A tutorial on Bayesian multi-model linear regression with BAS and JASP.

Linear regression analyses commonly involve two consecutive stages of statistical inquiry. In the first stage, a single 'best' model is defined by a specific selection of relevant predictors; in the second stage, the regression coefficients of the winning model are used for prediction and for inference concerning the importance of the predictors. However, such second-stage inference ignores the model uncertainty from the first stage, resulting in overconfident parameter estimates that generalize poorly. These drawbacks can be overcome by model averaging, a technique that retains all models for inference, weighting each model's contribution by its posterior probability. Although conceptually straightforward, model averaging is rarely used in applied research, possibly due to the lack of easily accessible software. To bridge the gap between theory and practice, we provide a tutorial on linear regression using Bayesian model averaging in JASP, based on the BAS package in R. Firstly, we provide theoretical background on linear regression, Bayesian inference, and Bayesian model averaging. Secondly, we demonstrate the method on an example data set from the World Happiness Report. Lastly, we discuss limitations of model averaging and directions for dealing with violations of model assumptions.

Age modification of ozone associations with cardiovascular disease risk in adults: a potential role for soluble P-selectin and blood pressure.

BACKGROUND: Studies have suggested that age increases susceptibility to ozone-associated mortality, but the underlying mechanisms are unclear. In a previous study, personal exposure to ozone was significantly associated with a platelet activation biomarker, plasma soluble P-selectin (sCD62P), and blood pressure in 89 healthy adults, aged 22-52 years. The present study examines whether age modifies these associations in the same adults and in additional adults. METHODS: Interaction terms of age and exposure were analyzed using hierarchical Bayesian mixed effects ridge regressions. Data from a similar additional study involving 71 healthy participants, aged 19-26 years, were pooled with the data from the first study to evaluate age effect modification when more young adults were added to the analysis. RESULTS: In the 89 adults, significant age interactions were observed for past 24-hour and 2-week ozone exposures and sCD62P. Based on the pooled data (89 plus 71 adults), a 10 ppb increase in 24-hour ozone exposure was associated with increases in sCD62P and systolic blood pressure (SBP) by 22.3% (95% CI: 14.3%, 31.2%) and 1.35 (-0.18, 2.84) mmHg, respectively, at age 25; these values increased to 48.6% (32.7%, 65.1%) and 4.98 (2.56, 7.35) mmHg, respectively, at age 40. CONCLUSIONS: These results mechanistically suggest that increasing age enhances cardiovascular effects of ozone.

Risk Prediction for Epithelial Ovarian Cancer in 11 United States-Based Case-Control Studies: Incorporation of Epidemiologic Risk Factors and 17 Confirmed Genetic Loci.

Previously developed models for predicting absolute risk of invasive epithelial ovarian cancer have included a limited number of risk factors and have had low discriminatory power (area under the receiver operating characteristic curve (AUC) < 0.60). Because of this, we developed and internally validated a relative risk prediction model that incorporates 17 established epidemiologic risk factors and 17 genome-wide significant single nucleotide polymorphisms (SNPs) using data from 11 case-control studies in the United States (5,793 cases; 9,512 controls) from the Ovarian Cancer Association Consortium (data accrued from 1992 to 2010). We developed a hierarchical logistic regression model for predicting case-control status that included imputation of missing data. We randomly divided the data into an 80% training sample and used the remaining 20% for model evaluation. The AUC for the full model was 0.664. A reduced model without SNPs performed similarly (AUC = 0.649). Both models performed better than a baseline model that included age and study site only (AUC = 0.563). The best predictive power was obtained in the full model among women younger than 50 years of age (AUC = 0.714); however, the addition of SNPs increased the AUC the most for women older than 50 years of age (AUC = 0.638 vs. 0.616). Adapting this improved model to estimate absolute risk and evaluating it in prospective data sets is warranted.

Functional annotation signatures of disease susceptibility loci improve SNP association analysis.

BACKGROUND: Genetic association studies are conducted to discover genetic loci that contribute to an inherited trait, identify the variants behind these associations and ascertain their functional role in determining the phenotype. To date, functional annotations of the genetic variants have rarely played more than an indirect role in assessing evidence for association. Here, we demonstrate how these data can be systematically integrated into an association study's analysis plan. RESULTS: We developed a Bayesian statistical model for the prior probability of phenotype-genotype association that incorporates data from past association studies and publicly available functional annotation data regarding the susceptibility variants under study. The model takes the form of a binary regression of association status on a set of annotation variables whose coefficients were estimated through an analysis of associated SNPs in the GWAS Catalog (GC). The functional predictors examined included measures that have been demonstrated to correlate with the association status of SNPs in the GC and some whose utility in this regard is speculative: summaries of the UCSC Human Genome Browser ENCODE super-track data, dbSNP function class, sequence conservation summaries, proximity to genomic variants in the Database of Genomic Variants and known regulatory elements in the Open Regulatory Annotation database, PolyPhen-2 probabilities and RegulomeDB categories. Because we expected that only a fraction of the annotations would contribute to predicting association, we employed a penalized likelihood method to reduce the impact of non-informative predictors and evaluated the model's ability to predict GC SNPs not used to construct the model. We show that the functional data alone are predictive of a SNP's presence in the GC. Further, using data from a genome-wide study of ovarian cancer, we demonstrate that their use as prior data when testing for association is practical at the genome-wide scale and improves power to detect associations. CONCLUSIONS: We show how diverse functional annotations can be efficiently combined to create 'functional signatures' that predict the a priori odds of a variant's association to a trait and how these signatures can be integrated into a standard genome-wide-scale association analysis, resulting in improved power to detect truly associated variants.

BAYESIAN MODEL SEARCH AND MULTILEVEL INFERENCE FOR SNP ASSOCIATION STUDIES.

Technological advances in genotyping have given rise to hypothesis-based association studies of increasing scope. As a result, the scientific hypotheses addressed by these studies have become more complex and more difficult to address using existing analytic methodologies. Obstacles to analysis include inference in the face of multiple comparisons, complications arising from correlations among the SNPs (single nucleotide polymorphisms), choice of their genetic parametrization and missing data. In this paper we present an efficient Bayesian model search strategy that searches over the space of genetic markers and their genetic parametrization. The resulting method for Multilevel Inference of SNP Associations, MISA, allows computation of multilevel posterior probabilities and Bayes factors at the global, gene and SNP level, with the prior distribution on SNP inclusion in the model providing an intrinsic multiplicity correction. We use simulated data sets to characterize MISA's statistical power, and show that MISA has higher power to detect association than standard procedures. Using data from the North Carolina Ovarian Cancer Study (NCOCS), MISA identifies variants that were not identified by standard methods and have been externally "validated" in independent studies. We examine sensitivity of the NCOCS results to prior choice and method for imputing missing data. MISA is available in an R package on CRAN.

Association between DNA damage response and repair genes and risk of invasive serous ovarian cancer.

BACKGROUND: We analyzed the association between 53 genes related to DNA repair and p53-mediated damage response and serous ovarian cancer risk using case-control data from the North Carolina Ovarian Cancer Study (NCOCS), a population-based, case-control study. METHODS/PRINCIPAL FINDINGS: The analysis was restricted to 364 invasive serous ovarian cancer cases and 761 controls of white, non-Hispanic race. Statistical analysis was two staged: a screen using marginal Bayes factors (BFs) for 484 SNPs and a modeling stage in which we calculated multivariate adjusted posterior probabilities of association for 77 SNPs that passed the screen. These probabilities were conditional on subject age at diagnosis/interview, batch, a DNA quality metric and genotypes of other SNPs and allowed for uncertainty in the genetic parameterizations of the SNPs and number of associated SNPs. Six SNPs had Bayes factors greater than 10 in favor of an association with invasive serous ovarian cancer. These included rs5762746 (median OR(odds ratio)(per allele) = 0.66; 95% credible interval (CI) = 0.44-1.00) and rs6005835 (median OR(per allele) = 0.69; 95% CI = 0.53-0.91) in CHEK2, rs2078486 (median OR(per allele) = 1.65; 95% CI = 1.21-2.25) and rs12951053 (median OR(per allele) = 1.65; 95% CI = 1.20-2.26) in TP53, rs411697 (median OR (rare homozygote) = 0.53; 95% CI = 0.35 - 0.79) in BACH1 and rs10131 (median OR( rare homozygote) = not estimable) in LIG4. The six most highly associated SNPs are either predicted to be functionally significant or are in LD with such a variant. The variants in TP53 were confirmed to be associated in a large follow-up study. CONCLUSIONS/SIGNIFICANCE: Based on our findings, further follow-up of the DNA repair and response pathways in a larger dataset is warranted to confirm these results.

Single nucleotide polymorphisms in the TP53 region and susceptibility to invasive epithelial ovarian cancer.

The p53 protein is critical for multiple cellular functions including cell growth and DNA repair. We assessed whether polymorphisms in the region encoding TP53 were associated with risk of invasive ovarian cancer. The study population includes a total of 5,206 invasive ovarian cancer cases (2,829 of which were serous) and 8,790 controls from 13 case-control or nested case-control studies participating in the Ovarian Cancer Association Consortium (OCAC). Three of the studies performed independent discovery investigations involving genotyping of up to 23 single nucleotide polymorphisms (SNP) in the TP53 region. Significant findings from this discovery phase were followed up for replication in the other OCAC studies. Mixed effects logistic regression was used to generate posterior median per allele odds ratios (OR), 95% probability intervals (PI), and Bayes factors (BF) for genotype associations. Five SNPs showed significant associations with risk in one or more of the discovery investigations and were followed up by OCAC. Mixed effects analysis confirmed associations with serous invasive cancers for two correlated (r(2) = 0.62) SNPs: rs2287498 (median per allele OR, 1.30; 95% PI, 1.07-1.57) and rs12951053 (median per allele OR, 1.19; 95% PI, 1.01-1.38). Analyses of other histologic subtypes suggested similar associations with endometrioid but not with mucinous or clear cell cancers. This large study provides statistical evidence for a small increase in risk of ovarian cancer associated with common variants in the TP53 region.

Polymorphism in the IL18 gene and epithelial ovarian cancer in non-Hispanic white women.

Over 22,000 cases of ovarian cancer were diagnosed in 2007 in the United States, but only a fraction of them can be attributed to mutations in highly penetrant genes such as BRCA1. To determine whether low-penetrance genetic variants contribute to ovarian cancer risk, we genotyped 1,536 single nucleotide polymorphisms (SNP) in several candidate gene pathways in 848 epithelial ovarian cancer cases and 798 controls in the North Carolina Ovarian Cancer Study (NCO) using a customized Illumina array. The inflammation gene interleukin-18 (IL18) showed the strongest evidence for association with epithelial ovarian cancer in a gene-by-gene analysis (P = 0.002) with a <25% chance of being a false-positive finding (q value = 0.240). Using a multivariate model search algorithm over 11 IL18 tagging SNPs, we found that the association was best modeled by rs1834481. Further, this SNP uniquely tagged a significantly associated IL18 haplotype and there was an increased risk of epithelial ovarian cancer per rs1834481 allele (odds ratio, 1.24; 95% confidence interval, 1.06-1.45). In a replication stage, 12 independent studies from the Ovarian Cancer Association Consortium (OCAC) genotyped rs1834481 in an additional 5,877 cases and 7,791 controls. The fixed effects estimate per rs1834481 allele was null (odds ratio, 0.99; 95% confidence interval, 0.94-1.05) when data from the 12 OCAC studies were combined. The effect estimate remained unchanged with the addition of the initial North Carolina Ovarian Cancer Study data. This analysis shows the importance of consortia, like the OCAC, in either confirming or refuting the validity of putative findings in studies with smaller sample sizes. (Cancer Epidemiol Biomarkers Prev 2008;17(12):3567-72).

Clinical response to varying the stimulus parameters in deep brain stimulation for essential tremor.

Deep brain stimulation (DBS) of the ventral intermediate nucleus of the thalamus for essential tremor is sometimes limited by side effects. The mechanisms by which DBS alleviates tremor or causes side effects are unclear; thus, it is difficult to select stimulus parameters that maximize the width of the therapeutic window. The goal of this study was to quantify the impact on side effect intensity (SE), tremor amplitude, and the therapeutic window of varying stimulus parameters. Tremor amplitude and SE were recorded at 40 to 90 combinations of pulse width, frequency, and voltage across 14 thalami. Posterior variable inclusion probabilities indicated that frequency and voltage were the most important predictors of both SE and tremor amplitude. The amount of tremor suppression achieved at frequencies of 90 to 100 Hz was not different from that at 160 to 170 Hz. However, the width of the therapeutic window decreased significantly and power consumption increased as frequency was increased above 90 to 100 Hz. Improved understanding of the relationships between stimulus parameters and clinical responses may lead to improved techniques of stimulus parameter adjustment.

Minimum energy single-shock internal atrial defibrillation in sheep.

Well-tolerated internal atrial defibrillation shocks must be below the pain threshold, which has been estimated to be less than 1 Joule. Defibrillation of the atria with low energy is made possible by delivering shocks at the low end of the defibrillation dose-response curve. We studied low-energy defibrillation in sheep to test the hypothesis that the energy that defibrillates the atria 10% of the time (ED10) is less than 1 Joule. The ED10 was estimated in seven sheep with rapid pacing induced chronic atrial fibrillation (AF). Low-energy defibrillation shocks were delivered from coronary sinus (CS) to superior vena cava (SVC) and the ED10 and ED50 (energy that defibrillates the atria 50% of the time) were then calculated using logistic regression. The mean ratio of ED10 to ED50 was 0.50, indicating that on average, the ED10 was equal to half of the ED50. ED10 shocks had energies ranging from 1.2 to 5.8 Joules. These results suggest that painless single-shock low-energy defibrillation may not be feasible.

EFFECTS OF AVIAN SEED DISPERSAL ON THE GENETIC STRUCTURE OF WHITEBARK PINE POPULATIONS.

We used allozyme analysis to examine family structure, the spatial patterning of related individuals, in two populations of whitebark pine (Pinus albicaulis), a subalpine conifer that commonly displays a multistem form. The individual stems within clumps are genetically distinct individuals, having arisen from separate seeds. Individuals within a clump are genetically more similar than individuals in different clumps, but individuals in neighboring clumps do not appear to be more similar than individuals in distant clumps. This family structure appears to be a direct result of the seed-caching behavior of Clark's nutcrackers (Nucifraga columbiana), the primary dispersal agent for whitebark pine seeds.