Multi-pollutant exposure profiles associated with breast cancer risk: A Bayesian profile regression analysis in the French E3N cohort.

BACKGROUND: Human exposure to air pollution involves complex mixtures of multiple correlated air pollutants. To date, very few studies have assessed the combined effects of exposure to multiple air pollutants on breast cancer (BC) risk. OBJECTIVES: We aimed to assess the association between combined exposures to multiple air pollutants and breast cancer risk. METHODS: The study was based on a case-control study nested within the French E3N cohort (5222 incident BC cases/5222 matched controls). For each woman, the average of the mean annual exposure to eight pollutants (benzo(a)oyrene, cadmium, dioxins, polychlorinated biphenyls (PCB153), nitrogen dioxide (NO2), ozone, particulate matter and fine particles (PMs)) was estimated from cohort inclusion in 1990 to the index date. We used the Bayesian Profile Regression (BPR) model, which groups individuals according to their exposure and risk levels, and assigns a risk to each cluster identified. The model was adjusted on a combination of matching variables and confounders to better consider the design of the nested case-control study. Odds ratios (OR) and their 95 % credible intervals (CrI) were estimated. RESULTS: Among the 21 clusters identified, the cluster characterised by low exposures to all pollutants, except ozone, was taken as reference. A consistent increase in BC risk compared to the reference cluster was observed for 3 clusters: cluster 9 (OR=1.61; CrI=1.13,2.26), cluster 16 (OR=1.59; CrI=1.10,2.30) and cluster 15 (OR=1.38; CrI=1.00,1.88) characterised by high levels of NO2, PMs and PCB153. The other clusters showed no consistent association with BC. DISCUSSION: This is the first study assessing the effect of exposure to a mixture of eight air pollutants on BC risk, using the BPR approach. Overall, results showed evidence of a positive joint effect of exposure to high levels to most pollutants, particularly high for NO2, PMs and PCB153, on the risk of BC.

Bayesian hierarchical profile regression for binary covariates.

Dysphagia, a common result of other medical conditions, is caused by malfunctions in swallowing physiology resulting in difficulty eating and drinking. The Modified Barium Swallow Study (MBSS), the most commonly used diagnostic tool for evaluating dysphagia, can be assessed using the Modified Barium Swallow Impairment Profile (MBSImP™). The MBSImP assessment tool consists of a hierarchical grouped data structure with multiple domains, a set of components within each domain which characterize specific swallowing physiologies, and a set of tasks scored on a discrete scale within each component. We lack sophisticated approaches to extract patterns of physiologic swallowing impairment from the MBSImP task scores within a component while still recognizing the nested structure of components within a domain. We propose a Bayesian hierarchical profile regression model, which uses a Bayesian profile regression model in conjunction with a hierarchical Dirichlet process mixture model to (1) cluster subjects into impairment profile patterns while respecting the hierarchical grouped data structure of the MBSImP, and (2) simultaneously determine associations between latent profile cluster membership for all components and the outcome of dysphagia severity. We apply our approach to a cohort of patients referred for an MBSS and assessed using the MBSImP. Our research results can be used to inform appropriate intervention strategies, and provide tools for clinicians to make better multidimensional management and treatment decisions for patients with dysphagia.

Disease Modelling of Cognitive Outcomes and Biomarkers in the European Prevention of Alzheimer's Dementia Longitudinal Cohort.

A key challenge for the secondary prevention of Alzheimer's dementia is the need to identify individuals early on in the disease process through sensitive cognitive tests and biomarkers. The European Prevention of Alzheimer's Dementia (EPAD) consortium recruited participants into a longitudinal cohort study with the aim of building a readiness cohort for a proof-of-concept clinical trial and also to generate a rich longitudinal data-set for disease modelling. Data have been collected on a wide range of measurements including cognitive outcomes, neuroimaging, cerebrospinal fluid biomarkers, genetics and other clinical and environmental risk factors, and are available for 1,828 eligible participants at baseline, 1,567 at 6 months, 1,188 at one-year follow-up, 383 at 2 years, and 89 participants at three-year follow-up visit. We novelly apply state-of-the-art longitudinal modelling and risk stratification approaches to these data in order to characterise disease progression and biological heterogeneity within the cohort. Specifically, we use longitudinal class-specific mixed effects models to characterise the different clinical disease trajectories and a semi-supervised Bayesian clustering approach to explore whether participants can be stratified into homogeneous subgroups that have different patterns of cognitive functioning evolution, while also having subgroup-specific profiles in terms of baseline biomarkers and longitudinal rate of change in biomarkers.

Profile of exposures and lung function in adults with asthma: An exposome approach in the EGEA study.

BACKGROUND: Environmental research on multifactorial health outcomes calls for exposome approaches able to assess the joint effect of multiple exposures. OBJECTIVE: Our aim was to identify profiles of exposure to lifestyle/environmental factors associated with lung function in adults with asthma using a cluster-based approach. METHODS: We used data from 599 adults of the Epidemiological study on the Genetics and Environment of Asthma, bronchial hyperresponsiveness and atopy (EGEA) (mean age 39.0 years, 52% men) who ever had asthma. Exposures to 53 lifestyle/environmental factors were assessed by questionnaires or geographic information systems-based models. A two-step approach was developed: 1) exposome dimension reduction by selecting factors showing association with forced expiratory volume in 1 s (FEV1) (p < 0.20) in an exposome-wide association study (ExWAS), 2) clustering analysis using the supervised Bayesian Profile Regression (sBPR) to group individuals according to FEV1 level and to their profile of exposure to a reduced set of uncorrelated exposures (each paired correlation<0.70) identified in step 1. RESULTS: The ExWAS identified 21 factors showing suggestive association with FEV1 (none significant when controlling for multiple tests). The sBPR conducted on 15 uncorrelated exposures identified in step 1, revealed 3 clusters composed of 30, 115 and 454 individuals with a mean ± SD FEV1(%pred) of 79% ± 21, 90% ± 19 and 93% ± 16, respectively. Cluster 1 was composed of individuals with heavy smoking, poor diet, higher outdoor humidity and proximity to traffic, while cluster 2 and 3 included individuals with moderate/low levels of exposure to these factors. DISCUSSION: This exposome study identified a specific profile of joint lifestyle and environmental factors, associated with a low FEV1 in adults with asthma. None of the exposures revealed significant association when considered independently.

Bayesian Profile Regression to Deal With Multiple Highly Correlated Exposures and a Censored Survival Outcome. First Application in Ionizing Radiation Epidemiology.

As multifactorial and chronic diseases, cancers are among these pathologies for which the exposome concept is essential to gain more insight into the associated etiology and, ultimately, lead to better primary prevention strategies for public health. Indeed, cancers result from the combined influence of many genetic, environmental and behavioral stressors that may occur simultaneously and interact. It is thus important to properly account for multifactorial exposure patterns when estimating specific cancer risks at individual or population level. Nevertheless, the risk factors, especially environmental, are still too often considered in isolation in epidemiological studies. Moreover, major statistical difficulties occur when exposures to several factors are highly correlated due, for instance, to common sources shared by several pollutants. Suitable statistical methods must then be used to deal with these multicollinearity issues. In this work, we focused on the specific problem of estimating a disease risk from highly correlated environmental exposure covariates and a censored survival outcome. We extended Bayesian profile regression mixture (PRM) models to this context by assuming an instantaneous excess hazard ratio disease sub-model. The proposed hierarchical model incorporates an underlying truncated Dirichlet process mixture as an attribution sub-model. A specific adaptive Metropolis-Within-Gibbs algorithm-including label switching moves-was implemented to infer the model. This allows simultaneously clustering individuals with similar risks and similar exposure characteristics and estimating the associated risk for each group. Our Bayesian PRM model was applied to the estimation of the risk of death by lung cancer in a cohort of French uranium miners who were chronically and occupationally exposed to multiple and correlated sources of ionizing radiation. Several groups of uranium miners with high risk and low risk of death by lung cancer were identified and characterized by specific exposure profiles. Interestingly, our case study illustrates a limit of MCMC algorithms to fit full Bayesian PRM models even if the updating schemes for the cluster labels incorporate label-switching moves. Then, although this paper shows that Bayesian PRM models are promising tools for exposome research, it also opens new avenues for methodological research in this class of probabilistic models.

A spatial joint analysis of metal constituents of ambient particulate matter and mortality in England.

Few studies have investigated associations between metal components of particulate matter on mortality due to well-known issues of multicollinearity. Here, we analyze these exposures jointly to evaluate their associations with mortality on small area data. We fit a Bayesian profile regression (BPR) to account for the multicollinearity in the elemental components (iron, copper, and zinc) of PM10 and PM2.5. The models are developed in relation to mortality from cardiovascular and respiratory disease and lung cancer incidence in 2008-2011 at a small area level, for a population of 13.6 million in the London-Oxford area of England. From the BPR, we identified higher risks in the PM10 fraction cluster likely to represent the study area, excluding London, for cardiovascular mortality relative risk (RR) 1.07 (95% credible interval [CI] 1.02, 1.12) and for respiratory mortality RR 1.06 (95%CI 0.99, 1.31), compared with the study mean. For PM2.5 fraction, higher risks were seen for cardiovascular mortality RR 1.55 (CI 95% 1.38, 1.71) and respiratory mortality RR 1.51 (CI 95% 1.33, 1.72), likely to represent the "highways" cluster. We did not find relevant associations for lung cancer incidence. Our analysis showed small but not fully consistent adverse associations between health outcomes and particulate metal exposures. The BPR approach identified subpopulations with unique exposure profiles and provided information about the geographical location of these to help interpret findings.

Prenatal phthalate, paraben, and phenol exposure and childhood allergic and respiratory outcomes: Evaluating exposure to chemical mixtures.

BACKGROUND: Chemicals found in personal care products and plastics have been associated with asthma, allergies, and lung function, but methods to address real life exposure to mixtures of these chemicals have not been applied to these associations. METHODS: We quantified urinary concentrations of eleven phthalate metabolites, four parabens, and five other phenols in mothers twice during pregnancy and assessed probable asthma, aeroallergies, and lung function in their age seven children. We implemented Bayesian Profile Regression (BPR) to cluster women by their exposures to these chemicals and tested the clusters for differences in outcome measurements. We used Bayesian Kernel Machine Regression (BKMR) to fit biomarkers into one model as joint independent variables. RESULTS: BPR clustered women into seven groups characterized by patterns of personal care product and plastic use, though there were no significant differences in outcomes across clusters. BKMR showed that monocarboxyisooctyl phthalate and 2,4-dichlorophenol were associated with probable asthma (predicted probability of probable asthma per IQR of biomarker z-score (standard deviation) = 0.08 (0.09) and 0.11 (0.12), respectively) and poorer lung function (predicted probability per IQR = -0.07 (0.05) and -0.07 (0.06), respectively), and that mono(3-carboxypropyl) phthalate and bisphenol A were associated with aeroallergies (predicted probability per IQR = 0.13 (0.09) and 0.11 (0.08), respectively). Several biomarkers demonstrated positive additive effects on other associations. CONCLUSIONS: BPR and BKMR are useful tools to evaluate associations of biomarker concentrations within a mixture of exposure and should supplement single-chemical regression models when data allow.

Modeling tails for collinear data with outliers in the English Longitudinal Study of Ageing: Quantile profile regression.

Research has shown that high blood glucose levels are important predictors of incident diabetes. However, they are also strongly associated with other cardiometabolic risk factors such as high blood pressure, adiposity, and cholesterol, which are also highly correlated with one another. The aim of this analysis was to ascertain how these highly correlated cardiometabolic risk factors might be associated with high levels of blood glucose in older adults aged 50 or older from wave 2 of the English Longitudinal Study of Ageing (ELSA). Due to the high collinearity of predictor variables and our interest in extreme values of blood glucose we proposed a new method, called quantile profile regression, to answer this question. Profile regression, a Bayesian nonparametric model for clustering responses and covariates simultaneously, is a powerful tool to model the relationship between a response variable and covariates, but the standard approach of using a mixture of Gaussian distributions for the response model will not identify the underlying clusters correctly, particularly with outliers in the data or heavy tail distribution of the response. Therefore, we propose quantile profile regression to model the response variable with an asymmetric Laplace distribution, allowing us to model more accurately clusters that are asymmetric and predict more accurately for extreme values of the response variable and/or outliers. Our new method performs more accurately in simulations when compared to Normal profile regression approach as well as robustly when outliers are present in the data. We conclude with an analysis of the ELSA.

Spatial analysis of drug poisoning deaths in the American west: A comparison study using profile regression to adjust for collinearity and spatial correlation.

BACKGROUND: The USA has seen dramatic increases in drug poisoning deaths (DPD) recently. State-level rates have responded to federal and state initiatives, yet the counties with the highest rates are stable. Spatial analysis enables investigators to identify the highest risk counties and most important risk factors, although results are often confounded by spatial autocorrelation and multicollinearity. METHODS: Profile regression (PR) is an integrated method for cluster and regression analysis, which adjusts for spatial-autocorrelation and multi-collinearity. RESULTS: With PR, three clusters were identified in the Western USA with most of NM, NV and UT and several counties in AZ, CO, ID and WY being high-risk. Cluster analysis in a previous study only identified high-risk counties in northern CA, NM and NV. Elevation, suicide and LDS population were positively, and population density was negatively linked with DPD for PR and standard regression (SR) showing differences between the mountain west and coastal areas. Complex relationships between DPD and several variables were identified by PR which was not possible with SR. CONCLUSIONS: Statistically principled methods like PR are needed for appropriate identification of the highest risk counties and important risk factors given the complex relationships with DPD. Funding for prevention, education and medical services should be targeted at rural, mountain communities in the west which have high %LDS and suicide rates. Counties with high %poverty and %Hispanic were also at high-risk. Individual-level studies are needed to confirm important risk factors in high-risk counties.

How Short Is Long Enough? Modeling Temporal Aspects of Human Mobility Behavior Using Mobile Phone Data.

Time-location data collected from location-sensing technologies have the potential to advance our understanding of human mobility. Existing human activity studies tend to ignore a critical issue in data collection-the time period for which the activity data will be collected. Our study investigated this significant gap in the literature on temporal aspects of human mobility behavior-how many days constitute a period long enough to capture individuals' highly organized activity episodes and how they vary among individuals with heterogeneous demographic and social-economic characteristics. To determine a minimum number of days to capture individuals' highly organized activity episodes in activity space, we examined a distribution of Kullback-Leibler divergence indexes. To evaluate the differences in the minimal number of observation days per subgroup whose demographic and economic characteristics are heterogenous, we used a Bayesian profile regression model. Our study showed that the estimated minimum number of days required to capture routine activity patterns was 13.5 days with a standard deviation of 6.64. We found that participant's age, employment status, size of household, and accessibility to downtown, food, and physical activity, as well as economic status of residential environment, are important factors that affect temporal aspects of mobility behavior.

Datos sobre la relación tiempo­localización generados a partir de tecnologías de percepción de la localización son potencialmente aptos para avanzar nuestra comprensión de la movilidad humana. Los estudios existentes sobre actividad humana tienden a ignorar un punto crítico en la recolección de datos­el período de tiempo para el cual se recogerán los datos de la actividad­. Nuestro estudio investigó esta brecha significativa en la literatura sobre los aspectos temporales de la conducta de movilidad humana­de cuántos días estará constituido un período lo suficientemente largo para captar los episodios altamente organizados de actividad de los individuos, y cómo varían aquellos entre individuos con características demográficas y socio-económicas heterogéneas­. Para determinar un número mínimo de días necesarios para captar los episodios altamente organizados de actividad de los individuos en un espacio de actividad, examinamos una distribución de los índices de divergencia Kullback­Leibler. Para evaluar las diferencias en el número mínimo de días de observación por subgrupo cuyas características demográficas y económicas son heterogéneas, usamos un modelo de regresión de perfil bayesiano. Nuestro estudio mostró que el número mínimo estimado de días requeridos para captar patrones de actividad rutinaria era de 13.5 días con una desviación estándar de 6.64. Descubrimos que la edad del participante, el estatus de empleo, el tamaño del hogar y la accesibilidad al centro de la ciudad, la alimentación y la actividad física, lo mismo que el estatus económico de los entornos residenciales, son factores importantes que afectan aspectos temporales de la conducta de movilidad.

Determinants of medication adherence in older people with dementia from the caregivers' perspective.

ABSTRACTBackground:Adherence to treatment is a primary determinant of treatment success. Caregiver support can influence medication adherence in people with cognitive impairment. This study sought to characterize medication adherence in older people with dementia from the caregivers' perspective, and to identify influencing factors. METHODS: Caregivers caring for a person with dementia and living in the community were eligible to complete the survey. Bayesian profile regression was applied to identify determinants of medication adherence measured using the Adherence to Refills and Medication Scale. RESULTS: Out of the 320 caregivers who participated in the survey, Bayesian profile regression on 221 participants identified two groups: Profile 1 (55 caregivers) with a mean adherence rate of 0.69 (80% Credible Interval (CrI): 0.61-0.77), and Profile 2 (166 caregivers) with a mean adherence rate of 0.80 (80% CrI: 0.77-0.84). Caregivers in Profile 1 were characterized with below data average scores for the following: cognitive functioning, commitment or intention, self-efficacy, and health knowledge, which were all above the data average in Profile 2, except for health knowledge. Caregivers in Profile 1 had a greater proportion of care recipients taking more than five medications and with late-stage dementia. Trade, technical, or vocational training was more common among the caregivers in Profile 1. Profile 2 caregivers had a better patient-provider relationship and less medical problems. CONCLUSIONS: Bayesian profile regression was useful in understanding caregiver factors that influence medication adherence. Tailored interventions to the determinants of medication adherence can guide the development of evidence-based interventions.

Multi-pollutant Modeling Through Examination of Susceptible Subpopulations Using Profile Regression.

PURPOSE OF REVIEW: The inter-correlated nature of exposure-based risk factors in environmental health studies makes it a challenge to determine their combined effect on health outcomes. As such, there has been much research of late regarding the development and utilization of methods in the field of multi-pollutant modeling. However, much of this work has focused on issues related to variable selection in a regression context, with the goal of identifying which exposures are the "bad actors" most responsible for affecting the health outcome of interest. However, the question addressed by these approaches does not necessarily represent the only or most important questions of interest in a multi-pollutant modeling context, where researchers may be interested in health effects from co-exposure patterns and in identifying subpopulations associated with patterns defined by different levels of constituent exposures. RECENT FINDINGS: One approach to analyzing multi-pollutant data is to use a method known as Bayesian profile regression, which aids in identifying susceptible subpopulations associated with exposure mixtures defined by different levels of each exposure. Identification of exposure-level patterns that correspond to a location may provide a starting point for policy-based exposure reduction. Also, in a spatial context, identification of locations with the most health-relevant exposure-mixture profiles might provide further policy relevant information. In this brief report, we review and describe an approach that can be used to identify exposures in subpopulations or locations known as Bayesian profile regression. An example is provided in which we examine associations between air pollutants, an indicator of healthy food retailer availability, and indicators of poverty in Los Angeles County. A general tread suggesting that vulnerable individuals are more highly exposed and have limited access to healthy food retailers is observed, though the associations are complex and non-linear.

Analysing malaria incidence at the small area level for developing a spatial decision support system: A case study in Kalaburagi, Karnataka, India.

Spatial decision support systems have already proved their value in helping to reduce infectious diseases but to be effective they need to be designed to reflect local circumstances and local data availability. We report the first stage of a project to develop a spatial decision support system for infectious diseases for Karnataka State in India. The focus of this paper is on malaria incidence and we draw on small area data on new cases of malaria analysed in two-monthly time intervals over the period February 2012 to January 2016 for Kalaburagi taluk, a small area in Karnataka. We report the results of data mapping and cluster detection (identifying areas of excess risk) including evaluating the temporal persistence of excess risk and the local conditions with which high counts are statistically associated. We comment on how this work might feed into a practical spatial decision support system.

Modelling collinear and spatially correlated data.

In this work we present a statistical approach to distinguish and interpret the complex relationship between several predictors and a response variable at the small area level, in the presence of (i) high correlation between the predictors and (ii) spatial correlation for the response. Covariates which are highly correlated create collinearity problems when used in a standard multiple regression model. Many methods have been proposed in the literature to address this issue. A very common approach is to create an index which aggregates all the highly correlated variables of interest. For example, it is well known that there is a relationship between social deprivation measured through the Multiple Deprivation Index (IMD) and air pollution; this index is then used as a confounder in assessing the effect of air pollution on health outcomes (e.g. respiratory hospital admissions or mortality). However it would be more informative to look specifically at each domain of the IMD and at its relationship with air pollution to better understand its role as a confounder in the epidemiological analyses. In this paper we illustrate how the complex relationships between the domains of IMD and air pollution can be deconstructed and analysed using profile regression, a Bayesian non-parametric model for clustering responses and covariates simultaneously. Moreover, we include an intrinsic spatial conditional autoregressive (ICAR) term to account for the spatial correlation of the response variable.

Multidimensional analysis of the effect of occupational exposure to organic solvents on lung cancer risk: the ICARE study.

BACKGROUND: The association between lung cancer and occupational exposure to organic solvents is discussed. Since different solvents are often used simultaneously, it is difficult to assess the role of individual substances. OBJECTIVES: The present study is focused on an in-depth investigation of the potential association between lung cancer risk and occupational exposure to a large group of organic solvents, taking into account the well-known risk factors for lung cancer, tobacco smoking and occupational exposure to asbestos. METHODS: We analysed data from the Investigation of occupational and environmental causes of respiratory cancers (ICARE) study, a large French population-based case-control study, set up between 2001 and 2007. A total of 2276 male cases and 2780 male controls were interviewed, and long-life occupational history was collected. In order to overcome the analytical difficulties created by multiple correlated exposures, we carried out a novel type of analysis based on Bayesian profile regression. RESULTS: After analysis with conventional logistic regression methods, none of the 11 solvents examined were associated with lung cancer risk. Through a profile regression approach, we did not observe any significant association between solvent exposure and lung cancer. However, we identified clusters at high risk that are related to occupations known to be at risk of developing lung cancer, such as painters. CONCLUSIONS: Organic solvents do not appear to be substantial contributors to the occupational risk of lung cancer for the occupations known to be at risk.

Multi-pollutant exposure profiles associated with term low birth weight in Los Angeles County.

Research indicates that multiple outdoor air pollutants and adverse neighborhood conditions are spatially correlated. Yet health risks associated with concurrent exposure to air pollution mixtures and clustered neighborhood factors remain underexplored. Statistical models to assess the health effects from pollutant mixtures remain limited, due to problems of collinearity between pollutants and area-level covariates, and increases in covariate dimensionality. Here we identify pollutant exposure profiles and neighborhood contextual profiles within Los Angeles (LA) County. We then relate these profiles with term low birth weight (TLBW). We used land use regression to estimate NO2, NO, and PM2.5 concentrations averaged over census block groups to generate pollutant exposure profile clusters and census block group-level contextual profile clusters, using a Bayesian profile regression method. Pollutant profile cluster risk estimation was implemented using a multilevel hierarchical model, adjusting for individual-level covariates, contextual profile cluster random effects, and modeling of spatially structured and unstructured residual error. Our analysis found 13 clusters of pollutant exposure profiles. Correlations between study pollutants varied widely across the 13 pollutant clusters. Pollutant clusters with elevated NO2, NO, and PM2.5 concentrations exhibited increased log odds of TLBW, and those with low PM2.5, NO2, and NO concentrations showed lower log odds of TLBW. The spatial patterning of pollutant cluster effects on TLBW, combined with between-pollutant correlations within pollutant clusters, imply that traffic-related primary pollutants influence pollutant cluster TLBW risks. Furthermore, contextual clusters with the greatest log odds of TLBW had more adverse neighborhood socioeconomic, demographic, and housing conditions. Our data indicate that, while the spatial patterning of high-risk multiple pollutant clusters largely overlaps with adverse contextual neighborhood cluster, both contribute to TLBW while controlling for the other.

Sampling from Dirichlet process mixture models with unknown concentration parameter: mixing issues in large data implementations.

We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter [Formula: see text]. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45-54, 2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169-186, 2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37-57, 2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on [Formula: see text]. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484-498, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples.

PReMiuM: An R Package for Profile Regression Mixture Models Using Dirichlet Processes.

PReMiuM is a recently developed R package for Bayesian clustering using a Dirichlet process mixture model. This model is an alternative to regression models, non-parametrically linking a response vector to covariate data through cluster membership (Molitor, Papathomas, Jerrett, and Richardson 2010). The package allows binary, categorical, count and continuous response, as well as continuous and discrete covariates. Additionally, predictions may be made for the response, and missing values for the covariates are handled. Several samplers and label switching moves are implemented along with diagnostic tools to assess convergence. A number of R functions for post-processing of the output are also provided. In addition to fitting mixtures, it may additionally be of interest to determine which covariates actively drive the mixture components. This is implemented in the package as variable selection.