate multiple effects is of long-standing interest in social science and biomedical research. In this paper, we focus on revisiting a classical scenario closely related to meta-analysis, which combines a relatively small (finite and fixed) number of p-values while the sample size for generating each p-value is large (asymptotically goes to infinity). We evaluate a list of traditional and recently developed modified Fisher's methods to investigate their asymptotic efficiencies and finite-sample numerical performance. The result concludes Fisher and adaptively weighted Fisher method to have top performance and complementary advantages across different proportions of true signals. Finally, we propose an ensemble method, namely Fisher ensemble, to combine the two top-performing Fisher-related methods using a robust truncated Cauchy ensemble approach. We show that Fisher ensemble achieves asymptotic Bahadur optimality and integrates the strengths of Fisher and adaptively weighted Fisher methods in simulations. We subsequently extend Fisher ensemble to a variant with emphasized power for concordant effect size directions. A transcriptomic meta-analysis application confirms the theoretical and simulation conclusions, generates intriguing biomarker and pathway findings and demonstrates strengths and strategy of using proposed Fisher ensemble methods., Comment: 25 pages, 1 table, 5figures

Studying phenotype-gene association can uncover mechanism of diseases and develop efficient treatments. In complex disease where multiple phenotypes are available and correlated, analyzing and interpreting associated genes for each phenotype respectively may decrease statistical power and lose intepretation due to not considering the correlation between phenotypes. The typical approaches are many global testing methods, such as multivariate analysis of variance (MANOVA), which tests the overall association between phenotypes and each gene, without considersing the heterogeneity among phenotypes. In this paper, we extend and evaluate two p-value combination methods, adaptive weighted Fisher's method (AFp) and adaptive Fisher's method (AFz), to tackle this problem, where AFp stands out as our final proposed method, based on extensive simulations and a real application. Our proposed AFp method has three advantages over traditional global testing methods. Firstly, it can consider the heterogeneity of phenotypes and determines which specific phenotypes a gene is associated with, using phenotype specific 0-1 weights. Secondly, AFp takes the p-values from the test of association of each phenotype as input, thus can accommodate different types of phenotypes (continuous, binary and count). Thirdly, we also apply bootstrapping to construct a variability index for the weight estimator of AFp and generate a co-membership matrix to categorize (cluster) genes based on their association-patterns for intuitive biological investigations. Through extensive simulations, AFp shows superior performance over global testing methods in terms of type I error control and statistical power, as well as higher accuracy of 0-1 weights estimation over AFz. A real omics application with transcriptomic and clinical data of complex lung diseases demonstrates insightful biological findings of AFp.

The issue of combining individual $p$-values to aggregate multiple small effects is prevalent in many scientific investigations and is a long-standing statistical topic. Many classical methods are designed for combining independent and frequent signals in a traditional meta-analysis sense using the sum of transformed $p$-values with the transformation of light-tailed distributions, in which Fisher's method and Stouffer's method are the most well-known. Since the early 2000, advances in big data promoted methods to aggregate independent, sparse and weak signals, such as the renowned higher criticism and Berk-Jones tests. Recently, Liu and Xie(2020) and Wilson(2019) independently proposed Cauchy and harmonic mean combination tests to robustly combine $p$-values under "arbitrary" dependency structure, where a notable application is to combine $p$-values from a set of often correlated SNPs in genome-wide association studies. The proposed tests are the transformation of heavy-tailed distributions for improved power with the sparse signal. It calls for a natural question to investigate heavy-tailed distribution transformation, to understand the connection among existing methods, and to explore the conditions for a method to possess robustness to dependency. In this paper, we investigate the regularly varying distribution, which is a rich family of heavy-tailed distribution and includes Pareto distribution as a special case. We show that only an equivalent class of Cauchy and harmonic mean tests have sufficient robustness to dependency in a practical sense. We also show an issue caused by large negative penalty in the Cauchy method and propose a simple, yet practical modification. Finally, we present simulations and apply to a neuroticism GWAS application to verify the discovered theoretical insights and provide practical guidance., Comment: 34 pages, 3 figures

The discovery of disease subtypes is an essential step for developing precision medicine, and disease subtyping via omics data has become a popular approach. While promising, subtypes obtained from existing approaches are not necessarily associated with clinical outcomes. With the rich clinical data along with the omics data in modern epidemiology cohorts, it is urgent to develop an outcome-guided clustering algorithm to fully integrate the phenotypic data with the high-dimensional omics data. Hence, we extended a sparse K-means method to an outcome-guided sparse K-means (GuidedSparseKmeans) method. An unified objective function was proposed, which was comprised of (i) weighted K-means to perform sample clusterings; (ii) lasso regularizations to perform gene selection from the high-dimensional omics data; (iii) incorporation of a phenotypic variable from the clinical dataset to facilitate biologically meaningful clustering results. By iteratively optimizing the objective function, we will simultaneously obtain a phenotype-related sample clustering results and gene selection results. We demonstrated the superior performance of the GuidedSparseKmeans by comparing with existing clustering methods in simulations and applications of high-dimensional transcriptomic data of breast cancer and Alzheimer's disease. Our algorithm has been implemented into an R package, which is publicly available on GitHub (https://github.com/LingsongMeng/GuidedSparseKmeans).

High-throughput microarray and sequencing technology have been used to identify disease subtypes that could not be observed otherwise by using clinical variables alone. The classical unsupervised clustering strategy concerns primarily the identification of subpopulations that have similar patterns in gene features. However, as the features corresponding to irrelevant confounders (e.g. gender or age) may dominate the clustering process, the resulting clusters may or may not capture clinically meaningful disease subtypes. This gives rise to a fundamental problem: can we find a subtyping procedure guided by a pre-specified disease outcome? Existing methods, such as supervised clustering, apply a two-stage approach and depend on an arbitrary number of selected features associated with outcome. In this paper, we propose a unified latent generative model to perform outcome-guided disease subtyping constructed from omics data, which improves the resulting subtypes concerning the disease of interest. Feature selection is embedded in a regularization regression. A modified EM algorithm is applied for numerical computation and parameter estimation. The proposed method performs feature selection, latent subtype characterization and outcome prediction simultaneously. To account for possible outliers or violation of mixture Gaussian assumption, we incorporate robust estimation using adaptive Huber or median-truncated loss function. Extensive simulations and an application to complex lung diseases with transcriptomic and clinical data demonstrate the ability of the proposed method to identify clinically relevant disease subtypes and signature genes suitable to explore toward precision medicine., Comment: 29 pages in total, 4 figures, 2 tables and 1 supplement

Clustering with variable selection is a challenging yet critical task for modern small-n-large-p data. Existing methods based on sparse Gaussian mixture models or sparse K-means provide solutions to continuous data. With the prevalence of RNA-seq technology and lack of count data modeling for clustering, the current practice is to normalize count expression data into continuous measures and apply existing models with Gaussian assumption. In this paper, we develop a negative binomial mixture model with lasso or fused lasso gene regularization to cluster samples (small n) with high-dimensional gene features (large p). EM algorithm and Bayesian information criterion are used for inference and determining tuning parameters. The method is compared with existing methods using extensive simulations and two real transcriptomic applications in rat brain and breast cancer studies. The result shows superior performance of the proposed count data model in clustering accuracy, feature selection and biological interpretation in pathways.

Estimating the number of clusters (K) is a critical and often difficult task in cluster analysis. Many methods have been proposed to estimate K, including some top performers using resampling approach. When performing cluster analysis in high-dimensional data, simultaneous clustering and feature selection is needed for improved interpretation and performance. To our knowledge, none has investigated simultaneous estimation of K and feature selection in an exploratory cluster analysis. In this paper, we propose a resampling method to meet this gap and evaluate its performance under the sparse K-means clustering framework. The proposed target function balances between sensitivity and specificity of clustering evaluation of pairwise subjects from clustering of full and subsampled data. Through extensive simulations, the method performs among the best over classical methods in estimating K in low-dimensional data. For high-dimensional simulation data, it also shows superior performance to simultaneously estimate K and feature sparsity parameter. Finally, we evaluated the methods in four microarray, two RNA-seq, one SNP and two non-omics datasets. The proposed method achieves better clustering accuracy with fewer selected predictive genes in almost all real applications., Comment: The earlier version won 2019 ENAR distinguished student paper award

Meta-analysis is a statistical method to combine results from multiple clinical or genomic studies with the same or similar research problems. It has been widely use to increase statistical power in finding clinical or genomic differences among different groups. One major category of meta-analysis is combining p-values from independent studies and the Fisher's method is one of the most commonly used statistical methods. However, due to heterogeneity of studies, particularly in the field of genomic research, with thousands of features such as genes to assess, researches often desire to discover this heterogeneous information when identify differentially expressed genomic features. To address this problem, \citet{Li2011adaptively} proposed very interesting statistical method, adaptively weighted (AW) Fisher's method, where binary weights, $0$ or $1$, are assigned to the studies for each feature to distinguish potential zero or none-zero effect sizes. \citet{Li2011adaptively} has shown some good properties of AW fisher's method such as the admissibility. In this paper, we further explore some asymptotic properties of AW-Fisher's method including consistency of the adaptive weights and the asymptotic Bahadur optimality of the test., Comment: 17 pages, 2 figures

Advancement in technology has generated abundant high-dimensional data that allows integration of multiple relevant studies. Due to their huge computational advantage, variable screening methods based on marginal correlation have become promising alternatives to the popular regularization methods for variable selection. However, all these screening methods are limited to single study so far. In this paper, we consider a general framework for variable screening with multiple related studies, and further propose a novel two-step screening procedure using a self-normalized estimator for high-dimensional regression analysis in this framework. Compared to the one-step procedure and rank-based sure independence screening (SIS) procedure, our procedure greatly reduces false negative errors while keeping a low false positive rate. Theoretically, we show that our procedure possesses the sure screening property with weaker assumptions on signal strengths and allows the number of features to grow at an exponential rate of the sample size. In addition, we relax the commonly used normality assumption and allow sub-Gaussian distributions. Simulations and a real transcriptomic application illustrate the advantage of our method as compared to the rank-based SIS method., Comment: 25 pages, 1 figure

Meta-analysis methods have been widely used to combine results from multiple clinical or genomic studies to increase statistical power and ensure robust and accurate conclusion. Adaptively weighted Fisher's method (AW-Fisher) is an effective approach to combine p-values from $K$ independent studies and to provide better biological interpretation by characterizing which studies contribute to meta-analysis. Currently, AW-Fisher suffers from lack of fast, accurate p-value computation and variability estimate of AW weights. When the number of studies $K$ is large, the $3^K - 1$ possible differential expression pattern categories can become intractable. In this paper, we apply an importance sampling technique with spline interpolation to increase accuracy and speed of p-value calculation. Using resampling techniques, we propose a variability index for the AW weight estimator and a co-membership matrix to characterize pattern similarities between genes. The co-membership matrix is further used to categorize differentially expressed genes based on their meta-patterns for further biological investigation. The superior performance of the proposed methods is shown in simulations. These methods are also applied to two real applications to demonstrate intriguing biological findings., Comment: adaptive weights, Fisher's method, meta-analysis, differential expression analysis

Due to the rapid development of high-throughput experimental techniques and fast-dropping prices, many transcriptomic datasets have been generated and accumulated in the public domain. Meta-analysis combining multiple transcriptomic studies can increase the statistical power to detect disease-related biomarkers. In this paper, we introduce a Bayesian latent hierarchical model to perform transcriptomic meta-analysis. This method is capable of detecting genes that are differentially expressed (DE) in only a subset of the combined studies, and the latent variables help quantify homogeneous and heterogeneous differential expression signals across studies. A tight clustering algorithm is applied to detected biomarkers to capture differential meta-patterns that are informative to guide further biological investigation. Simulations and three examples, including a microarray dataset from metabolism-related knockout mice, an RNA-seq dataset from HIV transgenic rats, and cross-platform datasets from human breast cancer, are used to demonstrate the performance of the proposed method., Comment: 29 pages, 4 figures

Microarray analysis to monitor expression activities in thousands of genes simultaneously has become routine in biomedical research during the past decade. A tremendous amount of expression profiles are generated and stored in the public domain and information integration by meta-analysis to detect differentially expressed (DE) genes has become popular to obtain increased statistical power and validated findings. Methods that aggregate transformed $p$-value evidence have been widely used in genomic settings, among which Fisher's and Stouffer's methods are the most popular ones. In practice, raw data and $p$-values of DE evidence are often not available in genomic studies that are to be combined. Instead, only the detected DE gene lists under a certain $p$-value threshold (e.g., DE genes with $p$-value${}<0.001$) are reported in journal publications. The truncated $p$-value information makes the aforementioned meta-analysis methods inapplicable and researchers are forced to apply a less efficient vote counting method or na\"{i}vely drop the studies with incomplete information. The purpose of this paper is to develop effective meta-analysis methods for such situations with partially censored $p$-values. We developed and compared three imputation methods - mean imputation, single random imputation and multiple imputation - for a general class of evidence aggregation methods of which Fisher's and Stouffer's methods are special examples. The null distribution of each method was analytically derived and subsequent inference and genomic analysis frameworks were established. Simulations were performed to investigate the type I error, power and the control of false discovery rate (FDR) for (correlated) gene expression data. The proposed methods were applied to several genomic applications in colorectal cancer, pain and liquid association analysis of major depressive disorder (MDD). The results showed that imputation methods outperformed existing na\"{i}ve approaches. Mean imputation and multiple imputation methods performed the best and are recommended for future applications., Comment: Published in at http://dx.doi.org/10.1214/14-AOAS747 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Meta-analysis techniques have been widely developed and applied in genomic applications, especially for combining multiple transcriptomic studies. In this paper we propose an order statistic of $p$-values ($r$th ordered $p$-value, rOP) across combined studies as the test statistic. We illustrate different hypothesis settings that detect gene markers differentially expressed (DE) 'in all studies," "in the majority of studies"' or "in one or more studies," and specify rOP as a suitable method for detecting DE genes "in the majority of studies." We develop methods to estimate the parameter $r$ in rOP for real applications. Statistical properties such as its asymptotic behavior and a one-sided testing correction for detecting markers of concordant expression changes are explored. Power calculation and simulation show better performance of rOP compared to classical Fisher's method, Stouffer's method, minimum $p$-value method and maximum $p$-value method under the focused hypothesis setting. Theoretically, rOP is found connected to the na\"{i}ve vote counting method and can be viewed as a generalized form of vote counting with better statistical properties. The method is applied to three microarray meta-analysis examples including major depressive disorder, brain cancer and diabetes. The results demonstrate rOP as a more generalizable, robust and sensitive statistical framework to detect disease-related markers., Comment: Published in at http://dx.doi.org/10.1214/13-AOAS683 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)