ERIC ED609907: Impact of Adolescent Obesity on Middle-Age Health of Women Given Data MAR

2020·0.46 MB·English

by  ERIC

Impact of Adolescent Obesity on Middle-Age Health of
Women Given Data MAR
Yongyun Shin, Shumei S. Sun and Dipankar Bandyopadhyay
Biometrical Journal, v62 n7 p1702-1716, 2020
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Abstract
We analyze adolescent BMI and middle-age systolic blood pressure (SBP) repeat-
edly measured on women enrolled in the Fels longitudinal study (FLS) between 1929
and 2010 to address three questions: Do adolescent-specific growth rates in BMI and
menarche affect middle-age SBP? Do they moderate the aging effect on middle-age
SBP? Have the effects changed over historical time? To address the questions, we
propose analyzing a growth curve model (GCM) that controls for age, birth-year co-
hort and historical time. However, several complications in the data make the GCM
analysis non-standard. First, the person-specific adolescent BMI and middle-age SBP
trajectories are unobservable. Second, missing data are substantial on BMI, SBP and
menarche. Finally, modeling the latent trajectories for BMI and SBP, repeatedly mea-
sured on two distinct sets of unbalanced time points, are computationally intensive.
We adopt a bivariate GCM for BMI and SBP with correlated random coefficients. To
efficiently handle missing values of BMI, SBP and menarche assumed missing at ran-
dom, weestimatetheirjointdistributionbymaximumlikelihoodviatheEMalgorithm
where the correlated random coefficients and menarche are multivariate normal. The
estimated distribution will be transformed to the desired GCM for SBP that includes
the random coefficients of BMI and menarche as covariates. We demonstrate unbiased
estimation by simulation. We find that adolescent growth rates in BMI and menarche
are positively associated with and moderate the aging effect on SBP in middle age,
controlling for age, cohort and historical time, but the effect sizes are at most modest.
The aging effect is significant on SBP, controlling for cohort and historical time, but
not vice versa.
KEY WORDS: longitudinal data; growth curve model; mixed model; hierarchical
model; maximum likelihood.
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1 Introduction
The Fels longitudinal study (FLS) collected the lifetime repeated measurements on growth,
health and body composition of 2,567 participants enrolled in yearly cohorts of 20 to 35 from
1929 to 2010. The participants were scheduled to be examined every six months for the first
18 years and every two years for the rest of their life spans. Each examination took extensive
anthropometric measurements and recorded a health inventory (Sun et al. 2007, 2008). Via
a growth curve model (GCM) with person-specific growth trajectories, also known as a hier-
archical, multilevel, random coefficients or linear mixed model (Raudenbush and Bryk 2002;
Goldstein 2003), we link growth in adolescent BMI and person-specific (age at) menarche
to the course of middle-age systolic blood pressure (SBP) for 835 female participants where
459 women were enrolled at birth producing 85% of the repeated measurements while others
were family members, spouses and relatives. In this paper, we consider BMI and SBP as the
biomarkers of obesity and health, respectively.
Figure 1 plots all observed BMI and SBP of the participants against age where 5,694
adolescent BMIs and 1,196 middle-age SBPs are nested within 552 and 436 individuals, re-
spectively. It superimposes the scatterplots with person-specific longitudinal spaghetti plots.
BMI and SBP are standardized to have mean 0 and variance 1. Every participant has at
least one observation in adolescence or middle age. The spaghetti plots reveal that some
growth patterns in BMI are higher overall, accelerate earlier or decelerate earlier than others.
We analyze the impact of the growth patterns on progression of middle-age SBP. Because
the 835 female participants consist of 98% European Americans and a tiny fraction of mi-
nority participants including African American, Asian, multiracial and other individuals and
because the minority effect is not significant on either BMI or SBP outcome in preliminary
analysis, we do not consider the race covariate.
The examination schedule implies that those enrolled at age 10 or earlier can have up
to 18 adolescent measurements between 10 and 19 years of age and up to 10 middle-age
measurements between 45 and 65 years of age, depending on how old they are as of 2010.
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Some relatives, spouses and family members of the participants enrolled in the FLS during
adolescence or later produced their measurements afterward. It is, however, unreasonable to
take all these repeated measurements as complete data for analysis because only 18% of the
835 participants have at least one measurement in both adolescence and middle age. Instead,
we realistically take the measurements at the time points of actual visits to a clinic for exam-
ination and menarche as complete data and assume data missing at random (MAR, Rubin
1976). As the spaghetti plots in Figure 1 show, the actual timings of examinations varied,
and BMI appears semi-regularly measured while SBP looks more sparse and unbalanced.
Family relocations due to job transfers or newly acquired jobs, sickness of participants
or family members accompanied to scheduled examinations, or other family emergencies or
situations may have caused participants to miss scheduled examinations or attrite. During
the scheduled examinations, 12% of the adolescents and 18% of adults failed to have their
BMI and SBP measured, respectively, for some reasons. Menarche was either identified at a
scheduled examination or obtained from participants’ recollections, but missing otherwise.
It seems plausible that the missing patterns are not related to missing values themselves.
As a reviewer pointed out, however, it is possible for an adolescent to have a high BMI
worsenedsincethelastexaminationandfailtoparticipateinthenextscheduledexamination
becauseofembarrassmentoranxiety. Astherevieweralsocommented,participantswhohave
adolescent BMI measurements but who have no SBP measurement as adults are likely to be
unhealthy and have high SBP in middle age. The missing BMI and SBP will be associated
withmissingprobabilitytoviolatetheMARassumption. However, becausethemissingrates
on BMI and SBP are modest, because 40% of the female participants having adolescent BMI
measurements are younger than middle age as of 2010, and because most adolescents appear
to have attended most follow-ups in the spaghetti plots, even if some missing BMI and SBP
are NMAR, they will not seriously affect our analysis.
We leverage the FLS data collected longitudinally on individuals enrolled over time to
estimate the age, cohort and historical time effects separately. In large-scale single cohort
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studies such as the Dunedin Longitudinal Study and the Environmental-Risk Longitudinal
Twin Study (Caspi et al. 2016; Moffitt et al. 2011), age and historical time are perfectly
correlated. The confounded effects of age and childhood factors may explain only modest
effect sizes of childhood risk factors on adult health outcomes reported (Felitti et al. 1998;
Roberts et al. 2007; Moffitt et al. 2011; Caspi et al. 2016). On the contrary, age and
cohort are perfectly collinear in a cross-sectional study of multiple birth year cohorts such as
National Growth and Health Studies (Ren and Shin 2016). The multi-cohort FLS enables
us to control for temporal, cohort and historical sources of SBP, and estimate the main
effects of growth rates in adolescent BMI and menarche, and the growth rates-by-age and
menarche-by-age interaction effects on middle-age SBP by a multilevel GCM. These effects
may also change over cohorts and time.
Researchers have studied the impact of longitudinal growth patterns in childhood obesity
onadulthealthortheeffectsofchildhoodcovariatesonalongitudinaladultoutcomeinlinear
and nonlinear mixed models (Eriksson et al. 1999; Law et al. 2002; Ferreira et al. 2005;
Nooyens et al. 2007; Sabo et al. 2012; McLeod et al. 2018; Sabo et al. 2014). Kim et
al. (2016) analyzed FLS longitudinal childhood BMI to predict the person-specific timing
of the BMI rebound, and subsequently analyzed the impact of the timing on a longitudinal
adult cardiac outcome. Sabo et al. (2017) analyzed the FLS childhood BMI to predict child-
specific ages and BMIs at the BMI rebound and maximum BMI growth, and subsequently
estimated the effects of the predicted childhood covariates on longitudinal adulthood blood
pressure outcomes. These studies have assessed the impact of either adult characteristics on
a longitudinal childhood outcome or childhood covariates on a longitudinal adult outcome.
In structural equation models (Bollen 1989), a longitudinal model for the repeated mea-
surements of an outcome nested within individuals may be efficiently estimated by latent
growth modeling where age variables are considered as fixed factor loadings and random co-
efficients are latent variables (Willett and Sayer 1994; MacCallum et al. 1997; Bauer 2003;
Bollenand Curan2006; Preacheretal. 2008; Grimmand Ram2009; Ramand Grimm2015).
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Growth-mixture modeling extends the latent growth model to identify unobserved subpopu-
lations exhibiting different growth trajectories (Wang and Bodner 2007). These approaches
model longitudinal outcomes measured on a common set of time points.
In the joint modeling approach (Gueorguieva 2001; Ivanova et al. 2016), a mixture of
discrete and continuous longitudinal outcomes may be modeled in a joint mixed model. Data
MARinthemodelmaybeimputedbyunivariatesequentialregressionmodels(Raghunathan
etal. 2001), alsoknownasmultipleimputationbyfullyconditionalspecification(vanBuuren
et al. 2006; van Buuren 2011). A multilevel GCM given data MAR may also be efficiently
estimated by maximum likelihood (ML) or Bayesian methods (Liu et al. 2000; Schafer and
Yucel 2002; Goldstein and Browne 2002; Goldstein et al. 2009; Goldstein and Kounali 2009;
Shin and Raudenbush 2007, 2010, 2020; Ren and Shin 2016). These approaches typically
apply to longitudinal outcomes measured at a common set of time points.
In this paper, we estimate a nonstandard multilevel GCM for middle-age SBP that in-
troduces challenges. First, the key covariates are unobservable adolescent-specific growth
rates in BMI that have to be estimated from sample data. Sample average growth rates, for
example, are unreliable measurements of the true growth rates that are known to introduce
bias in the estimated effects of the growth rates (Lu¨dtke et al. 2008; Shin and Rauden-
bush 2010, 2020; Grilli and Rampichini 2011). Furthermore, missing data are substantial
on BMI, SBP and menarche. Finally, longitudinal m BMIs in adolescence and n SBPs in
j j
middle age are measured at two separate sets of unbalanced time points nested within each
person, and subjects may have measurements in either adolescence or middle age, or both.
To handle missing data efficiently, we may express, as a special case of the joint modeling
approach, a bivariate multilevel GCM for BMI and SBP and a linear model for menarche
jointly where correlated person-specific random coefficients and menarche are multivariate
normal and the variance covariance structure is appropriately constrained. Computationally
efficient estimation of the joint model, however, involves derivation of new estimators and
considerable amount of programming in a way that fully leverage the longitudinal structure.
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Our method is tailored to the structure with one set of outcomes repeatedly measured at un-
balanced time points during childhood and another set longitudinally measured at separate
unbalanced time points as an adult within each individual, thereby achieving computational
efficiency.
Viewing BMI, SBP, menarche and random coefficients as complete data, we analyze all
observed data to estimate the joint model efficiently by ML via the EM algorithm (Dempster
et al. 1977; Dempster et al. 1981; Shin and Raudenbush 2007). At convergence, we compute
standard errors by the approximate Fisher score (Hedeker and Gibbons 1994; Raudenbush
et al. 2000; Olsen and Schafer 2001). Subsequently, by the delta method (Casella and Berger
2002), we transform the estimated joint model to the desired GCM for SBP that includes
the random coefficients of adolescent BMI and menarche as covariates. An alternative is
to draw multiple imputation of completed data, including the random coefficients, from the
estimatedjointmodelandestimatethedesiredGCMgiventhemultipleimputation(Shinand
Raudenbush 2007). This approach requires a cumbersome extra step of multiple imputation.
We choose the delta method that demanded less programming than the alternative. We will
demonstrate unbiased estimation by simulation. Our findings may provide important policy
implications to promote adult health based on juvenile obesity history.
AlthoughFLShasnotselectedparticipantswithrespecttofactorsknowntobeassociated
with body composition, health, and other related conditions (Roche 1992), the participants
are far from being randomly assigned to levels of key covariates, BMI growth rates and
menarche. Furthermore, with adolescence far apart from middle age in time, there can
be a number of confounders of the key covariates that we have not considered such as
socioeconomic and environmental covariates unavailable in FLS. Consequently, the effect we
mention in this paper is associational, not causal.
The Fels data set is not publically available. However, the data set analyzed in this
paper will be available from the second author upon reasonable request. The next section
introduces our model. Section 3 explains how to estimate the model and compute standard
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errors given data MAR. Section 4 evaluates the accuracy and precision of the estimation
by simulation. Section 5 presents analysis of the multi-cohort FLS sample data. The final
section discusses the limitations and future extensions of the approach.
2 Model
Following Raudenbush and Bryk (2002), we express the repeated measurements of middle-
age SBP R and adolescent BMI C for adult occasion i and adolescent occasion t within
ij tj
person j in a level-1 model
R = AT β +BT γ +(cid:15) , (cid:15) ∼ N(0,σ ), (1)
ij Rij Rj Rij R1 Rij Rij RR
C = AT β +BT γ +(cid:15) , (cid:15) ∼ N(0,σ ) (2)
tj Ctj Cj Ctj C1 Ctj Cij CC
where AT β is a polynomial in adult age a for vectors A of age terms and β of
Rij Rj ij Rij Rj
person-specific age effects (e.g. AT = [1 a ] and βT = [β β ]), B is a vector of
Rij ij Rj R0j R1j Rij
known covariates (e.g. historical time) having fixed effects γ , AT β is a polynomial in
R1 Ctj Cj
adolescent age a for vectors A of age terms and β of adolescent-specific age effects (e.g.
tj Ctj Cj
AT = [1 a a2 ] and βT = [β β β ]), B is a vector of known covariates having
Ctj tj tj Cj C0j C1j C2j Ctj
fixed effects γ (e.g. a3 and historical time), and occasion-specific random errors (cid:15) and
C1 tj Rij
(cid:15) are independent for i = 1,···,n , t = 1,···,m and j = 1,···,J. With adolescence and
Ctj j j
middle age far apart within each person, it appears reasonable that occasion-specific random
errors (cid:15) and (cid:15) are uncorrelated given the person-specific growth trajectories β and
Rij Ctj Rj
β and time-varying covariates. That is, given the covariates, SBP and BMI are dependent
Cj
on each other only through the dependence between the trajectories.
The aging effects β on SBP vary between individuals according to a level-2 model
Rj
β = Γ U +v , v ∼ N(0,τ) (3)
Rj R0 j j j
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for a matrix Γ of fixed effects; a vector U = [WT YT βT ]T of known covariates W
R0 j 2j 2j Cj 2j
(e.g. cohort), partially observed covariates Y (e.g. menarche) and unobservable growth
2j
rates β in adolescent BMI; and a vector of random effects v independent of random errors
Cj j
and U . Conditional on v and covariates, SBP and BMI are independent. Although the
j j
growth trajectories may also vary differently across subpopulations to produce non-constant
variances of random coefficients, for example, between males and females, we believe that the
constant variance covariance matrix τ is plausible within our analysis of females, controlling
for U . Equations (1) and (3) imply our desired GCM (Raudenbush and Bryk 2002)
j
R = AT Γ U +BT γ +AT v +(cid:15) (4)
ij Rij R0 j Rij R1 Rij j Rij
for a matrix of fixed effects Γ including the effects of aging moderated by U , the fixed
R0 j
effects γ of B , and the random effects v of A independent of (cid:15) .
R1 Rij j Rij Rij
Estimation of GCM (4) is difficult because (R ,C ,Y ) are partially observed and be-
ij tj 2j
cause latent trajectories β are unobservable. We introduce efficient and unbiased estima-
Cj
tion below. For a positive integer n, let I be an n-by-n identity matrix, 1 be a vector of n
n n
unities, a diagonal matrix (cid:76)n A = diag{A ,···,A }, probability density function (pdf)
i=1 i 1 n
f(A) and conditional pdf f(A|B).
3 Efficient Estimation
Forefficientestimationbyallobserveddata,weestimatethejointdistributionof(R ,C ,Y )
ij tj 2j
MAR given known covariates. We view q random coefficients β = [βT βT ]T and p vari-
1j Rj Cj 2
ables Y as level-2 complete data in a model f(Y∗)
2j 2j
Y∗ = X γ +b , b ∼ N(0,T(φ )) (5)
2j 2j 2 j j T
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for Y∗ = [βT YT]T, a matrix of covariates X = diag{X ,X } having fixed effects
2j 1j 2j 2j 21j 22j
     
γ b T T
21 1j 11 12
γ =  , random effects b =   and T(φ ) =   having distinct
2   j   T  
γ b T T
22 2j 21 22
elements φ for X = I ⊗WT and X = I ⊗WT. Each outcome in Y∗ may control
T 21j q 2j 22j p2 2j 2j
for a different subset of W as we do in Section 6. Let θ = [γT φT]T.
2j 2 2 T
We aggregate R = [R ···R ]T and C = [C ···C ]T to view Y = [RT CT]T and
j 1j njj j 1j mjj 1j j j
Y∗ as complete data and observed values of Y = [YT YT]T as observed data for person j,
2j j 1j 2j
and estimate the joint model f(Y ,Y∗) = f(Y |β )f(Y∗) given known covariates by the
1j 2j 1j 1j 2j
EM algorithm. We aggregate Equations (1) and (2) as the level-1 model f(Y |β )
1j 1j
Y = A β +B γ +(cid:15) , (cid:15) ∼ N(0,ψ ) (6)
1j 1j 1j 1j 1 1j 1j j
of person j where A = diag{A ,A }, β = [βT βT ]T, B = diag{B ,B }, γ =
1j Rj Cj 1j Rj Cj 1j Rj Cj 1
[γT γT ]T, (cid:15) = [(cid:15)T (cid:15)T ]T and ψ = diag{I σ ,I σ } for A = [A ···A ]T,
R1 C1 1j Rj Cj j nj RR mj CC Rj R1j Rnjj
A = [A ···A ]T,B = [B ···B ]T,B = [B ···B ]T,(cid:15) = [(cid:15) ···(cid:15) ]T
Cj C1j Cmjj Rj R1j Rnjj Cj C1j Cmjj Rj R1j Rnjj
and (cid:15) = [(cid:15) ···(cid:15) ]. Let θ = [γT σ ]T and θ = [γT σ ]T.
Cj C1j Cmjj R R1 RR C C1 CC
Equations (5) and (6) imply a linear mixed model
Y = X γ +Z b +(cid:15) (7)
j j j j j
         
Y X 0 γ A 0 (cid:15)
1j 1j y1 1j 1j
for Y =  , X =  , γ =  , Z =   and (cid:15) =  
j   j     j   j  
Y 0 X γ 0 I 0
2j 22j 22 p2
where X = [B A X ] and γ = [γT γT ]T. The parameters are θ = (θ ,θ ,θ ).
1j 1j 1j 21j y1 1 21 R C 2
3.1 The EM Algorithm
We view complete data as (Y ,b ,Y ) for individual j, equivalent to ((cid:15) ,b ,b ) = ((cid:15) ,b )
1j 1j 2j 1j 1j 2j 1j j
given known covariates and θ for (cid:15) = Y −A β −B γ and b = Y −X γ . This
1j 1j 1j 1j 1j 1 2j 2j 22j 22
approachsimplifiesexpressionsforthecompletedatalikelihood, E-step, M-stepandstandard
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