Biometrics 59, 1–3
DOI: 000
December 2013
Multilevel Quantile Function Modeling with Application to Birth Outcomes
Luke B. Smith1,∗ , Montserrat Fuentes1 , Brian J. Reich 1 , Amy H. Herring2 , and Peter H. Langlois3
1
Department of Statistics, North Carolina State University, Raleigh, North Carolina, 27695-8203, U.S.A.
2
Department of Biostatistics, University of North Carolina at Chapel Hill,
Chapel Hill, North Carolina, 27599-7420, U.S.A.
3
Texas Department of State Health Services, Austin, Texas 78714-9347, U.S.A.
*email: luke [email protected]
Summary:
Infants born preterm or small for gestational age have elevated rates of morbidity and mortality. Using
birth certificate records in Texas from 2002-2004 and Environmental Protection Agency air pollution estimates, we
relate the quantile functions of birth weight and gestational age to ozone exposure and multiple predictors, including
parental age, race, and education level. We introduce a semi-parametric Bayesian quantile approach that models
the full quantile function rather than just a few quantile levels. Our multilevel quantile function model establishes
relationships between birth weight and the predictors separately for each week of gestational age and between
gestational age and the predictors separately across Texas Public Health Regions. We permit these relationships to
vary nonlinearly across gestational age, spatial domain and quantile level and we unite them in a hierarchical model
via a basis expansion on the regression coefficients that preserves interpretability. Very low birth weight is a primary
concern, so we leverage extreme value theory to supplement our model in the tail of the distribution. Gestational
ages are rounded into weekly values, so we present methodology for modeling quantile functions of discrete response
data. In a simulation study we show that pooling information across gestational age and quantile level substantially
reduces MSE of predictor effects relative to standard frequentist quantile regression. We find that ozone is negatively
associated with the lower tail of gestational age in south Texas and across the distribution of birth weight for high
gestational ages. Our methods are available in the R package BSquare.
Key words:
Birth weight; Discrete; Extremes; Gestational Age; Graphics processing units; Ozone; Quantile.
This paper has been submitted for consideration for publication in Biometrics
1
Supplementary Materials
[Figure 1 about here.]
[Figure 2 about here.]
[Figure 3 about here.]
[Figure 4 about here.]
[Figure 5 about here.]
[Figure 6 about here.]
[Figure 7 about here.]
[Figure 8 about here.]
[Figure 9 about here.]
[Figure 10 about here.]
[Figure 11 about here.]
[Figure 12 about here.]
[Figure 13 about here.]
[Figure 14 about here.]
[Figure 15 about here.]
[Figure 16 about here.]
[Figure 17 about here.]
[Figure 18 about here.]
[Figure 19 about here.]
[Figure 20 about here.]
2
Biometrics, December 2013
[Figure 21 about here.]
[Figure 22 about here.]
[Figure 23 about here.]
[Figure 24 about here.]
[Figure 25 about here.]
[Figure 26 about here.]
[Figure 27 about here.]
[Figure 28 about here.]
[Figure 29 about here.]
[Figure 30 about here.]
[Figure 31 about here.]
[Figure 32 about here.]
[Figure 33 about here.]
[Figure 34 about here.]
[Figure 35 about here.]
[Figure 36 about here.]
[Figure 37 about here.]
[Figure 38 about here.]
[Figure 39 about here.]
[Figure 40 about here.]
[Figure 41 about here.]
3
[Figure 42 about here.]
[Figure 43 about here.]
Received October 2013.
Appendix MCMC
Posterior distributions of θmjg were updated singly via random walk Metropolis using Gaussian candidate distributions with variances tuned to have acceptance probability between 30
and 40%. Posterior distributions of the autocorrelation parameters ρmj and range parameters
ϕmj were updated using independent Metropolis updates. The basis function means µmj were
given Gaussian priors with mean µ0 and precision δ0 . The basis function precisions γmj were
given Gamma priors with shape parameter a and scale parameters b. These means and
precisions have conjugate posteriors and were updated using Gibbs sampling.
(
)
γmj 1T Σ−1
θ
mj
mj
−1
µmj |rest ∼ N
, (γmj 1T Σ−1
mj 1)
γmj 1T Σ−1
1
+
δ
0
mj
−1
γmj |rest ∼ Gamma(G/2 + a, (0.5(θmj − 1µmj )Σ−1
)
mj (θmj − 1µmj ) + 1/b)
where θmj = θmj1 , ..., θmjG .
Biometrics, December 2013
4
n = 200
0.95
0.85
0.80
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
Quantile Level
Quantile Level
n = 400
n = 400
0.8
1.0
0.8
1.0
5
1.00
0.0
0.95
0.90
0.80
2
0.85
3
Coverage
4
Pareto CP
Exp CP
Pareto IP
Exp IP
Freq
1
Freq/Bayes MSE Ratio
0.90
Coverage
4
3
2
1
Freq/Bayes MSE Ratio
5
1.00
n = 200
0.0
0.2
0.4
0.6
Quantile Level
0.8
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 1. MSE ratios of quantile function estimator over frequentist quantile estimator
and coverage probabilities for beta quantile functions. The maximum Monte Carlo standard
error of MSE was 0.2 for Bayesian estimators and 0.3 for the frequentist estimator. The
maximum Monte Carlo standard error for the MSE ratio was 0.09.
5
0.2
0.4
0.6
0.8
1.0
42
38
30
34
38
30
34
38
34
Effect (weeks)
30
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
42
30
34
38
42
34
0.8
1.0
PHR 9
30
0.6
0.8
42
0.4
38
42
38
34
0.4
0.6
34
0.2
PHR 8
30
0.2
0.4
30
0.0
PHR 7
0.0
0.2
38
42
38
30
0.2
0.0
PHR 6
34
38
34
Effect (weeks)
30
0.0
Effect (weeks)
0.4
PHR 5
42
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
38
34
30
30
34
38
42
PHR 11
42
PHR 10
Effect (weeks)
PHR 3
42
PHR 2
42
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 2. 95% credible limits for the posterior distribution of the intercept process for
gestational age by Public Health Region (PHR). The intercept process represents the
information regarding gestational age not explained by the predictors and was permitted
to vary by PHR. The intercept process is not interpretable, as we had binary variables that
were valued at either -1 or 1.
Biometrics, December 2013
6
0.2
0.4
0.6
0.8
1.0
0.0
−0.6
−0.4
−0.2
0.0
−0.2
−0.4
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.0
−0.6
−0.4
−0.2
0.0
−0.4
0.8
1.0
PHR 9
−0.6
0.6
0.8
0.0
0.4
−0.2
0.0
−0.2
−0.4
0.4
0.6
−0.4
0.2
PHR 8
−0.6
0.2
0.4
−0.6
0.0
PHR 7
0.0
0.2
−0.2
0.0
−0.4
−0.6
0.2
0.0
PHR 6
−0.2
0.0
−0.2
−0.4
0.0
Effect (weeks)
0.4
PHR 5
−0.6
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
−0.2
−0.4
−0.6
−0.6
−0.4
−0.2
0.0
PHR 11
0.0
PHR 10
Effect (weeks)
PHR 3
−0.6
−0.4
−0.2
0.0
PHR 2
−0.6
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 3. 95% credible limits for the posterior distribution of the effect of being male vs.
female on gestational age by Public Health Region (PHR).
7
0.2
0.4
0.6
0.8
1.0
1.5
−0.5
0.5
1.5
0.5
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.8
1.0
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
1.5
−0.5
0.5
1.5
−0.5
0.6
0.8
PHR 9
0.5
1.5
0.5
0.4
0.6
1.5
0.2
PHR 8
−0.5
0.2
0.4
−0.5
0.0
PHR 7
0.0
0.2
0.5
1.5
−0.5
0.2
0.0
PHR 6
0.5
1.5
0.5
0.0
Effect (weeks)
0.4
PHR 5
−0.5
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.5
−0.5
−0.5
0.5
1.5
PHR 11
1.5
PHR 10
Effect (weeks)
PHR 3
−0.5
0.5
1.5
PHR 2
−0.5
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 4. 95% credible limits for the posterior distribution of the effect of maternal parity
on gestational age by Public Health Region (PHR).
Biometrics, December 2013
8
0.2
0.4
0.6
0.8
1.0
0.0
−3.0
−2.0
−1.0
0.0
−1.0
−2.0
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.0
−3.0
−2.0
−1.0
0.0
−2.0
0.8
1.0
PHR 9
−3.0
0.6
0.8
0.0
0.4
−1.0
0.0
−1.0
−2.0
0.4
0.6
−2.0
0.2
PHR 8
−3.0
0.2
0.4
−3.0
0.0
PHR 7
0.0
0.2
−1.0
0.0
−2.0
−3.0
0.2
0.0
PHR 6
−1.0
0.0
−1.0
−2.0
0.0
Effect (weeks)
0.4
PHR 5
−3.0
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
−1.0
−2.0
−3.0
−3.0
−2.0
−1.0
0.0
PHR 11
0.0
PHR 10
Effect (weeks)
PHR 3
−3.0
−2.0
−1.0
0.0
PHR 2
−3.0
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 5. 95% credible limits for the posterior distribution of the effect of maternal age
40 and above on gestational age by Public Health Region (PHR).
9
0.2
0.4
0.6
0.8
1.0
0.2
−0.2
−0.6
−0.2
−0.6
−0.2
Effect (weeks)
−0.6
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.8
1.0
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.2
−0.6
−0.2
0.2
−0.6
0.6
0.8
PHR 9
−0.2
0.2
−0.2
0.4
0.6
0.2
0.2
PHR 8
−0.6
0.2
0.4
−0.6
0.0
PHR 7
0.0
0.2
−0.2
0.2
−0.2
0.2
0.0
PHR 6
−0.6
−0.2
Effect (weeks)
−0.6
0.0
Effect (weeks)
0.4
PHR 5
0.2
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
−0.2
−0.6
−0.6
−0.2
0.2
PHR 11
0.2
PHR 10
Effect (weeks)
PHR 3
0.2
PHR 2
0.2
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 6. 95% credible limits for the posterior distribution of the effect of paternal age 40
and above on gestational age by Public Health Region (PHR).
Biometrics, December 2013
10
0.2
0.4
0.6
0.8
1.0
0.3
−0.3
−0.1
0.1
0.3
0.1
−0.1
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.3
−0.3
−0.1
0.1
0.3
−0.1
0.8
1.0
PHR 9
−0.3
0.6
0.8
0.3
0.4
0.1
0.3
0.1
−0.1
0.4
0.6
−0.1
0.2
PHR 8
−0.3
0.2
0.4
−0.3
0.0
PHR 7
0.0
0.2
0.1
0.3
−0.1
−0.3
0.2
0.0
PHR 6
0.1
0.3
0.1
−0.1
0.0
Effect (weeks)
0.4
PHR 5
−0.3
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.1
−0.1
−0.3
−0.3
−0.1
0.1
0.3
PHR 11
0.3
PHR 10
Effect (weeks)
PHR 3
−0.3
−0.1
0.1
0.3
PHR 2
−0.3
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 7. 95% credible limits for the posterior distribution of the effect of the mother
finishing high school relative to not finishing high school on gestational age by Public Health
Region (PHR).
11
0.2
0.4
0.6
0.8
1.0
1.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.5
Effect (weeks)
0.0
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
1.5
0.0
0.5
1.0
1.5
0.5
0.8
1.0
PHR 9
0.0
0.6
0.8
1.5
0.4
1.0
1.5
1.0
0.5
0.4
0.6
0.5
0.2
PHR 8
0.0
0.2
0.4
0.0
0.0
PHR 7
0.0
0.2
1.0
1.5
1.0
0.0
0.2
0.0
PHR 6
0.5
1.0
0.5
Effect (weeks)
0.0
0.0
Effect (weeks)
0.4
PHR 5
1.5
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
1.0
0.5
0.0
0.0
0.5
1.0
1.5
PHR 11
1.5
PHR 10
Effect (weeks)
PHR 3
1.5
PHR 2
1.5
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 8. 95% credible limits for the posterior distribution of the effect of the mother
finishing education above high school relative to not finishing high school on gestational age
by Public Health Region (PHR).
Biometrics, December 2013
12
0.2
0.4
0.6
0.8
1.0
0.4
−0.2
0.0
0.2
0.4
0.2
0.0
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.4
−0.2
0.0
0.2
0.4
0.0
0.8
1.0
PHR 9
−0.2
0.6
0.8
0.4
0.4
0.2
0.4
0.2
0.0
0.4
0.6
0.0
0.2
PHR 8
−0.2
0.2
0.4
−0.2
0.0
PHR 7
0.0
0.2
0.2
0.4
0.0
−0.2
0.2
0.0
PHR 6
0.2
0.4
0.2
0.0
0.0
Effect (weeks)
0.4
PHR 5
−0.2
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.2
0.0
−0.2
−0.2
0.0
0.2
0.4
PHR 11
0.4
PHR 10
Effect (weeks)
PHR 3
−0.2
0.0
0.2
0.4
PHR 2
−0.2
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 9. 95% credible limits for the posterior distribution of the effect of the father
finishing high school relative to not finishing high school on gestational age by Public Health
Region (PHR).
13
0.2
0.4
0.6
0.8
1.0
0.6
−0.2
0.2
0.6
0.2
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.8
1.0
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.6
−0.2
0.2
0.6
−0.2
0.6
0.8
PHR 9
0.2
0.6
0.2
0.4
0.6
0.6
0.2
PHR 8
−0.2
0.2
0.4
−0.2
0.0
PHR 7
0.0
0.2
0.2
0.6
−0.2
0.2
0.0
PHR 6
0.2
0.6
0.2
0.0
Effect (weeks)
0.4
PHR 5
−0.2
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.2
−0.2
−0.2
0.2
0.6
PHR 11
0.6
PHR 10
Effect (weeks)
PHR 3
−0.2
0.2
0.6
PHR 2
−0.2
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 10. 95% credible limits for the posterior distribution of the effect of the father
finishing education above high school relative to not finishing high school on gestational age
by Public Health Region (PHR).
Biometrics, December 2013
14
0.2
0.4
0.6
0.8
1.0
0.0
−3.0
−2.0
−1.0
0.0
−1.0
−2.0
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.0
−3.0
−2.0
−1.0
0.0
−2.0
0.8
1.0
PHR 9
−3.0
0.6
0.8
0.0
0.4
−1.0
0.0
−1.0
−2.0
0.4
0.6
−2.0
0.2
PHR 8
−3.0
0.2
0.4
−3.0
0.0
PHR 7
0.0
0.2
−1.0
0.0
−2.0
−3.0
0.2
0.0
PHR 6
−1.0
0.0
−1.0
−2.0
0.0
Effect (weeks)
0.4
PHR 5
−3.0
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
−1.0
−2.0
−3.0
−3.0
−2.0
−1.0
0.0
PHR 11
0.0
PHR 10
Effect (weeks)
PHR 3
−3.0
−2.0
−1.0
0.0
PHR 2
−3.0
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 11. 95% credible limits for the posterior distribution of the effect of black nonHispanic maternal ethnicity relative to white non-Hispanic maternal ethnicity on gestational
age by Public Health Region (PHR).
15
0.2
0.4
0.6
0.8
1.0
1.0
0.5
−0.5
0.0
0.5
−0.5
0.0
0.5
0.0
Effect (weeks)
−0.5
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
1.0
−0.5
0.0
0.5
1.0
0.0
0.8
1.0
PHR 9
−0.5
0.6
0.8
1.0
0.4
0.5
1.0
0.5
0.0
0.4
0.6
0.0
0.2
PHR 8
−0.5
0.2
0.4
−0.5
0.0
PHR 7
0.0
0.2
0.5
1.0
0.5
−0.5
0.2
0.0
PHR 6
0.0
0.5
0.0
Effect (weeks)
−0.5
0.0
Effect (weeks)
0.4
PHR 5
1.0
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.5
0.0
−0.5
−0.5
0.0
0.5
1.0
PHR 11
1.0
PHR 10
Effect (weeks)
PHR 3
1.0
PHR 2
1.0
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 12. 95% credible limits for the posterior distribution of the effect of Hispanic
maternal ethnicity relative to white non-Hispanic maternal ethnicity on gestational age by
Public Health Region (PHR).
Biometrics, December 2013
16
0.2
0.4
0.6
0.8
1.0
0.8
−0.4
0.0
0.4
0.8
0.4
0.0
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.8
−0.4
0.0
0.4
0.8
0.0
0.8
1.0
PHR 9
−0.4
0.6
0.8
0.8
0.4
0.4
0.8
0.4
0.0
0.4
0.6
0.0
0.2
PHR 8
−0.4
0.2
0.4
−0.4
0.0
PHR 7
0.0
0.2
0.4
0.8
0.0
−0.4
0.2
0.0
PHR 6
0.4
0.8
0.4
0.0
0.0
Effect (weeks)
0.4
PHR 5
−0.4
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.4
0.0
−0.4
−0.4
0.0
0.4
0.8
PHR 11
0.8
PHR 10
Effect (weeks)
PHR 3
−0.4
0.0
0.4
0.8
PHR 2
−0.4
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 13. 95% credible limits for the posterior distribution of the effect of other maternal
ethnicity relative to white non-Hispanic maternal ethnicity on gestational age by Public
Health Region (PHR).
17
0.2
0.4
0.6
0.8
1.0
0.1
−0.5
−0.3
−0.1
0.1
−0.1
−0.3
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.2
1.0
0.4
0.6
0.8
1.0
0.8
1.0
0.1
−0.5
−0.3
−0.1
0.1
−0.3
0.8
1.0
PHR 9
−0.5
0.6
0.8
0.1
0.4
−0.1
0.1
−0.1
−0.3
0.4
0.6
−0.3
0.2
PHR 8
−0.5
0.2
0.4
−0.5
0.0
PHR 7
0.0
0.2
−0.1
0.1
−0.3
−0.5
0.2
0.0
PHR 6
−0.1
0.1
−0.1
−0.3
0.0
Effect (weeks)
0.4
PHR 5
−0.5
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
−0.1
−0.3
−0.5
−0.5
−0.3
−0.1
0.1
PHR 11
0.1
PHR 10
Effect (weeks)
PHR 3
−0.5
−0.3
−0.1
0.1
PHR 2
−0.5
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 14. 95% credible limits for the posterior distribution of the effect of a one-unit
increase in first trimester ozone exposure on gestational age by Public Health Region (PHR).
All ozone values were linearly transformed into [-1,1], so a one-unit increase can be roughly
thought of as an increase from low levels to middle levels of exposure, or middle to high.
Biometrics, December 2013
18
0.2
0.4
0.6
0.8
1.0
0.2
−0.4
0.0
0.2
0.0
0.0
0.0
0.2
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
0.8
1.0
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.2
−0.4
0.0
0.2
−0.4
0.6
0.8
PHR 9
0.0
0.2
0.0
0.4
0.6
0.2
0.2
PHR 8
−0.4
0.2
0.4
−0.4
0.0
PHR 7
0.0
0.2
0.0
0.2
−0.4
0.2
0.0
PHR 6
0.0
0.2
0.0
0.0
Effect (weeks)
0.4
PHR 5
−0.4
Effect (weeks)
PHR 4
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.2
0.4
0.6
0.0
−0.4
−0.4
0.0
0.2
PHR 11
0.2
PHR 10
Effect (weeks)
PHR 3
−0.4
0.0
0.2
PHR 2
−0.4
Effect (weeks)
PHR 1
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
Quantile Level
Figure 15. 95% credible limits for the posterior distribution of the effect of a one-unit
increase in second trimester ozone exposure on gestational age by Public Health Region
(PHR). All ozone values were linearly transformed into [-1,1], so a one-unit increase can be
roughly thought of as an increase from low levels to middle levels of exposure, or middle to
high.
19
0.2
0.4
0.6
0.8
1.0
4000
3000
1000
2000
3000
1000
2000
3000
2000
Effect (grams)
1000
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
4000
1000
2000
3000
4000
2000
0.8
1.0
33 weeks of GA
1000
0.6
0.8
4000
0.4
3000
4000
3000
2000
0.4
Quantile Level
0.6
2000
0.2
32 weeks of GA
1000
0.2
0.4
1000
0.0
31 weeks of GA
0.0
0.2
3000
4000
3000
1000
0.0
0.0
30 weeks of GA
2000
3000
2000
1000
Effect (grams)
0.2
29 weeks of GA
4000
28 weeks of GA
Effect (grams)
27 weeks of GA
4000
26 weeks of GA
4000
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 16. 95% credible limits for the posterior distribution of the intercept of birth weight
for gestational ages 25-33 weeks.
Biometrics, December 2013
20
0.2
0.4
0.6
0.8
1.0
4500
3500
1500
2500
3500
1500
2500
3500
2500
Effect (grams)
1500
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
4500
1500
2500
3500
4500
2500
0.8
1.0
42 weeks of GA
1500
0.6
0.8
4500
0.4
3500
4500
3500
2500
0.4
Quantile Level
0.6
2500
0.2
41 weeks of GA
1500
0.2
0.4
1500
0.0
40 weeks of GA
0.0
0.2
3500
4500
3500
1500
0.0
0.0
39 weeks of GA
2500
3500
2500
1500
Effect (grams)
0.2
38 weeks of GA
4500
37 weeks of GA
Effect (grams)
36 weeks of GA
4500
35 weeks of GA
4500
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 17. 95% credible limits for the posterior distribution of the intercept of birth weight
for gestational ages 34-42 weeks.
21
0.2
0.4
0.6
0.8
1.0
150
100
0
50
100
0
50
100
50
Effect (grams)
0
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
150
0
50
100
150
50
0.8
1.0
33 weeks of GA
0
0.6
0.8
150
0.4
100
150
100
50
0.4
Quantile Level
0.6
50
0.2
32 weeks of GA
0
0.2
0.4
0
0.0
31 weeks of GA
0.0
0.2
100
150
100
0
0.0
0.0
30 weeks of GA
50
100
50
0
Effect (grams)
0.2
29 weeks of GA
150
28 weeks of GA
Effect (grams)
27 weeks of GA
150
26 weeks of GA
150
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 18. 95% credible limits for the posterior distribution of the effect of male sex
relative to female on birth weight for gestational ages 25-33 weeks.
Biometrics, December 2013
22
0.2
0.4
0.6
0.8
1.0
200
150
50
100
150
50
100
150
100
Effect (grams)
50
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
200
50
100
150
200
100
0.8
1.0
42 weeks of GA
50
0.6
0.8
200
0.4
150
200
150
100
0.4
Quantile Level
0.6
100
0.2
41 weeks of GA
50
0.2
0.4
50
0.0
40 weeks of GA
0.0
0.2
150
200
150
50
0.0
0.0
39 weeks of GA
100
150
100
50
Effect (grams)
0.2
38 weeks of GA
200
37 weeks of GA
Effect (grams)
36 weeks of GA
200
35 weeks of GA
200
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 19. 95% credible limits for the posterior distribution of the effect of male sex
relative to female on birth weight for gestational ages 34-42 weeks.
23
0.2
0.4
0.6
0.8
1.0
300
200
0
100
200
0
100
200
100
Effect (grams)
0
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
300
0
100
200
300
100
0.8
1.0
33 weeks of GA
0
0.6
0.8
300
0.4
200
300
200
100
0.4
Quantile Level
0.6
100
0.2
32 weeks of GA
0
0.2
0.4
0
0.0
31 weeks of GA
0.0
0.2
200
300
200
0
0.0
0.0
30 weeks of GA
100
200
100
0
Effect (grams)
0.2
29 weeks of GA
300
28 weeks of GA
Effect (grams)
27 weeks of GA
300
26 weeks of GA
300
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 20. 95% credible limits for the posterior distribution of the effect of maternal parity
on birth weight for gestational ages 25-33 weeks.
Biometrics, December 2013
24
0.2
0.4
0.6
0.8
1.0
250
200
50
100
150
200
50
100
150
200
150
100
Effect (grams)
50
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
250
50
100
150
200
250
150
1.0
0.2
42 weeks of GA
100
0.8
1.0
250
0.6
50
0.6
0.8
150
0.4
200
250
200
150
100
0.4
Quantile Level
0.6
100
0.2
41 weeks of GA
50
0.2
0.4
50
0.0
40 weeks of GA
0.0
0.2
200
250
200
100
50
0.0
0.0
39 weeks of GA
150
200
150
100
50
Effect (grams)
0.2
38 weeks of GA
250
37 weeks of GA
Effect (grams)
36 weeks of GA
250
35 weeks of GA
250
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 21. 95% credible limits for the posterior distribution of the effect of maternal parity
on birth weight for gestational ages 34-42 weeks.
25
0.2
0.4
0.6
0.8
1.0
200
150
−100
0
50
100
150
−100
0
50
100
150
100
50
0
Effect (grams)
−100
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
200
0.8
1.0
0.0
0.4
0.6
0.8
1.0
200
−100
0
50
100
150
200
100
50
1.0
0.2
33 weeks of GA
0
0.8
1.0
100
0.6
−100
0.6
0.8
50
0.4
150
200
150
100
50
0
0.4
Quantile Level
0.6
0
0.2
32 weeks of GA
−100
0.2
0.4
−100
0.0
31 weeks of GA
0.0
0.2
150
200
150
50
0
−100
0.0
0.0
30 weeks of GA
100
150
100
50
0
−100
Effect (grams)
0.2
29 weeks of GA
200
28 weeks of GA
Effect (grams)
27 weeks of GA
200
26 weeks of GA
200
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 22. 95% credible limits for the posterior distribution of the effect of maternal age
greater than 40 relative to maternal age less than 40 on birth weight for gestational ages
25-33 weeks.
Biometrics, December 2013
26
0.2
0.4
0.6
0.8
1.0
100
50
−150
−50
0
50
−150
−50
0
50
0
−50
Effect (grams)
−150
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
100
−150
−50
0
50
100
0
1.0
0.2
42 weeks of GA
−50
0.8
1.0
100
0.6
−150
0.6
0.8
0
0.4
50
100
50
0
−50
0.4
Quantile Level
0.6
−50
0.2
41 weeks of GA
−150
0.2
0.4
−150
0.0
40 weeks of GA
0.0
0.2
50
100
50
−50
−150
0.0
0.0
39 weeks of GA
0
50
0
−50
−150
Effect (grams)
0.2
38 weeks of GA
100
37 weeks of GA
Effect (grams)
36 weeks of GA
100
35 weeks of GA
100
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 23. 95% credible limits for the posterior distribution of the effect of maternal age
greater than 40 relative to maternal age less than 40 on birth weight for gestational ages
34-42 weeks.
27
0.2
0.4
0.6
0.8
1.0
100
50
−50
0
50
−50
0
50
0
Effect (grams)
−50
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
100
−50
0
50
100
0
0.8
1.0
33 weeks of GA
−50
0.6
0.8
100
0.4
50
100
50
0
0.4
Quantile Level
0.6
0
0.2
32 weeks of GA
−50
0.2
0.4
−50
0.0
31 weeks of GA
0.0
0.2
50
100
50
−50
0.0
0.0
30 weeks of GA
0
50
0
−50
Effect (grams)
0.2
29 weeks of GA
100
28 weeks of GA
Effect (grams)
27 weeks of GA
100
26 weeks of GA
100
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 24. 95% credible limits for the posterior distribution of the effect of paternal age
greater than 40 relative to paternal age less than 40 on birth weight for gestational ages
25-33 weeks.
Biometrics, December 2013
28
0.2
0.4
0.6
0.8
1.0
150
100
−50
0
50
100
−50
0
50
100
50
0
Effect (grams)
−50
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
150
−50
0
50
100
150
50
1.0
0.2
42 weeks of GA
0
0.8
1.0
150
0.6
−50
0.6
0.8
50
0.4
100
150
100
50
0
0.4
Quantile Level
0.6
0
0.2
41 weeks of GA
−50
0.2
0.4
−50
0.0
40 weeks of GA
0.0
0.2
100
150
100
0
−50
0.0
0.0
39 weeks of GA
50
100
50
0
−50
Effect (grams)
0.2
38 weeks of GA
150
37 weeks of GA
Effect (grams)
36 weeks of GA
150
35 weeks of GA
150
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 25. 95% credible limits for the posterior distribution of the effect of paternal age
greater than 40 relative to paternal age less than 40 on birth weight for gestational ages
34-42 weeks.
29
0.2
0.4
0.6
0.8
1.0
40
20
−60
−40
−20
0
20
−60
−40
−20
0
20
0
−20
Effect (grams)
−40
−60
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
40
0.8
1.0
0.0
0.4
0.6
0.8
1.0
40
−60
−40
−20
0
20
40
0
−20
1.0
0.2
33 weeks of GA
−40
0.8
1.0
0
0.6
−60
0.6
0.8
−20
0.4
20
40
20
0
−20
−40
0.4
Quantile Level
0.6
−40
0.2
32 weeks of GA
−60
0.2
0.4
−60
0.0
31 weeks of GA
0.0
0.2
20
40
20
−20
−40
−60
0.0
0.0
30 weeks of GA
0
20
0
−20
−60
−40
Effect (grams)
0.2
29 weeks of GA
40
28 weeks of GA
Effect (grams)
27 weeks of GA
40
26 weeks of GA
40
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 26. 95% credible limits for the posterior distribution of the effect of the mother
finishing high school relative to not finishing high school on birth weight for gestational ages
25-33 weeks.
Biometrics, December 2013
30
0.2
0.4
0.6
0.8
1.0
40
20
−40
−20
0
20
−40
−20
0
20
0
Effect (grams)
−20
−40
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
40
−40
−20
0
20
40
0
1.0
0.2
42 weeks of GA
−20
0.8
1.0
40
0.6
−40
0.6
0.8
0
0.4
20
40
20
0
−20
0.4
Quantile Level
0.6
−20
0.2
41 weeks of GA
−40
0.2
0.4
−40
0.0
40 weeks of GA
0.0
0.2
20
40
20
−20
−40
0.0
0.0
39 weeks of GA
0
20
0
−20
−40
Effect (grams)
0.2
38 weeks of GA
40
37 weeks of GA
Effect (grams)
36 weeks of GA
40
35 weeks of GA
40
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 27. 95% credible limits for the posterior distribution of the effect of the mother
finishing high school relative to not finishing high school on birth weight for gestational ages
34-42 weeks.
31
0.2
0.4
0.6
0.8
1.0
50
0
−100
−50
0
−100
−50
0
Effect (grams)
−50
−100
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
50
−100
−50
0
50
−50
0.8
1.0
33 weeks of GA
−100
0.6
0.8
50
0.4
0
50
0
−50
0.4
Quantile Level
0.6
−50
0.2
32 weeks of GA
−100
0.2
0.4
−100
0.0
31 weeks of GA
0.0
0.2
0
50
0
−100
0.0
0.0
30 weeks of GA
−50
0
−50
−100
Effect (grams)
0.2
29 weeks of GA
50
28 weeks of GA
Effect (grams)
27 weeks of GA
50
26 weeks of GA
50
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 28. 95% credible limits for the posterior distribution of the effect of the mother
finishing education above high school relative to not finishing high school on birth weight
for gestational ages 25-33 weeks.
Biometrics, December 2013
32
0.2
0.4
0.6
0.8
1.0
100
50
−50
0
50
−50
0
50
Effect (grams)
0
−50
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
100
−50
0
50
100
0
0.8
1.0
42 weeks of GA
−50
0.6
0.8
100
0.4
50
100
50
0
0.4
Quantile Level
0.6
0
0.2
41 weeks of GA
−50
0.2
0.4
−50
0.0
40 weeks of GA
0.0
0.2
50
100
50
−50
0.0
0.0
39 weeks of GA
0
50
0
−50
Effect (grams)
0.2
38 weeks of GA
100
37 weeks of GA
Effect (grams)
36 weeks of GA
100
35 weeks of GA
100
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 29. 95% credible limits for the posterior distribution of the effect of the mother
finishing education above high school relative to not finishing high school on birth weight
for gestational ages 34-42 weeks.
33
0.2
0.4
0.6
0.8
1.0
0
−20
−60
−40
−20
−60
−40
−20
Effect (grams)
−40
−60
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
0
−60
−40
−20
0
−40
0.8
1.0
33 weeks of GA
−60
0.6
0.8
0
0.4
−20
0
−20
−40
0.4
Quantile Level
0.6
−40
0.2
32 weeks of GA
−60
0.2
0.4
−60
0.0
31 weeks of GA
0.0
0.2
−20
0
−20
−60
0.0
0.0
30 weeks of GA
−40
−20
−40
−60
Effect (grams)
0.2
29 weeks of GA
0
28 weeks of GA
Effect (grams)
27 weeks of GA
0
26 weeks of GA
0
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 30. 95% credible limits for the posterior distribution of the effect of the father
finishing high school relative to not finishing high school on birth weight for gestational ages
25-33 weeks.
Biometrics, December 2013
34
0.2
0.4
0.6
0.8
1.0
20
0
−60
−40
−20
0
−60
−40
−20
0
−20
Effect (grams)
−40
−60
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
20
−60
−40
−20
0
20
−20
1.0
0.2
42 weeks of GA
−40
0.8
1.0
20
0.6
−60
0.6
0.8
−20
0.4
0
20
0
−20
−40
0.4
Quantile Level
0.6
−40
0.2
41 weeks of GA
−60
0.2
0.4
−60
0.0
40 weeks of GA
0.0
0.2
0
20
0
−40
−60
0.0
0.0
39 weeks of GA
−20
0
−20
−40
−60
Effect (grams)
0.2
38 weeks of GA
20
37 weeks of GA
Effect (grams)
36 weeks of GA
20
35 weeks of GA
20
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 31. 95% credible limits for the posterior distribution of the effect of the father
finishing high school relative to not finishing high school on birth weight for gestational ages
34-42 weeks.
35
0.2
0.4
0.6
0.8
1.0
60
40
−80
−40
0
20
40
−80
−40
0
20
40
20
0
Effect (grams)
−40
−80
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
60
0.8
1.0
0.0
0.4
0.6
0.8
1.0
60
−80
−40
0
20
40
60
20
0
1.0
0.2
33 weeks of GA
−40
0.8
1.0
20
0.6
−80
0.6
0.8
0
0.4
40
60
40
20
0
−40
0.4
Quantile Level
0.6
−40
0.2
32 weeks of GA
−80
0.2
0.4
−80
0.0
31 weeks of GA
0.0
0.2
40
60
40
0
−40
−80
0.0
0.0
30 weeks of GA
20
40
20
0
−40
−80
Effect (grams)
0.2
29 weeks of GA
60
28 weeks of GA
Effect (grams)
27 weeks of GA
60
26 weeks of GA
60
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 32. 95% credible limits for the posterior distribution of the effect of the father
finishing education above high school relative to not finishing high school on birth weight
for gestational ages 25-33 weeks.
Biometrics, December 2013
36
0.2
0.4
0.6
0.8
1.0
60
40
−60
−40
−20
0
20
40
−60
−40
−20
0
20
40
20
0
Effect (grams)
−20
−40
−60
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
60
20
0.8
1.0
0.0
0.4
0.6
0.8
1.0
60
−60
−40
−20
0
20
40
60
20
0
−20
1.0
0.2
42 weeks of GA
−40
0.8
1.0
0
0.6
−60
0.6
0.8
−20
0.4
40
60
40
20
0
−20
−40
0.4
Quantile Level
0.6
−40
0.2
41 weeks of GA
−60
0.2
0.4
−60
0.0
40 weeks of GA
0.0
0.2
40
60
40
0
−20
−40
−60
0.0
0.0
39 weeks of GA
20
40
20
0
−20
−60
−40
Effect (grams)
0.2
38 weeks of GA
60
37 weeks of GA
Effect (grams)
36 weeks of GA
60
35 weeks of GA
60
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 33. 95% credible limits for the posterior distribution of the effect of the father
finishing education above high school relative to not finishing high school on birth weight
for gestational ages 34-42 weeks.
37
0.2
0.4
0.6
0.8
1.0
50
0
−150
−100
−50
0
−150
−100
−50
0
−50
−100
Effect (grams)
−150
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
50
−150
−100
−50
0
50
−50
1.0
0.2
33 weeks of GA
−100
0.8
1.0
50
0.6
−150
0.6
0.8
−50
0.4
0
50
0
−50
−100
0.4
Quantile Level
0.6
−100
0.2
32 weeks of GA
−150
0.2
0.4
−150
0.0
31 weeks of GA
0.0
0.2
0
50
0
−100
−150
0.0
0.0
30 weeks of GA
−50
0
−50
−100
−150
Effect (grams)
0.2
29 weeks of GA
50
28 weeks of GA
Effect (grams)
27 weeks of GA
50
26 weeks of GA
50
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 34. 95% credible limits for the posterior distribution of the effect of maternal
black non-Hispanic ethnicity relative to white non-Hispanic ethnicity on birth weight for
gestational ages 25-33 weeks.
Biometrics, December 2013
38
0.2
0.4
0.6
0.8
1.0
−100
−140
−220
−180
−140
−220
−180
−140
Effect (grams)
−180
−220
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
−100
−220
−180
−140
−100
−180
0.8
1.0
42 weeks of GA
−220
0.6
0.8
−100
0.4
−140
−100
−140
−180
0.4
Quantile Level
0.6
−180
0.2
41 weeks of GA
−220
0.2
0.4
−220
0.0
40 weeks of GA
0.0
0.2
−140
−100
−140
−220
0.0
0.0
39 weeks of GA
−180
−140
−180
−220
Effect (grams)
0.2
38 weeks of GA
−100
37 weeks of GA
Effect (grams)
36 weeks of GA
−100
35 weeks of GA
−100
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 35. 95% credible limits for the posterior distribution of the effect of maternal
black non-Hispanic ethnicity relative to white non-Hispanic ethnicity on birth weight for
gestational ages 34-42 weeks.
39
0.2
0.4
0.6
0.8
1.0
200
100
−100
0
100
−100
0
100
0
Effect (grams)
−100
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
200
−100
0
100
200
0
0.8
1.0
33 weeks of GA
−100
0.6
0.8
200
0.4
100
200
100
0
0.4
Quantile Level
0.6
0
0.2
32 weeks of GA
−100
0.2
0.4
−100
0.0
31 weeks of GA
0.0
0.2
100
200
100
−100
0.0
0.0
30 weeks of GA
0
100
0
−100
Effect (grams)
0.2
29 weeks of GA
200
28 weeks of GA
Effect (grams)
27 weeks of GA
200
26 weeks of GA
200
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 36. 95% credible limits for the posterior distribution of the effect of maternal
Hispanic ethnicity relative to white non-Hispanic ethnicity on birth weight for gestational
ages 25-33 weeks.
Biometrics, December 2013
40
0.2
0.4
0.6
0.8
1.0
150
100
−150
−50
0
50
100
−150
−50
0
50
100
50
0
Effect (grams)
−50
−150
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
150
0.8
1.0
0.0
0.4
0.6
0.8
1.0
150
−150
−50
0
50
100
150
50
0
1.0
0.2
42 weeks of GA
−50
0.8
1.0
50
0.6
−150
0.6
0.8
0
0.4
100
150
100
50
0
−50
0.4
Quantile Level
0.6
−50
0.2
41 weeks of GA
−150
0.2
0.4
−150
0.0
40 weeks of GA
0.0
0.2
100
150
100
0
−50
−150
0.0
0.0
39 weeks of GA
50
100
50
0
−50
−150
Effect (grams)
0.2
38 weeks of GA
150
37 weeks of GA
Effect (grams)
36 weeks of GA
150
35 weeks of GA
150
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 37. 95% credible limits for the posterior distribution of the effect of maternal
Hispanic ethnicity relative to white non-Hispanic ethnicity on birth weight for gestational
ages 34-42 weeks.
41
0.2
0.4
0.6
0.8
1.0
100
50
−200
−100
0
50
−200
−100
0
50
0
Effect (grams)
−100
−200
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
100
−200
−100
0
50
100
0
1.0
0.2
33 weeks of GA
−100
0.8
1.0
100
0.6
−200
0.6
0.8
0
0.4
50
100
50
0
−100
0.4
Quantile Level
0.6
−100
0.2
32 weeks of GA
−200
0.2
0.4
−200
0.0
31 weeks of GA
0.0
0.2
50
100
50
−100
−200
0.0
0.0
30 weeks of GA
0
50
0
−100
−200
Effect (grams)
0.2
29 weeks of GA
100
28 weeks of GA
Effect (grams)
27 weeks of GA
100
26 weeks of GA
100
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 38. 95% credible limits for the posterior distribution of the effect of maternal other
ethnicity relative to white non-Hispanic ethnicity on birth weight for gestational ages 25-33
weeks.
Biometrics, December 2013
42
0.2
0.4
0.6
0.8
1.0
0
−100
−300
−200
−100
−300
−200
−100
−200
Effect (grams)
−300
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
0
−300
−200
−100
0
−200
0.8
1.0
42 weeks of GA
−300
0.6
0.8
0
0.4
−100
0
−100
−200
0.4
Quantile Level
0.6
−200
0.2
41 weeks of GA
−300
0.2
0.4
−300
0.0
40 weeks of GA
0.0
0.2
−100
0
−100
−300
0.0
0.0
39 weeks of GA
−200
−100
−200
−300
Effect (grams)
0.2
38 weeks of GA
0
37 weeks of GA
Effect (grams)
36 weeks of GA
0
35 weeks of GA
0
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 39. 95% credible limits for the posterior distribution of the effect of maternal other
ethnicity relative to white non-Hispanic ethnicity on birth weight for gestational ages 34-42
weeks.
43
0.2
0.4
0.6
0.8
1.0
20
10
−20
−10
0
10
−20
−10
0
10
0
−10
Effect (grams)
−20
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
20
−20
−10
0
10
20
0
1.0
0.2
33 weeks of GA
−10
0.8
1.0
20
0.6
−20
0.6
0.8
0
0.4
10
20
10
0
−10
0.4
Quantile Level
0.6
−10
0.2
32 weeks of GA
−20
0.2
0.4
−20
0.0
31 weeks of GA
0.0
0.2
10
20
10
−10
−20
0.0
0.0
30 weeks of GA
0
10
0
−10
−20
Effect (grams)
0.2
29 weeks of GA
20
28 weeks of GA
Effect (grams)
27 weeks of GA
20
26 weeks of GA
20
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 40. 95% credible limits for the posterior distribution of the effect of a one unit
increase in first trimester ozone exposure on birth weight for gestational ages 25-33 weeks.
All ozone values were linearly transformed into [-1,1], so a one-unit increase can be roughly
thought of as an increase from low levels to middle levels of exposure, or middle to high.
Biometrics, December 2013
44
0.2
0.4
0.6
0.8
1.0
10
0
−20
−10
0
−20
−10
0
−10
Effect (grams)
−20
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
1.0
0.2
0.4
0.6
0.8
1.0
10
−20
−10
0
10
−10
0.8
1.0
42 weeks of GA
−20
0.6
0.8
10
0.4
0
10
0
−10
0.4
Quantile Level
0.6
−10
0.2
41 weeks of GA
−20
0.2
0.4
−20
0.0
40 weeks of GA
0.0
0.2
0
10
0
−20
0.0
0.0
39 weeks of GA
−10
0
−10
−20
Effect (grams)
0.2
38 weeks of GA
10
37 weeks of GA
Effect (grams)
36 weeks of GA
10
35 weeks of GA
10
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 41. 95% credible limits for the posterior distribution of the effect of a one unit
increase in first trimester ozone exposure on birth weight for gestational ages 34-42 weeks.
All ozone values were linearly transformed into [-1,1], so a one-unit increase can be roughly
thought of as an increase from low levels to middle levels of exposure, or middle to high.
45
0.2
0.4
0.6
0.8
1.0
10
0
−30
−20
−10
0
−30
−20
−10
0
−10
Effect (grams)
−20
−30
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
10
−30
−20
−10
0
10
−10
1.0
0.2
33 weeks of GA
−20
0.8
1.0
10
0.6
−30
0.6
0.8
−10
0.4
0
10
0
−10
−20
0.4
Quantile Level
0.6
−20
0.2
32 weeks of GA
−30
0.2
0.4
−30
0.0
31 weeks of GA
0.0
0.2
0
10
0
−20
−30
0.0
0.0
30 weeks of GA
−10
0
−10
−20
−30
Effect (grams)
0.2
29 weeks of GA
10
28 weeks of GA
Effect (grams)
27 weeks of GA
10
26 weeks of GA
10
25 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 42. 95% credible limits for the posterior distribution of the effect of a one unit
increase in second trimester ozone exposure on birth weight for gestational ages 25-33 weeks.
All ozone values were linearly transformed into [-1,1], so a one-unit increase can be roughly
thought of as an increase from low levels to middle levels of exposure, or middle to high.
Biometrics, December 2013
46
0.2
0.4
0.6
0.8
1.0
10
0
−30
−20
−10
0
−30
−20
−10
0
−10
Effect (grams)
−20
−30
0.0
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.0
0.4
0.6
0.8
1.0
10
−30
−20
−10
0
10
−10
1.0
0.2
42 weeks of GA
−20
0.8
1.0
10
0.6
−30
0.6
0.8
−10
0.4
0
10
0
−10
−20
0.4
Quantile Level
0.6
−20
0.2
41 weeks of GA
−30
0.2
0.4
−30
0.0
40 weeks of GA
0.0
0.2
0
10
0
−20
−30
0.0
0.0
39 weeks of GA
−10
0
−10
−20
−30
Effect (grams)
0.2
38 weeks of GA
10
37 weeks of GA
Effect (grams)
36 weeks of GA
10
35 weeks of GA
10
34 weeks of GA
0.0
0.2
0.4
0.6
0.8
Quantile Level
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Quantile Level
Figure 43. 95% credible limits for the posterior distribution of the effect of a one unit
increase in second trimester ozone exposure on birth weight for gestational ages 34-42 weeks.
All ozone values were linearly transformed into [-1,1], so a one-unit increase can be roughly
thought of as an increase from low levels to middle levels of exposure, or middle to high.