NEHA October 2022 Journal of Environmental Health

October 2022 • Journal of Environmental Health 9 5–9 µg/dL/children tested for BLL); m is the number of children <6 years who were tested for BLL; m . θ is the number of children with BLLs of 5–9 µg/dL; and p(z/θ) is the probability that there are z number of children <6 years with BLLs of 5–9 µg/dL under the assumption that θ is the rate of children with BLLs of 5–9 µg/dL. Clearly, θ is unknown or a parameter, and under the Bayesian principle, one tries to estimate it based on a reasonable assumption of its statistical distribution, called “prior distribution” or simply “prior.” It is reasonable to assume that a parameter coming from a Poisson distribution should follow a statistical distribution called gamma distribution. Thus, this model assumes that θ follows a gamma (α, β) prior: p(θ) = e–(β θ) βα θα-1/Γ(α) (2) Where θ > 0, and α and β are its unknowns or parameters. Then, according to Bayesian rule, actual or simply put, posterior distribution, p(θ/z) of θ, will be given by p(θ/z) = p(z/θ) × p(θ)/p(z), which is the distribution of the observed number multiplied by the prior of its parameter divided by the constant p(z). That is: p(θ/z) = e–(m . θ)(m . θ)z × e–(β θ)βα θα-1/z! Γ(α) p(z) (2a) or, p(θ/z) = e– θ(β + m) (θ)z+α-1 × constant (3) Here, the right-hand side of Equation 2 and that of the posterior distribution in Equation 3 are similar, which indicates that the posterior is also a gamma (α1, β1) distribution with parameters α1 and β1 where: α1 = z + α and β1 = β + m (3a) This equation means that if one assumes that the prior information about parameter θ (the rate of children with BLLs of 5-9 µg/ dL) can be obtained from a small group of counties in Georgia, each of which is believed to have the same rate (θ) of 5–9 µg/dL BLLs among children <6 years, then applying Bayesian rule, the posterior for θ can be estimated from a gamma distribution as shown in Equation 3. Moreover, if one supposes zj is the number of children <6 years with BLLs of 5–9 µg/ dL among xj children from county j, then, assuming zj follows a Poisson distribution, one would have, as in Equation 1: p(zj/θ) = e –(xj θ)(x jθ) zj/z j! (4) Where θis the same as defined earlier. Thus, the likelihood function for n counties with the same parameter θ is given as follows: L(∑zj/θ) = e –(∑xj θ)∏(x jθ) zj/z 1! z2 ! ….. zn! (5) This equation is obtained by multiplying density functions like Equation 4 for n counties. Omitting the constant terms, one has: L(∑zj/θ) ∝ e –(∑xj θ)(θ)∑zj (6) Where ∝ indicates proportionality. If for all these n counties, one assumes that θ follows a noninformative prior 1/θ (i.e., p(θ) = 1/θ), then as was done in Equation 2a and from Equation 6, the posterior distribution of θis given by the following: p(θ/∑zj) ∝ e –(∑xjθ)(θ)∑zj . 1/θ (i.e., p(θ/∑zj) ∝ e –(∑xjθ)(θ)∑zj-1) (7) This is a gamma (α2, β2), where: α2 = ∑zj and β2 = ∑xj (8) Here, ∑zj is the shape parameter and ∑xj is the rate parameter of this gamma distribution, where zj is the number of children <6 years with BLLs of 5–9 µg/dL in county j and xj is the number of children tested for BLL in county j. The assumption is that the rate of children with BLLs of 5–9 µg/dL among children <6 years in these counties is similar to that in a targeted county where one wants to estimate that rate. One can then use known α and β from Equation 8 in Equations 2 and 3 to evaluate the prior and posterior distributions of the parameter θin the targeted county. According to the multiplication rule of probability, the joint distribution of data z and the parameter θare given by the following: p(z,θ) = p(θ) × p(z/θ), and also p(z,θ) = p(z) × p(θ/z) Thus, p(z) × p(θ/z) = p(θ) × p(z/θ), giving: p(z) = p(θ) × p(z/θ)/p(θ/z) (9) Here, p(θ) and p(θ/z) are the known prior and posterior distributions, respectively, of the parameter θ. Thus, p(θ) is a gamma density with the known shape and rate parameters from Equation 8. Similarly, p(θ/z) is a gamma density with known shape and rate parameters from Equations 8 and 3a. Assuming that p(z/θ) is the sampling distribution of data in the targeted county, one can estimate the predictive density p(z) of z in the targeted county from Equation 9 before any data are observed, where p(z/θ) is a Poisson density with known mean (mθ) as shown in Equation 1. If our model assumptions for sampling distribution of data and prior density are valid, one can check the validity of the observed values of the number of children <6 years with BLLs of 5–9 µg/dL in the targeted county. Detailed information about this Bayesian model can be found at www.neha.org/jeh/ supplemental. County and Region Selection The model was applied by dividing Georgia into five di¨erent regions: North, South, East, West, and Central. Then 11 neighboring counties were arbitrarily selected in each region, assuming similarity of BLL rates of 5–9 µg/dL among children ages <6 years in these counties. For each region, the county with the lowest observed proportion of children with BLLs 5–9 µg/dL was selected as the targeted county. The remaining 10 counties from each region provided data for estimation of parameters α and β for the prior distribution. The parameter θ, the rate of children with BLLs of 5–9 µg/dL in the targeted county, was estimated from the mean value α/β of the gamma distribution, as the predictive density (Equation 9) is valid for all θ. Data Analysis Data were analyzed using statistical software SAS (version 9.4) and R package. For each region, predictive density was calculated for the targeted county from Equation 9 for all children, and separately for White and non-White children. We assumed that the observed value for the number of children with BLLs of 5–9 µg/dL among children <6 years within the three largest predictive probabilities was compatible. Additionally, the mean number of children with BLLs of 5–9 µg/dL was estimated in the

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