Bayesian importance parameter modeling of misaligned predictors: soil metal measures related to residential history and intellectual disability in children

Abstract

In this paper, we propose a novel spatial importance parameter hierarchical logistic regression modeling approach that includes measurement error from misalignment. We apply this model to study the relationship between the estimated concentration of soil metals at the residence of mothers and the development of intellectual disability (ID) in their children. The data consist of monthly computerized claims data about the prenatal experience of pregnant women living in nine areas within South Carolina and insured by Medicaid during January 1, 1996 and December 31, 2001 and the outcome of ID in their children during early childhood. We excluded mother-child pairs if the mother moved to an unknown location during pregnancy. We identified an association of the ID outcome with arsenic (As) and mercury (Hg) concentration in soil during pregnancy, controlling for infant sex, maternal race, mother’s age, and gestational weeks at delivery. There is some indication that Hg has a slightly higher importance in the third and fourth months of pregnancy, while As has a more uniform effect over all the months with a suggestion of a slight increase in risk in later months.

Appendices

Appendix 1: Metropolis-Hastings algorithm

Recent development in computer facilities made Markov chain Monte Carlo (MCMC) (Trasande et al. 2006) simulation techniques one of the main computational tools in Bayesian statistical inference. For this study, we implemented the Metropolis version of the Metropolis-Hastings algorithm (U.S. Department of Health And Human Services 2012), which is one of the most popular MCMC methods. The algorithm consists of the following steps repeated T times, where T is the number of iterations of the chain. Our parameter θ consists of the true chemical values Z*(s _j,i ), the regression coefficients and their standard deviations:

${\beta}_0,{\sigma}_{\beta_0},{\beta}_1,{\sigma}_{\beta_1},\left({w}_1,{w}_2,{w}_3,{w}_4,{w}_5,{w}_6,{w}_7,{w}_8,{w}_9\right),{\beta}_{21},{\sigma}_{\beta_{21}},\dots {\beta}_{2m},\dots {\sigma}_{\beta_{2m}}$, m being the number of individual level covariates.

In order to enhance computational efficiency block updates have been used for updating the true chemical values. Symmetric proposal distributions have been used for all the parameters.

• Step 0 Assign starting values to θ ⁰.

  For t = 1,…,T :

• Step 1 Propose new values θ ′ from symmetric proposal distributions (h(θ ′ |θ)). In this case, we have used Gaussian proposals centered on the previous value for continuous parameters on (−∞, ∞)

• Step 2 Calculate log (α) = min (0, R) then, as the symmetric proposals cancel in the ratio,

  $$ R= \log \left\{\frac{L\left(\left.y\right|\theta^{\prime}\right)\prod {g}_i\left(\theta^{\prime}\right)}{L\left(\left.y\right|\theta \right)\prod {g}_i\left({\theta}_i\right)}\right\}, $$

  where L is the likelihood function and g is the prior distribution. The following prior distributions have been assigned:

  $$ {\beta}_l\sim N\left(0,{\sigma}_{\beta_l}^2\right),l=0,1, $$
  $$ {\sigma}_{\beta_l}\sim U\left(0,2\right),l=0,1 $$

  w _i ∼ Bernoulli (0.5), i = 1,…9

  $$ \log \left({Z}^{*}\left({s}_{j,i}\right)\right)\sim N\left( \log \left({Z}^0\left({s}_{j,i}\right)\right),{\sigma}_{\log \left({Z}^0\left({s}_{j,i}\right)\right)}^2\right),j=1,\ldots 9,i=1,\dots n $$

• Step 3 Update θ ^(t) = θ ′ with probability α

Appendix 2: simulation

To test the ability of the model to estimate importance parameters when correlated with the outcome, we used simulated data with different model assumptions. The simulation was performed with two scenarios, one in which month six values were simulated from a different distribution compared to the rest of the months, with a higher association with the outcome (Model 1). The second scenario was there was no variation in the months related to the chemical concentrations, and they were all generated from the same distributions (Model 2). Each dataset had 500 observations. We included only a limited number of scenarios due to difficulty in obtaining convergence for other variety of distribution assumptions and parameters. In Model 1, the outcome was generated from a Bernoulli (0.5) distribution. The chemical values for each month except month six were generated from a lognormal distribution with log mean −8 and log standard deviation 0.1 to which, if the outcome was 1 a random lognormal with mean −8 and log standard deviation 0.1 was added for each individual in each month except month 6. In order to induce a higher association with the outcome, a random lognormal with mean −4.2 and log standard deviation 0.1 was added for month six. In Model 2, the outcome was generated from a Bernoulli (0.5) distribution. The chemical values for each month were generated from a lognormal distribution with log mean −8 and log standard deviation 0.0001, to which a uniform U (0, 0.0001) random number was added to each month values for each subject. The kriged standard deviation was fixed at 0.001 for both models. We considered a low standard deviation in order to have less impact on the estimated values. Model 1 was run for 20,000 iterations, with the first 5,000 iterations discarded as burn-in. Model convergence was checked using the Geweke criteria. The highest importance parameter was obtained for the month 6 values (0.99), with lower importance parameters for the rest of the months. For model 1, the credible intervals for the importance parameters were (0, 1) except for month 6 which was (1, 1). These results suggest a good capability of the model to identify the months that are correlated with the outcome. The mean intercept and 95 % credible interval was estimated to be 0.047 (−0.11, 0.21), while the common mean slope and 95 % credible interval was 2.57 (2.54, 2.61). Model 2 was run for 20,000 iterations, 5,000 burn-in and a thinning of 3. Similar importance parameters have been estimated for each month (Table 5). For Model 2 the credible intervals for the importance parameters were (0, 1).

Table 5 Simulation results