Direct formulation to Cholesky decomposition of a general nonsingular correlation matrix

We present two novel, explicit representations of Cholesky factor of a nonsingular correlation matrix. The first representation uses semi-partial correlation coefficients as its entries. The second, uses an equivalent form of the square roots of the differences between two ratios of successive determinants. Each of the two new forms enjoys parsimony of notations and offers a simpler alternative to both spherical factorization and the multiplicative partial correlation Cholesky matrix (Cooke et al 2011). Two relevant applications are offered for each form: a simple $t$-test for assessing the independence of a single variable in a multivariate normal structure, and a straightforward algorithm for generating random positive-definite correlation matrix. The second representation is also extended to any nonsingular hermitian matrix.Comment: Accepted to Statistics and Probability Letters, March 201

seeQTL: a searchable database for human eQTLs

Summary: seeQTL is a comprehensive and versatile eQTL database, including various eQTL studies and a meta-analysis of HapMap eQTL information. The database presents eQTL association results in a convenient browser, using both segmented local-association plots and genome-wide Manhattan plots

Gene Expression in Transformed Lymphocytes Reveals Variation in Endomembrane and HLA Pathways Modifying Cystic Fibrosis Pulmonary Phenotypes

Variation in cystic fibrosis (CF) phenotypes, including lung disease severity, age of onset of persistent Pseudomonas aeruginosa (P. aeruginosa) lung infection, and presence of meconium ileus (MI), has been partially explained by genome-wide association studies (GWASs). It is not expected that GWASs alone are sufficiently powered to uncover all heritable traits associated with CF phenotypic diversity. Therefore, we utilized gene expression association from lymphoblastoid cells lines from 754 p.Phe508del CF-affected homozygous individuals to identify genes and pathways. LPAR6, a G protein coupled receptor, associated with lung disease severity (false discovery rate q value = 0.0006). Additional pathway analyses, utilizing a stringent permutation-based approach, identified unique signals for all three phenotypes. Pathways associated with lung disease severity were annotated in three broad categories: (1) endomembrane function, containing p.Phe508del processing genes, providing evidence of the importance of p.Phe508del processing to explain lung phenotype variation; (2) HLA class I genes, extending previous GWAS findings in the HLA region; and (3) endoplasmic reticulum stress response genes. Expression pathways associated with lung disease were concordant for some endosome and HLA pathways, with pathways identified using GWAS associations from 1,978 CF-affected individuals. Pathways associated with age of onset of persistent P. aeruginosa infection were enriched for HLA class II genes, and those associated with MI were related to oxidative phosphorylation. Formal testing demonstrated that genes showing differential expression associated with lung disease severity were enriched for heritable genetic variation and expression quantitative traits. Gene expression provided a powerful tool to identify unrecognized heritable variation, complementing ongoing GWASs in this rare disease

Interaction models matter: an efficient, flexible computational framework for model-specific investigation of epistasis.

PURPOSE: Epistasis, the interaction between two or more genes, is integral to the study of genetics and is present throughout nature. Yet, it is seldom fully explored as most approaches primarily focus on single-locus effects, partly because analyzing all pairwise and higher-order interactions requires significant computational resources. Furthermore, existing methods for epistasis detection only consider a Cartesian (multiplicative) model for interaction terms. This is likely limiting as epistatic interactions can evolve to produce varied relationships between genetic loci, some complex and not linearly separable. METHODS: We present new algorithms for the interaction coefficients for standard regression models for epistasis that permit many varied models for the interaction terms for loci and efficient memory usage. The algorithms are given for two-way and three-way epistasis and may be generalized to higher order epistasis. Statistical tests for the interaction coefficients are also provided. We also present an efficient matrix based algorithm for permutation testing for two-way epistasis. We offer a proof and experimental evidence that methods that look for epistasis only at loci that have main effects may not be justified. Given the computational efficiency of the algorithm, we applied the method to a rat data set and mouse data set, with at least 10,000 loci and 1,000 samples each, using the standard Cartesian model and the XOR model to explore body mass index. RESULTS: This study reveals that although many of the loci found to exhibit significant statistical epistasis overlap between models in rats, the pairs are mostly distinct. Further, the XOR model found greater evidence for statistical epistasis in many more pairs of loci in both data sets with almost all significant epistasis in mice identified using XOR. In the rat data set, loci involved in epistasis under the XOR model are enriched for biologically relevant pathways. CONCLUSION: Our results in both species show that many biologically relevant epistatic relationships would have been undetected if only one interaction model was applied, providing evidence that varied interaction models should be implemented to explore epistatic interactions that occur in living systems