We now present a detailed analysis of the motivating example from the Southern California CHS. To illustrate the methods proposed in this paper, we use data from the first 4th-grade cohort of schoolchildren who were recruited in fall of 1993 from the 12 communities depicted in Figure S1 provided in the

supplementary material (available at

*Biostatistics* online,

http://www.biostatistics.oxfordjournals.org). Lung function tests were conducted annually on all children. Depending on whether subjects were part of annual retest samples (about 10% were retested each year, roughly 3 months apart), up to 13 lung function test measurements were available. We required that each subject has at least 2 measurements.

The growth trajectories of childhood lung function measurements are different in boys and girls due to the relatively earlier growth spurts in girls. (left panel) illustrates the growth trajectories of FEV 1 for female (dashed curve) and male (solid curve) participants in the first 4th-grade cohort of the CHS. The growth trajectories clearly differ by gender, with girls having their growth spurt earlier and also achieving their maximal FEV 1 levels at the end of the observation period, whereas boys show an increasing trend even at the end of their high school years. Hence, lung function growth trajectories should always be fitted separately by gender even though joint inference is possible through a combined analysis.

Note that the time at which maximal lung function levels are achieved (and the corresponding levels) could be considered as additional nonlinear functionals for ecologic inference by identifying the region of the growth trajectory at which the rate of growth is almost 0. However, males will still be growing, and this may lead to unstable estimates of their maximum attained lung function. Because our purpose here is to illustrate the methodology via gender-specific and combined analysis, we restrict our analyses to the examination of the effect of air pollutants on the maximum rate of growth. For this, we need both the growth curve and its first derivative in order to examine rates of change. The right panel of Figure 2 depicts the gender-specific rates of growth for female (dashed curve) and male (solid curve) study participants.

The first-level mixed-effects model (3.9) was fitted via community-specific natural spline basis functions with interior knots at 12 and 14 years of age and boundary knots at 10 and 18 years of age. All models were fitted using log(FEV_{1}) to satisfy the normality assumption of the model. The results from this first-level model are summarized in for female and male participants. In addition to the spline-based terms for age and log(height), both models included several design and adjustment variables: race/ethnicity, respiratory infection within the month prior to the spirometry testing, prevalent and incident asthma status, body mass index (BMI, including a squared term to model the well-documented U-shaped relationship), physical exercise, personal smoking, field technician, and the spirometer involved in the test.

| **Table 1.**Posterior mean estimates and Bayesian credible intervals (BCI) for the level-1 model adjustment covariates on FEV_{1} in female and male participants of the CHS |

For female participants, there was a significant deficit in FEV_{1} levels () for those who had a respiratory illness within the month prior to the spirometry test (− 0.9% deficit). Black children showed a significant deficit in FEV_{1} levels (−11.3%) as did Asian children (−5.8%) compared to their Caucasian counterparts. Marginally significant associations were also observed with diagnosis of asthma at study entry (−2.6% deficit) and Hispanic ethnicity (2.6% elevated) compared to their non-Hispanic white counterparts. There was also a significant quadratic relationship with BMI, but no significant relationship was found with personal smoking or incident asthma during follow-up.

For male participants, the first-level model results presented in indicate significant deficits with black ethnicity (−14.2%) and Asian ethnicity (−2.4%) compared to White ethnicity, asthma status at entry (−4.3%), and respiratory infection within a month of spirometry test (−1.3%). There was also a significant quadratic relationship with BMI levels. The only differences from the results for females were the findings of significant positive association with personal smoking (0.9% increase in those who reported smoking) and the significant association with exercising (0.6% increase in those who reported exercising). The anomalous finding related to personal smoking may reflect associations with the overall health status of the subjects and other correlated factors.

Based on the above-described first-level model for females (), 40 000 posterior realizations of the community-specific growth curves were sampled as described in Section 3.3 after a burn-in period of 10 000 samples. Figure S2 in the

supplementary material, available at

*Biostatistics* online, gives the posterior mean growth curves, 95% Bayesian confidence limits, and a random sample of 100 posterior realizations (in gray). Note that the gray curves are representative Gibbs sampling realizations from the population growth curves, not subject-specific ones.

Similarly, Figure S3 in the

supplementary material, available at

*Biostatistics* online, gives the first derivative of the growth curves for females in order to estimate rates of growth. From the Gibbs sampling realizations of these derivative curves, we estimate the posterior distribution of the maximum rate of growth to be modeled against multiyear averages of air pollution in the ecologic regressions that follow. Figures S4 and S5 in the

supplementary material, available at

*Biostatistics* online, give the community-specific growth and associated derivative curves for male participants.

For ecologic inference, we were interested in the association between long-term levels of air pollution and the nonlinear functional which depicts the maximum rate of growth in FEV

_{1}. For simplicity, we focus on air pollution only, although it is possible to consider other ecologic covariates as discussed in

Berhane *and others* (2004). Results from fitting the ecologic meta-analytic regression described in Section 3.4 are given in , with focus on O

_{3}, elemental carbon, and acid vapors, after transforming the maximum posterior rates of growth (and their corresponding posterior variances) to their original scale. To examine whether effects of air pollution are different in females and males, tests for pollution-by-gender interactions were conducted in each of the 3 pollutants under consideration. These tests for interaction revealed that the effects of air pollution were similar and not statistically different in females and males (

*p* values for interaction were 0.21, 0.59, and 0.86 for O

_{3}, elemental carbon, and acid vapors, respectively). Hence, we present results for both genders combined. indicates that significant deficits in the maximum rate of growth of FEV

_{1} are associated with long-term levels of elemental carbon and acid vapors. For comparisons that were scaled to contrast over the range between least and most polluted communities, there were significant deficits in the maximum rate of growth of FEV

_{1} of 27.2 and 32.8 mL for 1.2 μ g of elemental carbon and 9.6 ppb of acid vapor, respectively. Though not significant, the contrast of 37.5 ppb in O

_{3} levels (based on average daily levels between 10 AM and 6 PM, where children are mostly outside) was associated with a deficit of 5.9 mL in the maximum growth rate of FEV

_{1}.

| **Table 2.**Effects of various pollutants on the maximum rate of FEV_{1} growth in participants of the CHS |

The results outlined above assume normality of the posterior distributions of the maximum rate of growth. To check whether this claim was supported by the data, tests for normality were conducted for each of the communities. No evidence of nonnormality was found for the marginal posterior distributions of the maximum attained levels of FEV_{1}, but the marginal posterior distributions of the maximum rate of growth were found to be normally distributed only after a log transformation. The second-stage meta-analytic models for the maximum rate of growth of FEV_{1} were refitted using the community-specific mean and variance values of the log-transformed marginal posterior realization of the maximum rate of growth. The results were almost identical with those reported in .

To assess whether convergence was reached in the Gibbs sampler, the Gelman and Rubin diagnostic (GRD) measures described in Section 3.2 were calculated based on 3 different sets of initial values, that is 0, −1, and the maximum likelihood estimates. Based on 10 000 burn-in iterations with 10 000 more iterations saved for analysis, the estimates for the GRD measures for the residual variance (σ) were found to be equal to 1.0 for both males and females. The corresponding values for the fixed-coefficient vector associated with the natural spline (δ) were found to be 1.03 and 1.04 for females and males, respectively. These results indicate that convergence was reached for all of our models.

As noted in Section 3.2, sensitivity analysis was also conducted to check whether our results were sensitive to our choice of variance component priors. Our substantive findings were found to be robust to the use of alternative choices of priors. For example, using a *uniform* (0,100) prior for σ instead of IG(0.01,0.01) led to significant deficits in the maximum rate of growth of FEV_{1} of 26.0 and 31.3 mL for 1.2 μ g of elemental carbon and 9.6 ppb of acid vapor, respectively, for comparisons that were scaled to contrast over the range between least and most polluted communities. The effect estimate for O_{3} was also essentially identical to that obtained under the IG(0.01,0.01) prior for σ.