title: Addressing extrema and censoring in pollutant and exposure data using mixture of normal distributions.
abstract: BACKGROUND
  Volatile organic compounds (VOC), which include many hazardous chemicals, have been used extensively in personal, commercial and industrial products. Due to the variation in source emissions, differences in the settings and environmental conditions where exposures occur, and measurement issues, distributions of VOC concentrations can have multiple modes, heavy tails, and significant portions of data below the method detection limit (MDL). These issues challenge standard parametric distribution models needed to estimate the exposures, even after log-transformation of the data.
  
  
  METHODS
  This paper considers mixture of distributions that can be directly applied to concentration and exposure data. Two types of mixture distributions are considered: the traditional finite mixture of normal distributions, and a semi-parametric Dirichlet process mixture (DPM) of normal distributions. Both methods are implemented for a sample data set obtained from the Relationship between Indoor, Outdoor and Personal Air (RIOPA) study. Performance is assessed based on goodness-of-fit criteria that compare the closeness of the density estimates with the empirical density based on data. The goodness-of-fit for the proposed density estimation methods are evaluated by a comprehensive simulation study.
  
  
  RESULTS
  The finite mixture of normals and DPM of normals have superior performance when compared to the single normal distribution fitted to log-transformed exposure data. The advantages of using these mixture distributions are more pronounced when exposure data have heavy tails or a large fraction of data below the MDL. Distributions from the DPM provided slightly better fits than the finite mixture of normals. Additionally, the DPM method avoids certain convergence issues associated with the finite mixture of normals, and adaptively selects the number of components.
  
  
  CONCLUSIONS
  Compared to the finite mixture of normals, DPM of normals has advantages by characterizing uncertainty around the number of components, and by providing a formal assessment of uncertainty for all model parameters through the posterior distribution. The method adapts to a spectrum of departures from standard model assumptions and provides robust estimates of the exposure density even under censoring due to MDL.
authors.affiliation: Department of Biostatistics, School of Public Health, University of Michigan, Ann Arbor, MI, USA.
authors.forename: Shi
authors.initials: S
authors.lastname: Li
authors.affiliation: Department of Environmental Health Sciences, School of Public Health, University of Michigan, Ann Arbor, MI, USA.
authors.forename: Stuart
authors.initials: S
authors.lastname: Batterman
authors.affiliation: Department of Environmental Health Sciences, School of Public Health, University of Michigan, Ann Arbor, MI, USA.
authors.forename: Feng-Chiao
authors.initials: FC
authors.lastname: Su
authors.affiliation: Department of Biostatistics, School of Public Health, University of Michigan, Ann Arbor, MI, USA.
authors.forename: Bhramar
authors.initials: B
authors.lastname: Mukherjee
keywords: Air pollution; Dirichlet process mixture; density estimation; limits of detection; mixture of normal; volatile organic compounds