Air: Berkson measurement error

Whittemore and Keller (1988) use an approximate maximum likelihood approach to analyse the data shown below on reported respiratory illness versus exposure to nitrogen dioxide (NO₂) in 103 children. Stephens and Dellaportas (1992) later use Bayesian methods to analyse the same data.

A discrete covariate z_j (j = 1,2,3) representing NO₂ concentration in the child's bedroom classified into 3 categories is used as a surrogate for true exposure. The nature of the measurement error relationship associated with this covariate is known precisely via a calibration study, and is given by
   x_j = α + β z_j + ε_j

where α = 4.48, β = 0.76 and ε_j is a random element having normal distribution with zero mean and variance σ^{2 (}= 1/τ⁾ = 81.14. Note that this is a Berkson (1950) model of measurement error, in which the true values of the covariate are expressed as a function of the observed values. Hence the measurement error is independent of the latter, but is correlated with the true underlying covariate values. In the present example, the observed covariate z_j takes values 10, 30 or 50 for j = 1, 2, or 3 respectively (i.e. the mid-point of each category), whilst x_j is interpreted as the "true average value" of NO₂ in group j. The response variable is binary, reflecting presence/absence of respiratory illness, and a logistic regression model is assumed. That is

   y_j ~ Binomial(p_j, n_j)
logit(p_j) = θ₁ + θ₂ x_j
where p_j is the probability of respiratory illness for children in the jth exposure group. The regression coefficients θ₁ and θ₂ are given vague independent normal priors. The graphical model is shown below:
   model
   {
      for(j in 1 : J) {
         y[j] ~ dbin(p[j], n[j])
         logit(p[j]) <- theta[1] + theta[2] * X[j]
         X[j] ~ dnorm(mu[j], tau)
         mu[j] <- alpha + beta * Z[j]
      }
      theta[1] ~ dnorm(0.0, 0.001)
      theta[2] ~ dnorm(0.0, 0.001)
   }

   Results