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Controlling for confounding in the presence of measurement error in hierarchical models: a Bayesian approach

Journal of Exposure Science & Environmental Epidemiology volume 17, pages S20–S28 (2007)Cite this article

Abstract

A major concern in studies that address the health effects of air pollution is whether an observed association between concentrations of a pollutant and a health outcome is all, or in part, due to the correlation between that exposure and either a second pollutant or a confounder. The addition of exposure measurement error to such data complicates matters further. To account for measurement error when data come from a multi-city study, Schwartz and Coull (2003) proposed a two-stage estimator. These authors showed via both first principles and simulation that their approach yields unbiased estimates for the parameters of interest. However, these estimates have large variability. In this paper, we describe a fully Bayesian approach that yields estimators that are much more efficient than the existing two-stage measurement error correction yet still unbiased. The proposed approach can also incorporate additional exposures or confounders without requiring strict assumptions that are necessary in existing formulations of the model. We compare the properties of the Bayesian estimators to existing approaches via simulation.

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Acknowledgements

We thank Thomas Bateson and Jee-Young Kim from the U.S. EPA for the invitation to participate in the workshop on which this special issue is based. We also thank two referees and the Editor for helpful comments that improved the manuscript. This research was partially supported by NIEHS Grants ES012044, EPA Grant R832416, and American Chemistry Council Grant 2843. It has not been formally reviewed by the American Chemistry Council and the views expressed in this document are solely those of the authors.

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Authors and Affiliations

  1. Department of Biostatistics, Harvard School of Public Health, 655 Huntington Avenue, Boston, 02115, Massachusetts, USA

    Alexandros Gryparis & Brent A Coull

  2. Department of Environmental Health, Harvard School of Public Health, Landmark Center, Room 415, 401 Park Drive, Boston, 02115, Massachusetts, USA

    Joel Schwartz

Authors
  1. Alexandros Gryparis
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  2. Brent A Coull
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  3. Joel Schwartz
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Corresponding author

Correspondence to Alexandros Gryparis.

Appendix

Appendix

Here, we provide the WinBUGS code that one can use to fit a linear health effects model using two correlated exposures. The following code can be easily modified to accommodate different scenarios (e.g. adding more covariates with or without measurement error, categorical outcomes, etc.).

model

{

for (o in 1:n.all)

{

mean[o] <- beta0 + beta1*x[o,1] + beta2*x[o,2]

w[o,1] ∼ dnorm(x[o,1], tau.u1)

w[o,2] ∼ dnorm(x[o,2], tau.u2)

y[o] ∼ dnorm(mean[o],tau.y[city[o]])

w[o,1:2] ∼ dmnorm(mu.x[city[o],],

tau.all[(2*city[o] 1):(2*city[o]),])

}

for (i in 1:n.city)

{

mu.x[i,1] ∼ dnorm(0, 0.001)

mu.x[i,2] ∼ dnorm(0, 0.001)

oriz[i] <- sigma2.x1*sigma2.x2*(1-pow(rho[i],2))

tau.all[(2*i-1),1] <- sigma2.x2/oriz[i]

tau.all[(2*i),2] <- sigma2.x1/oriz[i]

diag.el[i] <- -pow(sigma2.x1,.5)*pow(sigma2.x2,.5)*

rho[i]/oriz[i]

tau.all[(2*i-1),2] <- diag.el[i]

tau.all[(2*i),1] <- diag.el[i]

rho[i] ∼ dunif(-.99,.99)

tau.y[i] ∼ dgamma(0.01,0.01)

}

beta0 ∼ dnorm(0,0.001)

beta1 ∼ dnorm(0,0.001)

beta2 ∼ dnorm(0,0.001)

tau.u1 ∼ dgamma(0.01,0.01)

tau.u2 ∼ dgamma(0.01,0.01)

tau.x1 ∼ dgamma(0.01,0.01)

tau.x2 ∼ dgamma(0.01,0.01)

sigma2.x1 <- 1/tau.x1

sigma2.x2 <- 1/tau.x2

}

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Gryparis, A., Coull, B. & Schwartz, J. Controlling for confounding in the presence of measurement error in hierarchical models: a Bayesian approach. J Expo Sci Environ Epidemiol 17 (Suppl 2), S20–S28 (2007). https://doi.org/10.1038/sj.jes.7500624

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  • DOI: https://doi.org/10.1038/sj.jes.7500624

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