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Biostatistics (2008), 9, 4, pp. 686–699

doi:10.1093/biostatistics/kxm059

Advance Access publication on March 18, 2008

A Bayesian approach to functional-based multilevel

modeling of longitudinal data: applications to

environmental epidemiology

KIROS BERHANE∗

Department of Preventive Medicine, University of Southern California,

Los Angeles, CA 90033-9987, USA

kiros@usc.edu

NUOO-TING MOLITOR

Department of Epidemiology and Public Health, Imperial College London,

Norfolk Place, London W2 1PG, UK

jassy.molitor@imperial.ac.uk

SUMMARY

Flexible multilevel models are proposed to allow for cluster-specific smooth estimation of growth curves

in a mixed-effects modeling format that includes subject-specific random effects on the growth parame-

ters. Attention is then focused on models that examine between-cluster comparisons of the effects of an

ecologic covariate of interest (e.g. air pollution) on nonlinear functionals of growth curves (e.g. maximum

rate of growth). A Gibbs sampling approach is used to get posterior mean estimates of nonlinear func-

tionals along with their uncertainty estimates. A second-stage ecologic random-effects model is used to

examine the association between a covariate of interest (e.g. air pollution) and the nonlinear functionals. A

unified estimation procedure is presented along with its computational and theoretical details. The models

are motivated by, and illustrated with, lung function and air pollution data from the Southern California

Children’s Health Study.

Keywords: Air pollution; Correlated data; Growth curves; Mixed-effects; Splines.

1. INTRODUCTION

Cohort studies on the long-term effects of air pollution, such as the Harvard Six-Cities study and the

Southern California Children’s Health Study (hereafter referred to as CHS), involve multiple measure-

ments on each of many subjects, in addition to possible geographic clustering due to inclusion of several

communities with a diverse pollution profile (Peters, Avol, Gauderman, and others, 1999). Proper analysis

∗To whom correspondence should be addressed.

c ? The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

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Bayesian approach to functional-based multilevel modeling

687

of the resulting longitudinal data should account for the clustering effects, as well as the serial correlation

that usually arises due to repeated measurements from the same subject. In such studies, the main interest

is in being able to estimate air pollution effects on outcomes such as lung function (i) between times within

subjects, (ii) between subjects within communities, and (iii) between communities. The multilevel mod-

eling paradigm (Berhane and others, 2004) gives a unified way of examining the effects of air pollution

at the different levels outlined above. These models fall under the general umbrella of the well-studied

mixed-effects models (Laird and Ware, 1982; Diggle and others, 1994).

Growth curve modeling of physiologic measures such as height and lung function in children has

always posed challenges because of the nonconstant growth rates as subjects move from preadolescence to

puberty and then to young adulthood. Several methods have been proposed to model these growth curves.

Fully parametric methods range from simple ones that assume a global polynomial curve (Kory and

others, 1961) to those that fit separate parametric curves in mutually exclusive pre- and postadolescence

age categories (Dockery and others, 1983). With the growing popularity of smoothing techniques (Hastie

and Tibshirani, 1990), more flexible methods have been proposed for modeling growth curves. Wypij

and others (1993) used generalized estimating equations–based methods (Liang and Zeger, 1986) that use

splines to model lung function growth curves in the Harvard Six-Cities study (Ferris and others, 1979).

Berhane and others (2000) used similar models in a mixed-effects modeling framework on data from the

CHS. Other related, but more general, models have also been proposed, along with extensive studies of the

ensuing theoretical and computational issues (Berhane and Tibshirani, 1998; Brumback and Rice, 1998;

Lin and Zhang, 1999).

While the use of smoothing techniques to model lung function growth has become the state of the

art, methods for relating these growth curves to pollution levels are not well established. The main chal-

lenge has been in deciding which aspects of the nonlinear growth curves to relate to air pollution and the

lack of appropriate statistical methods to conduct the analysis. In this paper, we propose a new technique

for modeling important features of nonlinear curves as functions of ecologic covariates. The methodol-

ogy is motivated by our desire to assess the impact of ambient air pollution on children’s lung function

growth.

The main features of the proposed methodology are the use of Gibbs sampling techniques to estimate

the posterior distribution of important features of the curves (known as functionals) and the use of meta-

analytic techniques to relate the posterior means and posterior variances to ambient air pollution levels.

The idea of using Gibbs sampling techniques to study posterior distributions of functionals has been used

before by Silverman (1985) and Hastie and Tibshirani (2000). Our methods differ from those previously

proposed in 2 important aspects: namely, (i) we fit cluster-specific curves with subject-specific random

effects to allow for subject-specific attributes and (ii) we introduce a meta-analytic second-stage model in

order to relate the posterior means of the functionals along with posterior variances, entering the mixed-

effects second-stage model either as inverse-variance weights or as known variances of the random effect

for modeling heterogeneity. The near equivalence of such multilevel models to a related unified mixed-

effects model has been discussed previously (Berhane and others, 2004). Our proposed strategy is similar

to those proposed for functional mixed-effects models by Guo (2002) and wavelet-based approaches of

Morris and Carroll (2006) and references therein. Our approach has the advantages of simplicity and

computational ease for analysis of large epidemiologic data sets. We plan to make our suite of computer

programs available to the research community.

We briefly describe the CHS in Section 2. Details of the proposed model are discussed in Section

3, followed by a detailed analysis of CHS data in Section 4. Further topics of interest, conclusions, and

areas for future research are discussed in Sections 5 and 6. Note that although the methods are moti-

vated by the CHS data, they are potentially applicable to a wide range of applications where the main

interest might be in drawing ecologic inference on scientifically relevant functionals of nonlinear growth

trajectories.

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688K. BERHANE AND N.-T. MOLITOR

2. THE SOUTHERN CALIFORNIA CHS

The CHS is a longitudinal study designed to assess the long-term effects of air pollution on children’s

health. The study included a sample of 12 Southern California communities out of 86 communities with

routine air pollution monitoring that were selected to exhibit maximum variability with respect to ambient

levels of ozone (O3), particulates (PM10), nitrogen dioxide (NO2), and acid vapor. Figure S1 in the sup-

plementary material (available at Biostatistics online, http://www.biostatistics.oxfordjournals.org) depicts

the geographic distribution of the study communities. Details of design- and power-related issues are dis-

cussed in Navidi and others (1994). In 1993, the study enrolled approximately 150 fourth graders and 75

seventh and tenth graders from each community, leading to a total of 3681 students who returned a signed

consent form. In 1994, 386 fifth and 111 eighth graders were added from the same schools. A second

4th-grade cohort of 2081 children was enrolled in 1996. Thus, a total of 6259 children have entered the

study for observation. Attrition amounted to about 8% per year, 95% of which was due to moving away

from participating schools. The models proposed in this paper will be illustrated by using data from all

participants of the first cohort of fourth graders.

A baseline questionnaire was completed by the primary caregiver of each child, covering residential

history, current residential characteristics (e.g. ventilation and sources of indoor pollution), personal risk

factors, respiratory symptoms, and usual activities. An abbreviated yearly follow-up questionnaire was

used to collect data on chronic respiratory symptoms and diseases and time-dependent covariates. For

more details, see Peters, Avol, Navidi, and others (1999).

Lung function tests, which are the focus of this paper, were conducted via field team visits to par-

ticipating schools in winter to spring of each year (January to June), a period of relatively low pollution

levels in the region, so as to minimize the effects of acute pollution episodes. The lung functions measured

included forced expiratory volume in 1 s (FEV1), the maximum mid-expiratory flow (MMEF), the peak

expiratory flow rate (PEFR), forced volume capacity (FVC), and forced expiratory flow (FEF75). More

details on the lung function testing techniques and quality control procedures are given in Peters, Avol,

Gauderman, and others (1999). For illustrative purposes, our examples in this paper are limited to FEV1,

as this is the main measure of lung volume and hence its maximum rate of growth is more likely to be

affected by effects of air pollution during a child’s growth spurt.

Air pollution levels were assessed via dedicated central site monitors in each study community, pro-

viding data on continuous hourly ambient O3, PM10, and NO2and 2-week measures of PM2.5and acid

vapors. Individual exposure predictions were made via the “microenvironmental” approach (Navidi and

Lurmann, 1995) based on ambient exposure levels, housing characteristics, and time–activity patterns.

Focusing on O3, elemental carbon, and acid vapors, our illustrative examples use multiyear average

exposure data obtained by averaging the hourly or 2-week integrated measures of exposure data from

central site monitors. An ubiquitous problem in air pollution research is the high correlation between

the various air pollutants. This leads to multicollinearity in models that simultaneously assess multiple

pollutants and hence precludes our ability to identify the pollutant with maximum impact on children’s

health. This issue and a potential solution that uses Bayesian model averaging have been discussed briefly

in Berhane and others (2004). We found out that all the pollutants were highly correlated with each other,

with the exception of O3which was uncorrelated with any of the other pollutants. In this paper, we only

consider models that assess effects of one pollutant at a time. Extensions of our proposed models to the

multipollutant problem in a manner that properly handles the multicollinearity in the pollutant effects,

while possible, are beyond the scope of this paper.

3. THE PROPOSED FUNCTIONAL-BASED MULTILEVEL MODEL

To establish notation, we first describe the multilevel linear model that has been used for assessing the

effects of air pollution on lung function growth (Berhane and others, 2004; Gauderman and others,

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Bayesian approach to functional-based multilevel modeling

689

2004). Denote by ycij the outcome measure (e.g. lung function, log transformed to ensure normality)

for subject i in community c at attained age tcij, where j indexes year. Predictors can be time-dependent,

time-constant subject-specific attributes, or community-specific. Uppercase and lowercase letters denote

community-specific and subject-specific quantities, respectively. Thus, wcij represents time-dependent

covariates (e.g. respiratory infection), wcitime-constant fixed covariates (e.g. ethnicity), Pcjthe commu-

nity annual-average levels of pollution, and Pcthe community-specific multiyear average levels of pollu-

tion. A 3-level linear model that allows for subject- and community-level variance components based on

subject-specific intercepts and slopes is the following linear mixed-effects model (Laird and Ware, 1982):

ycij= α0+ α3Pc+η η η2Twci+ ec+ eci+ [β0+ β3Pc+γ γ γ2Twci+ fc+ fci] tcij

+η η η1Twcij+ α1(Pcj− Pc) + ecij,

where eci ∼ N(0,σ2

tercepts and ecij ∼ N(0,σ2

the corresponding random slopes. The implicit assumption of independence between random effects in

Model (3.1) could be relaxed to allow for more complex temporal and/or spatial correlation structures.

See Berhane and others (2004) for more details.

(3.1)

e,ci), and ec ∼ N(0,σ2

cij) is an error term. Similarly, fci ∼ N(0,σ2

e,c) are random subject-specific and community-specific in-

f,ci) and fc ∼ N(0,σ2

f,c) are

3.1

Flexible models for modeling nonlinear trajectories

Exploratory analyses have shown that the relationship between log-transformed values of lung function

measures and height is linear in short age-intervals, but both the intercept and the slope change with

age during the adolescent growth period (Wypij and others, 1993). This observation reflects the complex

relationship between height growth, age, and lung function growth. It is well known that human lung

growth spurt lags by about 6 months from that of height growth spurt. We extend Model (3.1) to depict

this complex relationship. Focusing on the temporal level of the model for modeling growth trajectories,

a varying-coefficient model (Hastie and Tibshirani, 1993) has the following form:

ycij= aci+ f1(tcij) + f2(tcij) × hcij+η η η1Twcij+ α1(Pcj− Pc),

where “h” denotes the time-dependent log-transformed height measurement. In order to make f1(tcij) in-

terpretable as a growth curve, h is usually centered by a smooth function of “log (height)” on

“attained age.”

Options for modeling f1and f2range from completely nonparametric ones such as smoothing splines

to fully parametric functions such as polynomials. Here, we use natural spline basis functions. These are

constructed by using B-spline basis functions (DeBoor, 1974), leading to piecewise polynomials that are

joined together at a specified number of break points (or knots), and constrained to be linear beyond the

boundary knots. The advantage of such inherently parametric representation is that inference is based on

solid theoretical grounds and extracting features of growth curves via appropriate derivatives is straight-

forward. Based on this approach, a multilevel version of (3.1) that allows for nonlinear subject-specific

growth parameters could then be formulated as follows:

(3.2)

ycij=

K

?

k=1

b(k)

ciN(k)(tcij) +

K

?

k=1

˜b(k)

ciN(k)(tcij) × hcij+η η η1Twcij+ α1(Pcj− Pc) + ecij

(3.3)

b(k)

ci= B(k)

c

+η η η(k)

2

Twci+ e(k)

ci

(3.4)

˜b(k)

ci=˜B(k)

c

+γ γ γ(k)

2

Twci+ ˜ e(k)

ci

(3.5)

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690K. BERHANE AND N.-T. MOLITOR

B(k)

c

= ζ(k)

=˜ζ(k)

0

+ ζ(k)

+˜ζ(k)

3

Pc+ e(k)

c

(3.6)

˜B(k)

c

03

Pc+ ˜ e(k)

c,

(3.7)

where k = 1,..., K subscripts the K natural spline basis functions Nk(tcij). Models (3.3)–(3.7) could

be combined to give the following unified mixed-effects model:

ycij=

K

?

k=1

?

ζ(k)

0

+ ζ(k)

3

Pc+η η η(k)

2

Twci+ e(k)

ci

?

N(k)(tcij)

+

K

?

k=1

?

˜ζ(k)

0

+˜ζ(k)

3

Pc+γ γ γ(k)

2

Twci+ ˜ e(k)

ci

?

N(k)(tcij) × hcij

+η η η1Twcij+ α1(Pcj− Pc) + ecij.(3.8)

For settings where the number of observations per subject is limited, as in the CHS data, it may not be

advisable to fit complex (individual-specific and/or community-specific) interactions for the “N(k)(tcij)×

hcij” terms. It also makes more sense to fit separate smooth functions for each of the communities, while

subject-to-subject variability is captured via random error terms in order to avoid overparameterization.

Moreover, the biologically important features of the growth curves that may be affected by air pollution

may not be extractable from the above formulation. Hence, we propose the following 2-level model,

where (i) Models (3.3)–(3.5) are combined and simplified to form a new level-1 model, (ii) a Markov

Chain Monte Carlo (MCMC) procedure, such as the Gibbs sampler, is employed to estimate the posterior

distribution of functionals (e.g. maximum rate of growth) of the community-specific curves, and (iii) a

mixed-effects meta-analytic model is fitted to assess the relationship of the functionals with air pollution.

The new level-1 model is given as follows:

ycij=

K

?

k=1

B(k)

cN(k)(tcij) +

K

?

k=1

˜B(k)

cN(k)(tcij) × hcij+η η η(k)

2

Twci+η η η1Twcij+ α1(Pcj− Pc)

+

K

?

k=1

e(k)

ciN(k)(tcij) + ecij,

(3.9)

where eci ∼ MVN(0,?e,ci), eci =?e(1)

is a residual error term. The parametric model (3.9) can be fitted via popular software packages (e.g.

PROC MIXED in SAS, LME in Splus). We can then extract the community-specific growth curves, with

proper adjustments for subject-to-subject variability due to the rich random-effects structure. Growth rate

estimates at any given age can also be extracted by using derivatives of the growth curves, a relatively

straightforward process when dealing with natural splines.

When the interest is in predicted value(s) at any age of either the growth curve itself or one of its

derivatives, or linear combinations of them, these features are known as linear functionals and inference

on them is straightforward (Silverman, 1985) as they allow for closed-form estimation of variances. An

important recent substantive publication from the CHS uses this approach to model the relationship be-

tween air pollution and lung function growth over the childhood period (Gauderman and others, 2004).

The need for Gibbs sampling arises when the focus is on nonlinear functionals (e.g. maximum rate of

ci,...,e(K)

ci

?is the vector of subject-specific random error terms

associated with the growth parameters, included to allow for random subject effects, and ecij∼ N(0,σ2

cij)

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Bayesian approach to functional-based multilevel modeling

691

growth) mainly due to the difficulty of getting full access to uncertainty estimates for the functional. This

problem arises because φ(λ1f1+ λ2f2) ?= φ(λ1f1) + φ(λ2f2).

3.2

Estimating posterior distributions of functionals

The mixed-effects model (3.9) can be rewritten as

yci= Xciβ β β + Wciδ δ δ + Zcijbci+ ecij

= [Xci,Wci] ×

⎡

⎣

β β β

δ δ δ

⎤

⎦+ Zcibci+ eci

= X∗

ciβ β β∗+ Zcibci+ eci,

(3.10)

where c = 1,...,C, i = 1,..., I, and j = 1,...,ni, denote clusters, subjects, and repeated measure-

ments per subject, respectively, and N =?

growth curves, Wcij =

?Nk(tcij); k = 1,..., K?denotes a vector of covariates forming the random component of the model.

of random terms. In the last version of (3.10), β β β∗= [β β β,δ δ δ]Trepresents fixed coefficients associated with

X∗

of the community-specific growth curves, Wci denotes adjustment factors, and Zci denotes covariates

forming the random component of the model.

ini. Here, Xcij=?Nkc(tcij); c = 1,...,C; k = 1,..., K?

?wcij : wci : (Pcj− Pc)?denotes a vector of adjustment factors, and Zcij =

Additionally, we assume that ecij∼ N(0,σ2) and bi∼ MV N(0,?), where bidenotes the q × 1 vector

represents the fixed effects from the realizations of the B-spline basis functions of the community-specific

ci= [Xci,Wci], Xcidenotes the fixed effects from the realizations of the natural spline basis functions

Gibbs sampling. Following the Bayesian paradigm, all parameters in (3.10) are treated as random quan-

tities (with appropriate prior distributions) regardless of whether effects are fixed or random. Therefore,

we proceed by specifying the prior distributions for the parameters in Model (3.10) as follows:

• β β β∗∼ MVN(0,σ σ σβ β β∗), where σ σ σ−1

• bci∼ MVN(0,σ σ σb).

• σ2∼ IG(eσ2, fσ2), where IG denotes the inverse gamma distribution.

Hyperpriors also need to be specified for bci, leading to σ σ σb∼ W?(R)−1,ρ?, where W denotes the

d is the number of columns in matrix Zci, and ρ ? d is a scalar degree of freedom parameter. Given all

prior and hyperprior information, their full conditional posterior distributions are as follows:

• Given (y∗= y − Zb,σ σ σβ β β∗,σ2), the posterior distribution of β β β∗is MVN?d(σ−2X∗y∗),d?.

• Given (y∗

• Given (y∗∗= y − X∗β β β∗− Zb,eσ2, fσ2), the posterior distribution of σ2follows IG?n

• given (b∗

where k is the number of clusters.

Here, d−1= σ−2X∗TX∗+ σ σ σ−1

σ σ σ−1

b. In our Gibbs sampling, we use the restricted maximum likelihood (REML) estimates from the linear

β β β∗ = 0, indicates vague prior information about β β β∗.

Wishart distribution, a multivariate generalization of the gamma distribution. Here, R is a d × d matrix,

ci= yci− X∗

ciβ β β∗,σ σ σb,σ2), the posterior distribution of bciis MVN?dci(σ−2ZT

σ2]−1?.

ciy∗

ci),dci

?.

2+ eσ2,

[1

2(y∗∗Ty∗∗) + f−1

ci= bci− 0,ρ,R), the posterior distribution of ?bis W?k + ρ,?

β β β∗, b = [b11,...,b1m1,...,bC1,...,bCmC]T, and d−1

c

?

i(bT

cibci+ R−1)−1?,

= σ−2ZT

ci

ciZci+

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692 K. BERHANE AND N.-T. MOLITOR

mixed-effects model as initial values. Using the REML estimates shortens the burn-in period for the

Gibbs sampler. Noninformative priors are used in our model for appropriate parameters (e.g. adjustment

factors), and parameters for σ2were specified as (eσ2, fσ2) = (0.01,0.01). Since it has been suggested in

the literature (Gelman, 2006) that this choice of prior may prove to be informative, alternative choices of

priors are also considered to check for robustness of the substantive findings. Also, the hyperparameters

for σ σ σbwere specified as?(R)−1,ρ?

1. Update β β β∗and bci from their own full conditional distributions, that is the distribution of the

parameter in question given all other quantities in the model.

2. Update the variance components σ2, σ σ σbfrom their full conditional distribution.

3. Repeat (1) and (2) until convergence.

=

?(Id)−1,d?, where d = 4 in our case. Therefore, the Gibbs

sampling procedure is implemented as follows:

All analyses use WinBUGS. Issues related to convergence were examined by using convergence di-

agnostic measures as in Gelman and Rubin (1992), implemented using the CODA function in R. This

method uses 2 or more parallel chains with different starting values, essentially sampling from an overdis-

persed distribution. The Gelman and Rubin statistic is calculated based on the comparison of within- and

between-chain variances, giving a value close to 1 if the MCMC output from all chains is indistinguish-

able, and hence, convergence is considered to be reached.

Intermediate step: extracting community-specific functionals. In each cycle of the Gibbs sampling al-

gorithm, community-specific growth curves are estimated along with associated growth rate estimates

via derivatives of the community-specific growth curves. First, we extract the community-specific growth

curves viaˆfc= Xcˆ β β βc, c = 1,...,C, whereˆ β β βcis a g × 1 vector of the community-specific fixed coef-

ficient associated with the natural spline. Then, we obtain the community-specific growth rates by taking

the derivatives ofˆfcdescribed above. Finally, we construct the entire posterior distribution of function-

als of interest such as the maximum rate of growth. Figure 1 gives a pictorial presentation of the prior

specifications for the Bayesian modeling procedure outlined above.

3.3

Ecologic inference on functionals

Given estimates of functional(s) of interest from the algorithms outlined in Sections 3.1–3.2, along with

their entire posterior distributions, one could conduct inferences on the relationships between such

Fig. 1. Pictorial presentation of the setup for the Bayesian modeling paradigm.

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Bayesian approach to functional-based multilevel modeling

693

functionals and ecologic covariates (e.g. air pollution). To do this, we fit an ecologic weighted regres-

sion. This approach takes the uncertainty estimates of the functionals into account by using as weights

the inverse of the posterior variance estimates of the community-specific posterior means of the nonlinear

functional of interest.

Let Fcand Pcdenote a functional of interest (e.g. maximum rate of growth) and a level of an ecologic

covariate (e.g. air pollution) for community c, respectively. Let VFcdenote the variance estimate of Fcas

obtained from the Gibbs sampling algorithm. Then, a weighted ecologic regression model is

Fc= γ0+ γ1Pc+ ec,

(3.11)

where e = (e1,...,eC) ∼ N(0,τ2), with weights 1/VFc. Model (3.11) implicitly assumes that the

posterior distribution of the functional of interest is, at least approximately, Gaussian. It also assumes that

the estimates from the communities are uncorrelated with each other (i.e. there is no spatial correlation).

Each of these assumptions could be tested, and relaxed when necessary, to accommodate more general

classes of models. Specifically, a meta-analytic mixed linear–effects model could be used through an

iterative process as in Stram (1996) to allow for a nondiagonal weight matrix as discussed in Section 5.1.

A fully Bayesian approach that circumvents the Gaussian assumption is discussed in Section 5.2.

4. ANALYSIS OF THE LUNG FUNCTION DATA

We now present a detailed analysis of the motivating example from the Southern California CHS. To il-

lustrate 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 sup-

plementary 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 an-

nual 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. Figure 2 (left panel) illustrates the growth trajectories of FEV1

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 FEV1levels 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.

Fig. 2. Gender-specific growth curves (left panel) and rates of growth (right panel) for FEV1in 8- to 18-year-old

female (dashed curve) and male (solid curve) participants of the CHS. The curves are smooth functions estimated by

natural splines from a mixed-effects model with subject-specific random growth parameters.

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694 K. BERHANE AND N.-T. MOLITOR

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 func-

tions 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(FEV1) to satisfy the normality assumption of the model. The results from

this first-level model are summarized in Table 1 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.

For female participants, there was a significant deficit in FEV1levels (Table 1) 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 FEV1levels (−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 coun-

terparts. 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 Table 1 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

Table 1. Posterior mean estimates and Bayesian credible intervals (BCI) for the level-1 model adjustment

covariates on FEV1in female and male participants of the CHS

CovariateMean (BCI)(×100)

Females

0.67 (0.51,0.82)

−0.02 (−0.04,−0.01)

0.43 (−0.39,1.24)

0.33 (−0.17,0.83)

Males

BMI (kg/m2)

BMI2(kg/m2)

Personal smoking

Exercise

Ethnicity

Asian

Black

Hispanic

Mixed

Other

Ever asthma

Respiratory illness

Incidence asthma

0.35 (0.20,0.50)

−0.03 (−0.04,−0.02)

0.90 (0.20,1.64)

0.56 (0.11,1.05)

−6.02 (−10.57,−1.49)

−11.98 (−15.93,−7.62)

2.52 (−0.03,5.02)

−0.92 (−4.34,2.40)

−1.73 (−4.59,1.01)

−2.62 (−5.15,0.03)

−0.94 (−1.34,−0.49)

0.88 (−0.11,1.93)

−2.41 (−6.07,1.24)

−15.30 (−19.73,−11.26)

2.94 (0.71,5.19)

0.51 (−2.59,3.86)

−0.94 (−3.58,1.79)

−4.43 (−6.44,−2.24)

−1.28 (−1.74,−0.81)

0.42 (−0.49,1.35)

The models are fitted for log(FEV1) and also adjusted for the field technician who performed the tests and for

the spirometer used.

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Bayesian approach to functional-based multilevel modeling

695

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 (Table 1), 40000 posterior realizations of

the community-specific growth curves were sampled as described in Section 3.3 after a burn-in period

of 10000 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 FEV1. 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 Table 2, with focus on O3, 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 O3, elemental carbon, and acid vapors,

respectively). Hence, we present results for both genders combined. Table 2 indicates that significant

deficits in the maximum rate of growth of FEV1are 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 FEV1of 27.2 and

32.8 mL for 1.2 µg of elemental carbon and 9.6 ppb of acid vapor, respectively. Though not significant, the

contrast of 37.5 ppb in O3levels (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 FEV1.

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

Table 2. Effects of various pollutants on the maximum rate of FEV1growth in participants of the CHS

Pollutant

O3

Elemental carbon

Total acid

Difference (95%CI)

−5.93 (−35.48, 23.62)

−27.24 (−49.94, −4.53)

−32.81 (−58.07, −7.54)

p value

0.682

0.022

0.014

The effects are for comparisons between the least and the worst polluted communities, where the scaling

ranges were 37.5 ppb, 1.2 µg, and 9.6 ppb for O3(averaged from daily 10 AM to 6 PM levels), elemental

carbon, and total acid, respectively.

Page 11

696 K. BERHANE AND N.-T. MOLITOR

of the maximum attained levels of FEV1, 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 FEV1were 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 Table 2.

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 10000 burn-in iterations with 10000 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 δ 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 FEV1of 26.0 and 31.3 mL for

1.2 µg 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 O3was also

essentially identical to that obtained under the IG(0.01,0.01) prior for σ.

5. FURTHER TOPICS

In this section, we discuss several methodologic issues that could arise in fitting the proposed models with

respect to model building, inference, and model checking.

5.1

Meta-analytic inference on functionals

Given estimates of functional(s) of interest from the algorithms outlined in Sections 3.2–3.3, along with

their entire posterior distributions, one could conduct inferences on the relationships between such

functionals and ecologic covariates (e.g. air pollution) by using a meta-analytic mixed-effects model as

outlined in Stram (1996). This approach takes the uncertainty estimates of the functionals into account in

an appropriate manner. In fact, the near equivalence of this approach to a combined mixed-effects model

in the linear setting (as in Model (3.1)) has been discussed previously (see Berhane and others, 2004, for

details).

Let Fcand Pcdenote a functional of interest (e.g. maximum rate of growth) and a level of an ecologic

covariate (e.g. air pollution) for community c, respectively. Let VFcdenote the variance estimate of Fc

as obtained from the Gibbs sampling algorithm. Then, the following meta-analytic mixed-effects model

could be used:

Fc= γ0+ γ1Pc+ ec+ ψc,

(5.1)

where e = (e1,...,eC) ∼ N(0,τ2) and ψ ψ ψ = (ψ1,...,ψC) ∼ N(0,VF). Here, VFis a variance

matrix for the Fcs with VFcs as diagonal elements. Model (5.1) leads to an iterative process that alternates

between updates for (γ0,γ1) and τ2until convergence. It implicitly assumes the normality of the posterior

distribution of the functional of interest. The assumption of normality of the posterior distributions of

the functionals could be tested via standard procedures, and transformation techniques could be used to

achieve normality whenever possible.

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Bayesian approach to functional-based multilevel modeling

697

5.2

Bayesian inference on functionals

When the assumption of normality for the posterior distribution of the functional of interest is untenable,

even after transformation, an alternative Bayesian algorithm could be used. The algorithm involves 3

steps: namely, (i) fit a mixed-effects model (3.9), (ii) for each realization from the MCMC process (as

outlined in Section 3.3), fit the ecologic model

F(g)

c

= γ(g)

0

+ γ(g)

1

Pc+ e(g)

denotes the kth realization of Fcfrom the Gibbs

c ,

(5.2)

where e(g)=(e(g)

sampling process, and (iii) conduct Bayesian inference on the distribution of the parameter estimates γ(g)

and γ(g)

1.

1,...,e(g)

C) ∼ N(0,τ2(g)), and F(g)

c

0

5.3

Selection of knots in fitting growth curves

While natural spline–based models are quite flexible in depicting the nonlinear patterns in lung function,

there is still a need to determine the number and position of the knots. Several approaches have been

proposed to deal with this issue, which is essentially that of model selection. One could use substantive

judgment or a trial-and-error approach to determine the number and position of the knots. It may also be

possible to develop partially or fully automated methods such as BRUTO (Hastie, 1989), but such methods

need to pay attention to the effects of the intrasubject and intracommunity correlation on the procedures.

There is a rich literature on Bayesian approaches to selection of knots in spline-based models. See Biller

(2000), DiMatteo and others (2001), Zhou and Shen (2001), and references therein for details on related

approaches. However, this topic is beyond the scope of this paper.

6. DISCUSSION

We have presented a new method for analyzing the effects of ecologic covariates, in our case air pollu-

tion, on the biologically interesting aspects of nonlinear growth trajectories of physiologic measures. The

method was motivated by a desire to estimate the long-term effects of air pollution on the biologically

important and substantively relevant aspects of lung function growth trajectory. The long-term effects of

air pollution on the linear growth slope (in short follow-up periods) are well established. But effects of

air pollution on the entire growth trajectory (or important aspects of the trajectory) have not been well

studied. The methods presented in this paper are intended to bridge this gap.

While the immediate application of the methods is to environmental epidemiology, the methods could

also be applied in other settings. The application to air pollution described in Section 4, and hence the

methodology, could be extended in several ways, for example to enable more thorough assessment of the

multipollutant problem, measurement error in the exposure data, and simultaneous modeling of several

outcomes.

FUNDING

California Air Resources Board (A033-186), National Institute of Environmental Health Sciences

(5P30ES07048-04, 1PO1ESO939581-01), and US Environmental Protection Agency (CR824034-01-3,

R826708-01-0).

ACKNOWLEDGMENTS

We thank Duncan Thomas, Jim Gauderman, John Peters, Frank Gilliland, Rob McConnell, Ed Avol,

Fred Lurmann, and members of the external advisory committee (especially Scott Zeger) for valuable

discussions. Conflicts of Interest: None declared.

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698 K. BERHANE AND N.-T. MOLITOR

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[Received 31 January 2007; revised 7 November 2007; accepted for publication 17 December 2007]