Recent advances in technology for brain imaging and high-throughput genotyping have motivated studies examining the influence of genetic variation on brain structure. Wang et al. have developed an approach for the analysis of imaging genomic studies using penalized multi-task regression with regularization based on a novel group $l2,1$-norm penalty which encourages structured sparsity at both the gene level and SNP level. While incorporating a number of useful features, the proposed method only furnishes a point estimate of the regression coefficients; techniques for conducting statistical inference are not provided. A new Bayesian method is proposed here to overcome this limitation. We develop a Bayesian hierarchical modeling formulation where the posterior mode corresponds to the estimator proposed by Wang et al. and an approach that allows for full posterior inference including the construction of interval estimates for the regression parameters. We show that the proposed hierarchical model can be expressed as a three-level Gaussian scale mixture and this representation facilitates the use of a Gibbs sampling algorithm for posterior simulation. Simulation studies demonstrate that the interval estimates obtained using our approach achieve adequate coverage probabilities that outperform those obtained from the nonparametric bootstrap. Our proposed methodology is applied to the analysis of neuroimaging and genetic data collected as part of the Alzheimer’s Disease Neuroimaging Initiative (ADNI), and this analysis of the ADNI cohort demonstrates clearly the value added of incorporating interval estimation beyond only point estimation when relating SNPs to brain imaging endophenotypes. Software and sample data is available as an R package ‘bgsmtr’ that can be downloaded from The Comprehensive R Archive Network (CRAN). Supplementary data are available at Bioinformatics online.

With the rapid growth of modern technology, many biomedical studies are being conducted to collect massive datasets with volumes of multi‐modality imaging, genetic, neurocognitive and clinical information from increasingly large cohorts. Simultaneously extracting and integrating rich and diverse heterogeneous information in neuroimaging and/or genomics from these big datasets could transform our understanding of how genetic variants impact brain structure and function, cognitive function and brain‐related disease risk across the lifespan. Such understanding is critical for diagnosis, prevention and treatment of numerous complex brain‐related disorders (e.g., schizophrenia and Alzheimer's disease). However, the development of analytical methods for the joint analysis of both high‐dimensional imaging phenotypes and high‐dimensional genetic data, a big data squared (BD²) problem, presents major computational and theoretical challenges for existing analytical methods. Besides the high‐dimensional nature of BD², various neuroimaging measures often exhibit strong spatial smoothness and dependence and genetic markers may have a natural dependence structure arising from linkage disequilibrium. We review some recent developments of various statistical techniques for imaging genetics, including massive univariate and voxel‐wise approaches, reduced rank regression, mixture models and group sparse multi‐task regression. By doing so, we hope that this review may encourage others in the statistical community to enter into this new and exciting field of research. The Canadian Journal of Statistics 47: 108–131; 2019 © 2019 Statistical Society of Canada

Unlike other histological types of epithelial ovarian carcinoma, clear cell ovarian carcinoma (CCOC) has poor response to therapy. In many other carcinomas, expression of the hypoxia-related enzyme Carbonic anhydrase IX (CAIX) by cancer cells is associated with poor prognosis, while the presence of CD8 + tumor-infiltrating lymphocytes (TIL) is positively prognostic. We employed [¹⁸F]EF5-PET/CT imaging, transcriptome profiling, and spatially-resolved histological analysis to evaluate relationships between CAIX, CD8, and survival in CCOC. Tissue microarrays (TMAs) were evaluated for 218 cases in the Canadian COEUR study. Non-spatial relationships between CAIX and CD8 were investigated using Spearman rank correlation, negative binomial regression and gene set enrichment analysis. Spatial relationships at the cell level were investigated using the cross K-function. Survival analysis was used to assess the relationship of CAIX and CD8 with patient survival for 154 cases. CD8 + T cell infiltration positively predicted survival with estimated hazard ratio 0.974 (95% CI 0.950, 1000). The negative binomial regression analysis found a strong TMA effect (p-value < 0.0001). It also indicated a negative association between CD8 and CAIX overall (p-value = 0.0171) and in stroma (p-value = 0.0050) but not in tumor (p-value = 0.173). Examination of the spatial association between the locations of CD8 + T cells and CAIX cells found a significant amount of heterogeneity in the first TMA, while in the second TMA there was a clear signal indicating negative spatial association in stromal regions. These results suggest that hypoxia may contribute to immune exclusion, primarily mediated by effects in stroma.

We discuss hypothesis testing and compare different theories in light of observed or experimental data as fundamental endeavors in the sciences. Issues associated with the p-value approach and null hypothesis significance testing are reviewed, and the Bayesian alternative based on the Bayes factor is introduced, along with a review of computational methods and sensitivity related to prior distributions. We demonstrate how Bayesian testing can be practically implemented in several examples, such as the t-test, two-sample comparisons, linear mixed models, and Poisson mixed models by using existing software. Caveats and potential problems associated with Bayesian testing are also discussed. We aim to inform researchers in the many fields where Bayesian testing is not in common use of a well-developed alternative to null hypothesis significance testing and to demonstrate its standard implementation.