When analysing spatial data, it is important to account for spatial autocorrelation. In Bayesian statistics, spatial autocorrelation is commonly modelled by the intrinsic conditional autoregressive prior distribution. At the heart of this model is a spatial weights matrix which controls the behaviour and degree of spatial smoothing. The purpose of this study is to review the main specifications of the spatial weights matrix found in the literature, and together with some new and less common specifications, compare the effect that they have on smoothing and model performance. The popular BYM model is described, and a simple solution for addressing the identifiability issue among the spatial random effects is provided. Seventeen different definitions of the spatial weights matrix are defined, which are classified into four classes: adjacency-based weights, and weights based on geographic distance, distance between covariate values, and a hybrid of geographic and covariate distances. These last two definitions embody the main novelty of this research. Three synthetic data sets are generated, each representing a different underlying spatial structure. These data sets together with a real spatial data set from the literature are analysed using the models. The models are evaluated using the deviance information criterion and Moran’s I statistic. The deviance information criterion indicated that the model which uses binary, first-order adjacency weights to perform spatial smoothing is generally an optimal choice for achieving a good model fit. Distance-based weights also generally perform quite well and offer similar parameter interpretations. The less commonly explored options for performing spatial smoothing generally provided a worse model fit than models with more traditional approaches to smoothing, but usually outperformed the benchmark model which did not conduct spatial smoothing. The specification of the spatial weights matrix can have a colossal impact on model fit and parameter estimation. The results provide some evidence that a smaller number of neighbours used in defining the spatial weights matrix yields a better model fit, and may provide a more accurate representation of the underlying spatial random field. The commonly used binary, first-order adjacency weights still appear to be a good choice for implementing spatial smoothing.

Although many genetic variants have been associated with multiple sclerosis (MS) risk, they do not explain all the disease risk and there remains uncertainty as to how these variants contribute to disease. DNA methylation is an epigenetic mechanism that can influence gene expression and has the potential to mediate the effects of environmental factors on MS. In a previous study, we found a differentially methylation region (DMR) at MHC HLA-DRB1 that was associated within relapsing-remitting MS (RRMS) patients in CD4+ T cells. This study aimed to confirm this earlier finding in an independent RRMS cohort of treatment-naïve female patients. Total genomic DNA was extracted from CD4+ T cells of 28 female RRMS and 22 age-matched healthy controls subjects. DNA was bisulfite-converted and hybridised to Illumina 450K arrays. Beta values for all CpGs were analysed using the DMPFinder function in the MINFI program, and a follow-up prioritisation process was applied to identify the most robust MS-associated DMRs. This study confirmed our previous findings of a hypomethylated DMR at HLA-DRB1 and a hypermethylated DMR at HLA-DRB5 in this RRMS patient cohort. In addition, we identified a large independent DMR at MHC, whereby 11 CpGs in RNF39 were hypermethylated in MS cases compared to controls (max. ∆beta = 0.19, P = 2.1 × 10−4). We did not find evidence that SNP genotype was influencing the DMR in this cohort. A smaller MHC DMR was also identified at HCG4B, and two non-MHC DMRs at PM20D1 on chr1 and ERICH1 on chr8 were also identified. The findings from this study confirm our previous results of a DMR at HLA-DRB1 and also suggest hypermethylation in an independent MHC locus, RNF39, is associated with MS. Taken together, our results highlight the importance of epigenetic factors at the MHC locus in MS independent of treatment, age and sex. Prospective studies are now required to discern whether methylation at MHC is involved in influencing risk of disease onset or whether the disease itself has altered the methylation profile.

This paper proposes solutions to three issues pertaining to the estimation of finite mixture models with an unknown number of components: the non-identifiability induced by overfitting the number of components, the mixing limitations of standard Markov Chain Monte Carlo (MCMC) sampling techniques, and the related label switching problem. An overfitting approach is used to estimate the number of components in a finite mixture model via a Zmix algorithm. Zmix provides a bridge between multidimensional samplers and test based estimation methods, whereby priors are chosen to encourage extra groups to have weights approaching zero. MCMC sampling is made possible by the implementation of prior parallel tempering, an extension of parallel tempering. Zmix can accurately estimate the number of components, posterior parameter estimates and allocation probabilities given a sufficiently large sample size. The results will reflect uncertainty in the final model and will report the range of possible candidate models and their respective estimated probabilities from a single run. Label switching is resolved with a computationally light-weight method, Zswitch, developed for overfitted mixtures by exploiting the intuitiveness of allocation-based relabelling algorithms and the precision of label-invariant loss functions. Four simulation studies are included to illustrate Zmix and Zswitch, as well as three case studies from the literature. All methods are available as part of the R package Zmix, which can currently be applied to univariate Gaussian mixture models.