Multiplicative factor analysis with a latent mixed model structure for air pollution exposure assessment

Abstract

A primary objective of current air pollution research is the assessment of health effects related to specific sources of air particles, or particulate matter (PM). Because most PM health studies do not observe the activity of the pollution sources directly, investigators must infer pollution source contributions based on a complex mixture of exposure. Methods such as source apportionment and multivariate receptor modeling use standard factor analytic techniques to estimate the source-specific contributions from a large number of observed chemical components. In the interest of a more flexible source apportionment, we propose a multiplicative factor analysis with a latent mixed model structure on the latent source contributions. A factor analysis with multiplicative errors serves to maintain the non-negativity of the measured chemical concentrations. A mixed model on the latent source contributions provides for systematic effects on source contributions as well as an adjustment for residual correlation in the source-specific exposures. In a simulation study, we examine the impact of (1) accounting for meteorological covariates and (2) adjusting for temporal correlation in the exposures on the estimation of the source profiles and the source contributions. Finally, we explore the influence of meteorological conditions on source-specific exposures in an analysis of PM exposure data from an animal toxicology study.

Major PM10 source location by a spatial multivariate receptor model

“…, p − 1. Finally, setting precision hyperparameters is not straightforward for the proposed model, as small precisions (large variances) induce skew and very informative lognormal priors (Nikolov et al 2010). Therefore, after some fine tuning, precisions were assumed to be distributed far from zero and with large variances:…”

“…In applying the same model (without the compositional profiles assumption) to PM exposure data, Nikolov et al (2010) constrain the elements of P considering the sufficient set of identifiability conditions given in Park et al (2002b) for model (1). Such conditions imply the a priori choice of a marker and q − 1 non-marker monitoring stations for each source, setting the corresponding elements of P to 1 and 0 respectively.…”

“…Following Nikolov et al (2010), Wolbers and Stahel (2005), Billheimer (2001) we consider multiplicative factor analysis with log-normal source contributions and lognormal errors…”

“…[0] Three sources, compositional spatially correlated source profiles, time correlated source contributions; [1] Two sources, compositional spatially correlated source profiles, time correlated source contributions; [2] Four sources, compositional spatially correlated source profiles, time correlated source contributions; Nikolov et al (2010) and is also the multiplicative log-normal version of the model in Park et al (2004). We assumed p k j iid ∼ N (−0.5, 0.588) in models [4] and [5] (according to Nikolov et al 2010) and t i iid ∼ N (0, 0.01) in model [6].…”

“…Explicit consideration of the spatial variability implied by the location of the p receptors is not addressed by the former studies and is one of the purposes of this paper, where we adopt a multivariate receptor model in the form of mixed multiplicative log-normal factor analysis (Billheimer 2001;Wolbers and Stahel 2005;Nikolov et al 2010) for the geographic identification of PM 10 pollution sources. A similar model was recently proposed by Lopes et al (2008) in the context of dynamic factor models, to identify the temporal components and summarize the spatial variation reducing the dimensionality of a general univariate space-time model.…”