Normal Mixture Model Research Articles

Normal mixture method for stock daily returns over different sub-periods

ABSTRACTIn this paper, the normal mixture model, as an alternative distribution, is utilized to represent the characteristics of stock daily returns over different bull and bear markets. Firstly, we conduct the normality test for the return data and compare the Kolmogorov-Smirnov statistics of normal mixture models with different components. Secondly, we analyze the likely reasons why parameters change over different sub-periods. Our empirical examination proves that majority of the data series reject the normality assumption and mixture models with three components can model the behavior of daily returns more appropriately and steadily. This result has both statistical and economic significance.

Detectability of nonparametric signals: higher criticism versus likelihood ratio

We study the signal detection problem in high dimensional noise data (possibly) containing rare and weak signals. Log-likelihood ratio (LLR) tests depend on unknown parameters, but they are needed to judge the quality of detection tests since they determine the detection regions. The popular Tukey’s higher criticism (HC) test was shown to achieve the same completely detectable region as the LLR test does for different (mainly) parametric models. We present a novel technique to prove this result for very general signal models, including even nonparametric $p$-value models. Moreover, we address the following questions which are still pending since the initial paper of Donoho and Jin: What happens on the border of the completely detectable region, the so-called detection boundary? Does HC keep its optimality there? In particular, we give a complete answer for the heteroscedastic normal mixture model. As a byproduct, we give some new insights about the LLR test’s behaviour on the detection boundary by discussing, among others, Pitmans’s asymptotic efficiency as an application of Le Cam’s theory.

The Warburg effect: persistence of stem-cell metabolism in cancers as a failure of differentiation

Two recent observations regarding the Warburg effect are that (i) the metabolism of stem cells is constitutive (aerobic) glycolysis while normal cellular differentiation involves a transition to oxidative phosphorylation and (ii) the degree of glucose uptake of a malignancy as imaged by 18F-fluorodeoxyglucose positron emission tomography (FDG-PET) is associated with histologic measures of tumor differentiation. Combining these observations, we hypothesized that the high levels of glucose uptake observed in poorly differentiated cancers may reflect persistence of the glycolytic metabolism of stem cells in malignant cells that fail to fully differentiate. Tumor glucose uptake was measured by FDG-PET in 552 patients with histologically diverse cancers. We used normal mixture modeling to explore FDG-PET standardized uptake value (SUV) distributions and tested for associations between glucose uptake and histological differentiation, risk of lymph node metastasis, and survival. Using RNA-seq data, we carried out pathway and transcription factor analyses to compare tumors with high and low levels of glucose uptake. We found that well-differentiated tumors had low FDG uptake, while moderately and poorly differentiated tumors had higher uptake. The distribution of SUV for each histology was bimodal, with a low peak around SUV 2-5 and a high peak at SUV 8-14. The cancers in the two modes were clinically distinct in terms of the risk of nodal metastases and death. Carbohydrate metabolism and the pentose-related pathway were elevated in the poorly differentiated/high SUV clusters. Embryonic stem cell-related signatures were activated in poorly differentiated/high SUV clusters. Our findings support the hypothesis that the biological basis for the Warburg effect is a persistence of stem cell metabolism (i.e. aerobic glycolysis) in cancers as a failure to transition from glycolysis-utilizing undifferentiated cells to oxidative phosphorylation-utilizing differentiated cells. We found that cancers cluster along the differentiation pathway into two groups, utilizing either glycolysis or oxidative phosphorylation. Our results have implications for multiple areas of clinical oncology.

A latent class analysis of PTSD symptoms among inner city primary care patients

ObjectivesExamine evidence for different subclasses of posttraumatic stress disorder (PTSD) symptoms in a sample of trauma exposed, low-income, predominantly African American men and women. Assess the relationship between PTSD subclasses with major depressive disorder (MDD) and types of trauma experienced. MethodLatent class analysis (LCA) using a multivariate normal mixture model on the 17-item PTSD Symptom Scale (PSS) was used to identify latent subclasses of PTSD symptoms (N = 5063). ResultsLCA suggested four subclasses of PTSD symptoms: (1) High severity and comorbidity (n = 932, 92.2% current PTSD, 88.7% MDD, 82% both), characterized by high PTSD symptoms, depression, and comorbidity of PTSD and MDD; (2) Moderate severity (n = 1179, 56.5% current PTSD, 53.9% MDD, 34.5% both), which had high avoidance and hyper-vigilance symptoms compared to the other symptoms; (3) Low PTSD and high depression (n = 657, 12.8% current PTSD, 49.9% MDD, 8.8% both) which had high insomnia but otherwise low PTSD symptoms and high depression; and (4) Resilient (n = 2295, 2.0% current PTSD, 16.4% MDD, and 0.6% both) characterized by low mean scores on all PTSD symptoms and depression. ConclusionsThe results suggest avoidance and hyper-vigilance are important symptoms in PTSD development and insomnia may be an important indicator for depression. The combination of severe insomnia, avoidance, and hyper-vigilance may be key symptoms for comorbidity of PTSD and MDD. Future studies should focus on these symptoms to better target people at high risk for developing PTSD or MDD.

Weight Trajectories from Birth and Bone Mineralization at 7 Years of Age

To assess whether different trajectories of weight gain since birth influence bone mineral content (BMC) and areal bone mineral density (aBMD) at 7 years of age. We studied a subsample of 1889 children from the Generation XXI birth cohort who underwent whole-body dual-energy radiograph absorptiometry. Weight trajectories identified through normal mixture modeling for model-based clustering and labeled "normal weight gain," "weight gain during infancy," "weight gain during childhood," and "persistent weight gain" were used. Differences in subtotal BMC, aBMD, and size-corrected BMC (scBMC) at age 7 years according to weight trajectories were estimated through analysis of covariance. Compared with the "normal weight gain" trajectory, children in the remaining trajectories had significantly greater BMC, aBMD, and scBMC at age 7 years, with the strongest associations for "persistent weight gain" (girls [BMC: 674.0 vs 559.8 g, aBMD: 0.677 vs 0.588 g/cm2, scBMC: 640.7 vs 577.4 g], boys [BMC: 689.4 vs 580.8 g, aBMD: 0.682 vs 0.611 g/cm2, scBMC: 633.0 vs 595.6 g]). After adjustment for current weight, and alternatively for fat and lean mass, children with a "weight gain during childhood" trajectory had greater BMC and aBMD than those with a "normal weight gain" trajectory, although significant differences were restricted to girls (BMC: 601.4 vs 589.2 g, aBMD: 0.618 vs 0.609 g/cm2). Overall, children following a trajectory of persistent weight gain since birth had clearly increased bone mass at 7 years, but weight gain seemed slightly more beneficial when it occurred later rather than on a normal trajectory during the first 7 years of life.

Issues and Perspectives in Species Delimitation using Phenotypic Data: Atlantean Evolution in Darwin's Finches.

Progress in the development and use of methods for species delimitation employing phenotypic data lags behind conceptual and practical advances in molecular genetic approaches. The basic evolutionary model underlying the use of phenotypic data to delimit species assumes random mating and quantitative polygenic traits, so that phenotypic distributions within a species should be approximately normal for individuals of the same sex and age. Accordingly, two or more distinct normal distributions of phenotypic traits suggest the existence of multiple species. In light of this model, we show that analytical approaches employed in taxonomic studies using phenotypic data are often compromised by three issues: 1) reliance on graphical analyses that convey little information on phenotype frequencies; 2) exclusion of characters potentially important for species delimitation following reduction of data dimensionality; and 3) use of measures of central tendency to evaluate phenotypic distinctiveness. We outline approaches to overcome these issues based on statistical developments related to normal mixture models (NMMs) and illustrate them empirically with a reanalysis of morphological data recently used to claim that there are no morphologically distinct species of Darwin's ground-finches (Geospiza). We found negligible support for this claim relative to taxonomic hypotheses recognizing multiple species. Although species limits among ground-finches merit further assessments using additional sources of information, our results bear implications for other areas of inquiry including speciation research: because ground-finches have likely speciated and are not trapped in a process of "Sisyphean" evolution as recently argued, they remain useful models to understand the evolutionary forces involved in speciation. Our work underscores the importance of statistical approaches grounded on appropriate evolutionary models for species delimitation. We discuss how NMMs offer new perspectives in the kind of inferences available to systematists, with significant repercussions on ideas about the phenotypic structure of biodiversity.

The Bos taurus-Bos indicus balance in fertility and milk related genes.

Numerical approaches to high-density single nucleotide polymorphism (SNP) data are often employed independently to address individual questions. We linked independent approaches in a bioinformatics pipeline for further insight. The pipeline driven by heterozygosity and Hardy-Weinberg equilibrium (HWE) analyses was applied to characterize Bos taurus and Bos indicus ancestry. We infer a gene co-heterozygosity network that regulates bovine fertility, from data on 18,363 cattle with genotypes for 729,068 SNP. Hierarchical clustering separated populations according to Bos taurus and Bos indicus ancestry. The weights of the first principal component were subjected to Normal mixture modelling allowing the estimation of a gene’s contribution to the Bos taurus-Bos indicus axis. We used deviation from HWE, contribution to Bos indicus content and association to fertility traits to select 1,284 genes. With this set, we developed a co-heterozygosity network where the group of genes annotated as fertility-related had significantly higher Bos indicus content compared to other functional classes of genes, while the group of genes associated with milk production had significantly higher Bos taurus content. The network analysis resulted in capturing novel gene associations of relevance to bovine domestication events. We report transcription factors that are likely to regulate genes associated with cattle domestication and tropical adaptation. Our pipeline can be generalized to any scenarios where population structure requires scrutiny at the molecular level, particularly in the presence of a priori set of genes known to impact a phenotype of evolutionary interest such as fertility.

Modeling regime‐dependent agricultural commodity price volatilities

AbstractIn stark contrast to financial markets, relatively little attention has been given to modeling agricultural commodity price volatility. In recent years, numerous methodologies with various strengths have been proposed for modeling price volatility in financial markets. We propose using a mixture of normals with unique GARCH processes in each component for modeling agricultural commodity prices. While a normal mixture model is quite flexible and allows for time varying skewness and kurtosis, its biggest strength is that each component can be viewed as a different market regime and thus estimated parameters are more readily interpreted. We apply the proposed model to ten different agricultural commodity weekly cash prices. Both in‐sample fit and out‐of‐sample forecasting tests confirm that the two‐state NM‐GARCH approach performs better than the traditional normal GARCH model. A significant and state‐dependent inverse leverage effect is detected only for pork in the regime where the price is expected to drop, indicating the volatility in this regime tends to increase more following a realized price rise than a realized price drop.

MASTER: a model to improve and standardize clinical breakpoints for antimicrobial susceptibility testing using forecast probabilities.

The procedure for setting clinical breakpoints (CBPs) for antimicrobial susceptibility has been poorly standardized with respect to population data, pharmacokinetic parameters and clinical outcome. Tools to standardize CBP setting could result in improved antibiogram forecast probabilities. We propose a model to estimate probabilities for methodological categorization errors and defined zones of methodological uncertainty (ZMUs), i.e. ranges of zone diameters that cannot reliably be classified. The impact of ZMUs on methodological error rates was used for CBP optimization. The model distinguishes theoretical true inhibition zone diameters from observed diameters, which suffer from methodological variation. True diameter distributions are described with a normal mixture model. The model was fitted to observed inhibition zone diameters of clinical Escherichia coli strains. Repeated measurements for a quality control strain were used to quantify methodological variation. For 9 of 13 antibiotics analysed, our model predicted error rates of < 0.1% applying current EUCAST CBPs. Error rates were > 0.1% for ampicillin, cefoxitin, cefuroxime and amoxicillin/clavulanic acid. Increasing the susceptible CBP (cefoxitin) and introducing ZMUs (ampicillin, cefuroxime, amoxicillin/clavulanic acid) decreased error rates to < 0.1%. ZMUs contained low numbers of isolates for ampicillin and cefuroxime (3% and 6%), whereas the ZMU for amoxicillin/clavulanic acid contained 41% of all isolates and was considered not practical. We demonstrate that CBPs can be improved and standardized by minimizing methodological categorization error rates. ZMUs may be introduced if an intermediate zone is not appropriate for pharmacokinetic/pharmacodynamic or drug dosing reasons. Optimized CBPs will provide a standardized antibiotic susceptibility testing interpretation at a defined level of probability.

Statistical modeling of Optical Coherence Tomography images by asymmetric Normal Laplace mixture model.

Optical Coherence Tomography (OCT) is known as a non-invasive and high resolution imaging modality in ophthalmology. Effecting noise on the OCT images as well as other reasons cause a random behavior in these images. In this study, we introduce a new statistical model for retinal layers in healthy OCT images. This model, namely asymmetric Normal Laplace (NL), fits well the advent of asymmetry and heavy-tailed in intensity distribution of each layer. Due to the layered structure of retina, a mixture model is addressed. It is proposed to evaluate the fitness criteria called Kull-back Leibler Divergence (KLD) and chi-square test along visual results. The results express the well performance of proposed model in fitness of data except for 6th and 7th layers. Using a complicated model, e.g. a mixture model with two component, seems to be appropriate for these layers. The mentioned process for train images can then be devised for a test image by employing the Expectation Maximization (EM) algorithm to estimate the values of parameters in mixture model.

Multivariate normal mean-variance mixture distribution based on Lindley distribution

ABSTRACTThis article introduces a new asymmetric distribution constructed by assuming the multivariate normal mean-variance mixture model. Called normal mean-variance mixture of the Lindley distribution, we derive some mathematical properties of the new distribution. Also, a feasible maximum likelihood estimation procedure using the EM algorithm and the asymptotic standard errors of parameter estimates are developed. The performance of the proposed distribution is illustrated by means of real datasets and simulation analysis.

Diel periodicity and chronology of upstream migration in yellow-phase American eels (Anguilla rostrata)

Yellow-phase American eel (Anguilla rostrata) upstream migration is temporally punctuated, yet migration chronology within diel time periods is not well-understood. This study examined diel periodicity, chronology, and total length (TL) of six multi-day, high-count (285–1,868 eels) passage events of upstream migrant yellow-phase American eels at the Millville Dam eel ladder, lower Shenandoah River, West Virginia during 2011–2014. We categorized passage by diel periods (vespertine, nocturnal, matutinal, diurnal) and season (spring, summer, late summer/early fall, fall). We depicted passage counts as time-series histograms and used time-series spectral analysis (Fast Fourier Transformation) to identify cyclical patterns and diel periodicity of upstream migration. We created histograms to examine movement patterns within diel periods for each passage event and fit normal mixture models (2–9 mixtures) to describe multiple peaks of passage counts. Periodicity of movements for each passage event followed a 24-h activity cycle with mostly nocturnal movement. Multimodal models were supported by the data; most modes represented nocturnal movements, but modes at or near the transition between twilight and night were also common. We used mixed-model methodology to examine relationships among TL, diel period, and season. An additive-effects model of diel period + season was the best approximating model. A decreasing trend of mean TL occurred across diel movement periods, with the highest mean TL occurring during fall relative to similar mean values of TL for spring, summer, and late summer/early fall. This study increased our understanding of yellow-phase American eels by demonstrating the non-random nature of their upstream migration.

Probabilistic Record Linkage (PRL) advances between Surveillance Data and HES datasets using the Expectation-Maximization (EM) algorithm

ABSTRACT&#x0D; ObjectivesThis study shows how to improve PRL when using a limited number of personal identifiable information (PIIs) available in HES such as NHS number, DOB, Hospital Number, Gender and Postcode. The focus is on new approaches in categorisation of m-u probabilities for fields related to Postcode. In addition, a brief outline of our procedure to generate an appropriate training dataset will be presented as well as reasons for splitting HES dataset into single and multi-admissions, and proposed techniques to avoid computational cost.&#x0D; ApproachPIIs such as DOB and Postcode can be categorised in order to increase the number of potential comparison vectors which is necessary in order to deal with the multi-million size of the HES dataset. We propose a multinomial approach for comparison between postcodes both for agreement and disagreement case scenarios, based on geographical information and a similarity score resulting from a Probit model. In the absence of a ‘gold-standard’ dataset, generating an appropriate training set is possible by combining two-way Deterministic Linkage (DL) and simple random sampling. Splitting HES into single and multi-admissions is recommended for reasons related to computational performance and demographic differences of the underlying population. Selection of the lower and upper thresholds regarding linkage weights can be done in a practical way by using conditional Normal mixture modelling, or more formally by using sampling techniques based on Fellegi-Sunter decision rule. We show how the latter can be further improved avoiding the computational cost by introducing appropriate mapping functions.&#x0D; ResultsWe show the advantages of PRL over DL between Surveillance Laboratory Flu Data and HES. Preliminary results indicate an additional 10% of matches for &lt; 1% expected false positives. EM convergence problems when treating missing values as separate outcome category will be discussed comparing results with the suggested re-normalisation approach in the bibliography.&#x0D; ConclusionPRL between Laboratory Surveillance Data and Hospital Episode Statistics (HES) is a common practice and performed in a number of epidemiological studies conducted at Public Health England (PHE). Therefore, improving linkage will subsequently benefit those studies. Furthermore, this study proposes new approaches for Postcode comparison, and the utilisation of mapping functions to avoid the computational cost of the Fellegi-Sunter decision rule. These approaches could be used in a wide range of applications. Finally, we suggest a structural way of handling the HES dataset in order to achieve high efficiency and robustness of linkage.

Consistency of the MLE under Mixture Models

The large-sample properties of likelihood-based statistical inference under mixture models have received much attention from statisticians. Although the consistency of the nonparametric MLE is regarded as a standard conclusion, many researchers ignore the precise conditions required on the mixture model. An incorrect claim of consistency can lead to false conclusions even if the mixture model under investigation seems well behaved. Under a finite normal mixture model, for instance, the consistency of the plain MLE is often erroneously assumed in spite of recent research breakthroughs. This paper streamlines the consistency results for the nonparametric MLE in general, and in particular for the penalized MLE under finite normal mixture models.

不同类型数据下混合模型参数估计效果的对比研究

混合高斯模型在描述数据方面应用较多，但它易受离群点的影响，其参数的极大似然估计不是稳健估计。混合t-分布模型由于其重尾分布的特性，相对于混合高斯分布，在分析重尾数据上更具稳健性。文章首先研究一元混合t-分布模型，利用标准EM算法给出了该模型参数极大似然估计的迭代步骤，并分别在三类模拟数据下与混合高斯模型进行了对比分析，验证了该模型的有效性以及在拟合重尾数据上的优势。算法初始化采用k-means方法。 The normal mixture model has more applications in describing data. But it is easily influenced by the outlier, and the maximum likelihood estimation of parameters is not robust estimation. T-distribution mixture model has better robustness than Gauss mixture model to analyze data with longer time than normal tails because of its heavy-tails. In this paper, we studied a univariate t mixture model primarily. Based on EM algorithm, we derived the iteration steps of maximum li-kelihood estimation of the model’s unknown parameters. Furthermore, we did a comparative analysis by three types of simulated data. Simulation study shows that this model has an advantage in fitting data with longer time than normal tails. The initial value is given by k-means method.

A moment-based criterion for determining the number of components in a normal mixture model

A moment-based criterion for determining the number of components in a normal mixture model

Modeling Multimodal Continuous Heterogeneity in Conjoint Analysis—A Sparse Learning Approach

Consumers’ preferences can often be represented using a multimodal continuous heterogeneity distribution. One explanation for such a preference distribution is that consumers belong to a few distinct segments, with preferences of consumers in each segment being heterogeneous and unimodal. We propose an innovative approach for modeling such multimodal distributions that builds on recent advances in sparse learning and optimization. We apply the model to conjoint analysis where consumer heterogeneity plays a critical role in determining optimal marketing decisions. Our approach uses a two-stage divide-and-conquer framework, where we first divide the consumer population into segments by recovering a set of candidate segmentations using sparsity modeling, and then use each candidate segmentation to develop a set of individual-level heterogeneity representations. We select the optimal individual-level heterogeneity representation using cross-validation. Using extensive simulation experiments and three field data sets, we show the superior performance of our sparse learning model compared to benchmark models including the finite mixture model and the Bayesian normal component mixture model.Data, as supplemental material, are available at http://dx.doi.org/10.1287/mksc.2016.0992 .

P1.03-003 The Warburg Effect: Persistence of Stem Cell Metabolism in Lung Cancer as Failure of Differentiation

Two recent observations are relevant to explaining Warburg's observation the cancers constitutively utilize glycolysis in the presence of oxygen sufficient for oxidative phosphorylation. First, the metabolism of stem cells has been shown to be constitutive (‘aerobic’) glycolysis, with differentiation involving a transition to oxidative phosphorylation. Second, the degree of glucose uptake by a cancer has been associated with histologic differentiation. We hypothesized that the high levels of glucose uptake observed in poorly differentiated lung cancers may reflect persistence in cancers of the glycolytic metabolism of stem cells that fail to fully differentiate. Tumor glucose uptake was measured by FDG-PET in 859 patients with histologically diverse cancers including NSCLC. We used normal mixture modeling to explore SUV distributions and tested for association between glucose uptake and histological differentiation, risk of lymph node metastasis, and survival. Using microarray data, we performed pathway and transcription factor analyses to compare tumors with high/low glucose uptake. Well-differentiated NSCLC had low FDG uptake, and moderately/poorly differentiated tumors higher uptake. The distribution of FDG-PET uptake was modal with a low peak at SUV 2-4 and a high peak at SUV 8-11. The cancers in the two peaks were clinically distinct in terms of the risk of nodal metastases and of death. Carbohydrate metabolism-related and pentose/nucleotide synthesis-related genes were elevated in the high SUV clusters, Krebs cycle/glutamine metabolism-related genes were elevated in the low SUV mode samples. Expression of Myc target genes was associated with SUV mode, but Nanog, Sox2, Oct4 and PRC2 where not. The biological basis for the Warburg effect is persistence of stem cell metabolism in lung cancers as a failure to transition from glycolysis-utilizing undifferentiated cells to oxidative phosphorylation-utilizing differentiated cells. Lung cancers cluster along the differentiation pathway into two groups. Our results have implications for determining prognosis, cancer screening and surveillance after resection.

Model-based longitudinal clustering with varying cluster assignments

It is often of interest to perform clustering on longitudinal data, yet it is difficult to formulate an intuitive model for which estimation is computationally feasible. We propose a model-based clustering method for clustering objects that are observed over time. The proposed model can be viewed as an extension of the normal mixture model for clustering to longitudinal data. While existing models only account for clustering effects, we propose modeling the distribution of the observed values of each object as a blending of a cluster effect and an individual effect, hence also giving an estimate of how much the behavior of an object is determined by the cluster to which it belongs. Further, it is important to detect how explanatory variables affect the clustering. An advantage of our method is that it can handle multiple explanatory variables of any type through a linear modeling of the cluster transition probabilities. We implement the generalized EM algorithm using several recursive relationships to greatly decrease the computational cost. The accuracy of our estimation method is illustrated in a simulation study, and U.S. Congressional data is analyzed.

Detecting discordance enrichment among a series of two-sample genome-wide expression data sets

BackgroundWith the current microarray and RNA-seq technologies, two-sample genome-wide expression data have been widely collected in biological and medical studies. The related differential expression analysis and gene set enrichment analysis have been frequently conducted. Integrative analysis can be conducted when multiple data sets are available. In practice, discordant molecular behaviors among a series of data sets can be of biological and clinical interest.MethodsIn this study, a statistical method is proposed for detecting discordance gene set enrichment. Our method is based on a two-level multivariate normal mixture model. It is statistically efficient with linearly increased parameter space when the number of data sets is increased. The model-based probability of discordance enrichment can be calculated for gene set detection.ResultsWe apply our method to a microarray expression data set collected from forty-five matched tumor/non-tumor pairs of tissues for studying pancreatic cancer. We divided the data set into a series of non-overlapping subsets according to the tumor/non-tumor paired expression ratio of gene PNLIP (pancreatic lipase, recently shown it association with pancreatic cancer). The log-ratio ranges from a negative value (e.g. more expressed in non-tumor tissue) to a positive value (e.g. more expressed in tumor tissue). Our purpose is to understand whether any gene sets are enriched in discordant behaviors among these subsets (when the log-ratio is increased from negative to positive). We focus on KEGG pathways. The detected pathways will be useful for our further understanding of the role of gene PNLIP in pancreatic cancer research. Among the top list of detected pathways, the neuroactive ligand receptor interaction and olfactory transduction pathways are the most significant two. Then, we consider gene TP53 that is well-known for its role as tumor suppressor in cancer research. The log-ratio also ranges from a negative value (e.g. more expressed in non-tumor tissue) to a positive value (e.g. more expressed in tumor tissue). We divided the microarray data set again according to the expression ratio of gene TP53. After the discordance enrichment analysis, we observed overall similar results and the above two pathways are still the most significant detections. More interestingly, only these two pathways have been identified for their association with pancreatic cancer in a pathway analysis of genome-wide association study (GWAS) data.ConclusionsThis study illustrates that some disease-related pathways can be enriched in discordant molecular behaviors when an important disease-related gene changes its expression. Our proposed statistical method is useful in the detection of these pathways. Furthermore, our method can also be applied to genome-wide expression data collected by the recent RNA-seq technology.