Childhood body mass index trajectories: modeling, characterizing, pairwise correlations and sociodemographic predictors of trajectory characteristics
 Xiaozhong Wen^{1}Email author,
 Ken Kleinman^{1},
 Matthew W Gillman^{1},
 Sheryl L RifasShiman^{1} and
 Elsie M Taveras^{1}
DOI: 10.1186/147122881238
© Wen et al; licensee BioMed Central Ltd. 2012
Received: 3 August 2011
Accepted: 29 March 2012
Published: 29 March 2012
Abstract
Background
Modeling childhood body mass index (BMI) trajectories, versus estimating change in BMI between specific ages, may improve prediction of later bodysizerelated outcomes. Prior studies of BMI trajectories are limited by restricted age periods and insufficient use of trajectory information.
Methods
Among 3,289 children seen at 81,550 pediatric wellchild visits from infancy to 18 years between 1980 and 2008, we fit individual BMI trajectories using mixed effect models with fractional polynomial functions. From each child's fitted trajectory, we estimated age and BMI at infancy peak and adiposity rebound, and velocity and area under curve between 1 week, infancy peak, adiposity rebound, and 18 years.
Results
Among boys, mean (SD) ages at infancy BMI peak and adiposity rebound were 7.2 (0.9) and 49.2 (11.9) months, respectively. Among girls, mean (SD) ages at infancy BMI peak and adiposity rebound were 7.4 (1.1) and 46.8 (11.0) months, respectively. Ages at infancy peak and adiposity rebound were weakly inversely correlated (r = 0.09). BMI at infancy peak and adiposity rebound were positively correlated (r = 0.76). Blacks had earlier adiposity rebound and greater velocity from adiposity rebound to 18 years of age than whites. Higher birth weight zscore predicted earlier adiposity rebound and higher BMI at infancy peak and adiposity rebound. BMI trajectories did not differ by birth year or type of health insurance, after adjusting for other sociodemographics and birth weight zscore.
Conclusions
Childhood BMI trajectory characteristics are informative in describing childhood body mass changes and can be estimated conveniently. Future research should evaluate associations of these novel BMI trajectory characteristics with adult outcomes.
Background
Childhood body mass index (BMI) predicts adulthood obesity [1, 2] and other longterm health outcomes [3–5]. But previous studies have observed weak or moderate correlations (r = 0.20.5) between early childhood (< 7 years of age) and adulthood BMI [6, 7]. Most of these studies [2, 8–10] have used BMI at fixed ages or change in BMI between fixed ages as predictors. This fixedage approach assumes that individuals in the sample belong to a homogeneous group with similar developmental patterns, which seems unrealistic for childhood BMI [11]. Also, the biological meaning of childhood BMI at a given fixed age may differ among children who have different growth patterns (initiation, velocity, duration, etc.) in bone, muscle, and fat tissues. Instead, a more appealing way of examining childhood BMI is to model individual trajectories based on repeated BMI measures throughout childhood. The capacity of childhood BMI to predict adult BMI can potentially be improved by using a child's BMI trajectory, in addition to or in place of his or her BMI at specific ages.
Individual and groupbased approaches are the two distinct methods for studying childhood BMI trajectories in the literature. The groupbased approach tries to generate several groups or classes that share overall patterns of changes in BMI [12], BMI zscore [13], or risk of high BMI [11] across childhood, using methods such as latent growth mixture modeling. Despite its simplicity in summarizing overall patterns, the groupbased approach requires the investigator's subjective decisions on the number of groups, even after optimization by statistical software. It is also subject to arbitrary names or definitions of selected groups, substantial variations in patterns within each group, and unsatisfying generalizability (e.g., the number and patterns of groups often change among new samples). Alternatively, the individualbased approach examines the specific trajectory for each child and then estimates informative BMI characteristics, and thus allows for further links to individualspecific exposures or health outcomes. For example, from individualspecific trajectories, one can identify BMI milestones including infancy peak and adiposity rebound [14–17], and also estimate some novel features of BMI change, such as velocity and the area under a BMI trajectory curve. Modeling childhood BMI trajectory may reveal stronger ties between childhood and adulthood BMI, leading to a better rationale for childhood interventions to prevent obesity and other health outcomes in adulthood. However, previous studies using the individualbased approach are limited by restricted age periods, such as from birth to 3 years [14] or from 2 to 18 years [17, 18]. Consequently, the full picture on correlations between BMI milestones throughout childhood remains unclear [14], as does their independent and interactive impacts on longterm outcomes.
Our aims are: 1) to build parametric models to fit BMI trajectory throughout childhood; 2) to estimate BMI trajectory milestones and related characteristics; and 3) to examine pairwise correlations and sociodemographic predictors of BMI trajectory characteristics.
Methods
Study sample
As part of the Collecting Electronic Nutrition Trajectory Data Using eRecords of Youth (CENTURY) Study, we extracted length/height, weight, and demographic data from electronic medical records of wellchild visits from 1980 through 2008 at Harvard Vanguard Medical Associates (HVMA), a multisite group practice in eastern Massachusetts. Details of the data collection methods can be found elsewhere [19]. The study protocol was approved by the Institutional Review Board of Harvard Pilgrim Health Care.
Inclusion criteria
Characteristics of the analytic and excluded ageeligible sample born between October 1, 1979 and June 30, 1994
Characteristic  Analytic sample  Excluded sample 

Childlevel  
Total # of children  3289  139057 
Sex, n (%)  
Boys  1680 (51.1)  70216 (50.5) 
Girls  1609 (48.9)  68841 (49.5) 
Race/ethnicity, n (%)  
White  2362 (71.8)  59644 (42.9) 
Black  214 (6.5)  14341 (10.3) 
Other  168 (5.1)  7847 (5.6) 
Unknown  503 (15.3)  52443 (37.7) 
Year of birth, n (%)  
1979 ~ 1984  382 (11.6)  38343 (27.6) 
1985 ~ 1989  1315 (40.0)  48075 (34.6) 
1990 ~ 1994  1592 (48.4)  52639 (37.9) 
Birth weight in grams, mean (SD)  3442 (488)  3433 (507) 
Type of health insurance, %  
Medicaid  129 (3.9)  7,256 (5.2) 
NonMedicaid  3160 (96.1)  131801 (94.8) 
Visitlevel  
Total # of visits  81550  993687 
Age at visit (years), n (%)  
0 ~ 1  25188 (30.9)  294540 (29.6) 
2 ~ 5  17501 (21.5)  215744 (21.7) 
6 ~ 10  16681 (20.5)  175422 (17.7) 
11 ~ 14  12974 (15.9)  164369 (16.5) 
15 ~ 18  9206 (11.3)  143612 (14.5) 
Measures
At wellchild visits, medical assistants measured children's weight and length/height according to the written protocol of HVMA. Anthropometric equipment is calibrated annually at HVMA, and a master trainer conducts periodic quality checks of anthropometric measures by medical assistants. Using pediatric scales, medical assistants measured weight without heavy clothes and shoes, and rounded it to the nearest 0.25 pound (0.11 kg). Although the position for length measure was not documented in medical records, medical assistants usually measured length without shoes in recumbent position using a paperandpencil technique (see below) for children younger than 24 months, and height without shoes in standing position for those aged 24 months or older [22].
Briefly, for the paperandpencil technique, the child lay supine on a piece of paper atop an examination table. The medical assistant drew a tick mark abutting the top of the child's head, and then straightened the child's legs, flattened the child's knees, flexed the child's foot to be perpendicular to the table, and marked the paper again at the bottom of the child's heels. The medical assistant then measured the distance between the two marks with a flexible tape, and rounded it to the nearest quarter inch. However, in our previous validation study among 0 to 24 monthold infants conducted at one of the participating pediatric practice sites, we found that the paperandpencil method systematically overestimated children's length compared with a reference method [22]. We converted our paperandpencil lengths to 0.953 × length measured by paperandpencil method + 1.8 cm, as estimated in the validation study [22]. We applied this regression correction for all children younger than 24 months, and recognize that this universal correction might artificially introduce some errors in a small number of children who were measured in standing position before 24 months. We calculated BMI as, weight in kilograms/(height or length in meters)^{2}.
We extracted children's race/ethnicity from medical records, and then recoded it as nonHispanic white, nonHispanic black, or other race/ethnicity including Hispanic, Asian American, Native American, Alaskan Native, and Native Hawaiian or other Pacific Islander. We calculated internal zscore of birth weight as, (individual birth weight  mean value)/standard deviation, for boys and girls separately within the analytic sample. The type of health insurance, Medicaid vs. nonMedicaid, was retrieved from medical records.
Statistical analysis
We chose ages 3 months, 6 months, 1 year, 3 years, 4 years, 7 years, 11 years, and 18 years to check the normality of agespecific BMI distribution. QQ plots and KolmogorovSmirnov tests showed that BMI was approximately normally distributed at most of these age points, except for some right skewness at 18 years of age (skewness, 0.86 for boys and 0.90 for girls). So the normality assumption for agespecific BMI distribution is fairly acceptable in this sample.
We performed the main data analysis in three steps: modeling BMI trajectory, estimating trajectory characteristics, and examining correlations and predictors of trajectory characteristics. Given the wellknown sex differences [23] in childhood growth, we conducted steps 1 and 2 among boys and girls separately.
Step I
Mixed effect models with the best fractional polynomial function for childhood BMI trajectory, by model degree^{a}
Degree  No. of candidate models  Included age terms in the mixed effect model with the best fractional polynomial function  Goodness of fit (smaller is better)^{b}  

Age^{(2)}  Age^{(1)}  Age^{(0.5)}  log(Age)  Age^{0.5}  Age  Age^{2}  Age^{3}  2 Log likelihood  BIC  
Boys (N = 1,680)  
3 rd degree  56  ×  ×  ×  150196  150218  
4th degree  70  ×  ×  ×  ×  149688  149710  
5th degree  56  ×  ×  ×  ×  ×  147836  147858  
6th degree  28  ×  ×  ×  ×  ×  ×  148889  148911  
7th degree  8  ×  ×  ×  ×  ×  ×  ×  161668  161690  
8th degree  1  ×  ×  ×  ×  ×  ×  ×  ×  166173  166181 
Girls (N = 1,609)  
3 rd degree  56  ×  ×  ×  141787  141809  
4th degree  70  ×  ×  ×  ×  139990  140012  
5th degree  56  ×  ×  ×  ×  ×  138131  138153  
6th degree  28  ×  ×  ×  ×  ×  ×  140079  140101  
7th degree  8  ×  ×  ×  ×  ×  ×  ×  152402  152424  
8th degree  1  ×  ×  ×  ×  ×  ×  ×  ×  156241  156248 
We fit BMI trajectories with mixed effect models [26], specifying fixed effects of each fractional polynomial term, reflecting the populationaverage trend, and random effects of each term per child, modeling the deviation of each child from the populationaverage. We applied a twostage method [27] to select optimal mean and residual variancecovariance structures: first we used the most complex mean structure (m = 8, the model with all 8 candidate powers) to select the best variancecovariance structure from 8 candidates (autoregressive, spatial power, compound symmetry, heterogeneous, toeplitz, heterogeneous toeplitz, unstructured, and variance components); and then fixed this best variancecovariance structure to select the best mean structure from the 219 candidate models mentioned above. We used the Bayesian information criterion (BIC) [28] to make this selection.
We calculated individualspecific BMI trajectories by combining the estimated fixed effects, which are shared by all subjects within sex, with the predicted random effects, which are specific to each individual. This results in a unique predicted trajectory for each subject. To assess the goodness of fit for each individual BMI trajectory, we first calculated the residual between the observed BMI and the estimated individualspecific BMI trajectory, and then used these residuals to calculate the residual BMI variance for each child (note that a smaller value implies a better fit).
Step II
Based on the reported means and standard deviations (SD) of BMI trajectory milestones, or turning points on BMI curves, in the existing literature [14, 17], we defined their hypothetical age intervals as within 3 SD of from the mean: 3 to 17 months for infancy peak and 15 months to 9.5 years for adiposity rebound. Because of the relatively small sample size in previous studies, we combined both sexes for these age intervals, to assure a large probability of identifying plausible BMI milestones. Then we divided age from 1 week to 18 years into 8,632 evenly spaced "minor" points 0.025 months (about 1 day) apart. We then estimated the velocity at each of these points by taking the first derivative of the individualspecific BMI trajectory curve. The criteria for existence of a milestone within the corresponding age interval were that two consecutive minor age points had opposite signs of the first derivative [14]: for infancy peak, the derivative at minor point k > 0 and point k + 1 < 0; for adiposity rebound, derivative at k < 0 and at k + 1 > 0. Within each pair of consecutive ages meeting the criteria above, the minor point with derivative closer to zero was designated the age at the milestone. Note that some children did not have both BMI milestones: infancy peak did not exist for 2 girls, while adiposity rebound did not exist for 37 boys and 62 girls. This occurs when the individualspecific curves lack a local maximum (infancy peak) or a local minimum (adiposity rebound) in the specified age ranges.
The predicted BMI (i.e., the point on the curve) at the minor age point identified is the basis for our BMI trajectory measures. We calculated the linear BMI velocity (defined as 'difference in BMI/difference in age') for three time periods: between 1 week of age and infancy peak, between infancy peak and adiposity rebound, and between adiposity rebound and 18 years of age. If BMI values at 1 week and 18 years of age were not observed at wellchild visits, they were estimated from the fit individualspecific BMI trajectory models instead. The area under curve was estimated as the definite integral between the two age points. The SAS code used in Step II is available upon request.
Step III
We calculated pairwise Pearson correlation among pairs of BMI trajectory characteristics. Multivariable linear regression was used to examine predictors of the BMI trajectory characteristics; predictors included the child's sex, race/ethnicity, year of birth, zscore of birth weight, and the type of health insurance. Modeling was performed within a subsample with complete data on all these predictors.
Results
Sample characteristics
Table 1 shows characteristics of the analytic sample. Among the 3,289 children, 51.1% were boys; 71.8% nonHispanic whites, 6.5% nonHispanic blacks, 5.1% other race/ethnicity and 15.3% unknown race/ethnicity; 48.4% were born after 1990; the mean number of visits was 25 (range, 18 to 93). Among the total of 81,550 visits, over half occurred before 6 years of age.
Models for BMI trajectory
BMI trajectory characteristics and their correlations
Means and medians of childhood BMI trajectory characteristics, by sex
Boys (N = 1,680)  Girls (N = 1,609)  

BMI trajectory characteristics  n^{a}  Mean (SD)  Median (range)  n^{b}  Mean (SD)  Median (range) 
1 week to infancy peak  
Age at infancy peak, months  1680  7.2 (0.9)  7.1 (3.9, 12.5)  1607  7.4 (1.1)  7.3 (3.3, 14.2) 
BMI at infancy peak, kg/m^{2}  1680  17.8 (0.9)  17.7 (15.0, 20.6)  1607  17.3 (0.9)  17.2 (14.5, 20.3) 
Change in BMI, kg/m^{2}  1680  10.7 (5.4)  10.7 (8.2, 30.8)  1607  8.5 (5.1)  8.2 (12.6, 28.1) 
Velocity, kg/m^{2}/month  1680  1.58 (0.82)  1.57 (1.31, 5.10)  1607  1.21 (0.75)  1.15 (1.50, 5.65) 
Area under curve (kg/m^{2}months)  1680  114 (18)  112 (60, 207)  1607  116 (21)  113 (45, 242) 
Infancy peak to adiposity rebound  
Age at adiposity rebound, months  1643  49.2 (11.9)  50.0 (24.0, 84.2)  1547  46.8 (11.0)  47.1 (24.1, 85.3) 
BMI at adiposity rebound, kg/m^{2}  1643  15.6 (1.3)  15.5 (11.9, 19.9)  1547  15.5 (1.2)  15.4 (11.8, 19.4) 
Age difference, months  1643  42.0 (12.0)  42.8 (14.2, 76.7)  1547  39.4 (11.2)  40.0 (14.0, 77.0) 
Change in BMI, kg/m^{2}  1643  2.2 (0.8)  2.2 (4.7, 0.2)  1547  1.8 (0.7)  1.9 (4.0, 0.2) 
Velocity, kg/m^{2}/month  1643  0.05 (0.01)  0.05 (0.08, 0.01)  1547  0.04 (0.01)  0.05 (0.08, 0.01) 
Area under curve (kg/m^{2}months)  1643  680 (182)  690 (251, 1332)  1547  629 (170)  633 (239, 1301) 
Adiposity rebound to age 18 years  
Change in BMI, kg/m^{2}  1643  8.3 (3.0)  7.5 (2.9, 20.0)  1547  8.1 (2.6)  7.5 (2.6, 18.8) 
Velocity, kg/m^{2}/month  1643  0.02 (0.01)  0.02 (0.00, 0.05)  1547  0.02 (0.01)  0.02 (0.00, 0.05) 
Area under curve (kg/m^{2}months)  1643  3206 (590)  3115 (2010, 4985)  1547  3213 (534)  3139 (2008, 4725) 
Correlation matrix of childhood BMI trajectory characteristics (N = 3,289)
1 week to infancy peak  Infancy peak to adiposity rebound  Adiposity rebound to age 18 years  

Parameters  1  2  3  4  5  6  7  8  9  10  11  12  13  
1 week to infancy peak  
1  Age at infancy peak, months  
2  BMI at infancy peak, kg/m^{2}  0.39  
3  Change in BMI, kg/m^{2}  0.03  0.45  
4  Velocity, kg/m^{2}/month  0.25  0.34  0.96  
5  Area under curve (kg/m^{2}months)  0.96  0.63  0.07  0.14  
Infancy peak to adiposity rebound  
6  Age at adiposity rebound, months  0.09  0.15  0.12  0.11  0.08  
7  BMI at adiposity rebound, kg/m^{2}  0.59  0.76  0.26  0.13  0.76  0.48  
8  Age difference, months  0.17  0.12  0.13  0.13  0.16  0.99  0.53  
9  Change in BMI, kg/m^{2}  0.46  0.01  0.13  0.20  0.42  0.91  0.66  0.94  
10  Velocity, kg/m^{2}/month  0.84  0.23  0.10  0.27  0.79  0.54  0.71  0.61  0.82  
11  Area under curve (kg/m^{2}months)  0.06  0.33  0.21  0.19  0.01  0.98  0.32  0.97  0.87  0.51  
Adiposity rebound to age 18 years  
12  Change in BMI, kg/m^{2}  0.10  0.18  0.07  0.01  0.09  0.94  0.38  0.93  0.79  0.38  0.93  
13  Velocity, kg/m^{2}/month  0.08  0.18  0.07  0.02  0.07  0.95  0.39  0.93  0.80  0.40  0.93  0.99  
14  Area under curve (kg/m^{2}months)  0.22  0.16  0.03  0.00  0.28  0.94  0.71  0.95  0.91  0.64  0.87  0.91  0.92 
Predictors of BMI trajectory characteristics
Predictors of BMI trajectory characteristics, from multivariable linear regression models that include all covariates in the table
Mean difference in the BMI trajectory characteristic (95% confidence interval)  

Girls(vs boys)  Race/ethnicity (vs white)  Year of birth (vs 1979 ~ 1984)  Zscore of birth weight  Medicaid (vs nonMedicaid)  
BMI trajectory characteristics  n^{a}  Black  Other  1985 ~ 1989  1990 ~ 1994  
1 week to infancy peak  
Age at infancy peak, months  2128  0.2 (0.2, 0.3)  0.1 (0.2, 0.1)  0.1 (0.3, 0.0)  0.0 (0.2, 0.1)  0.0 (0.1, 0.2)  0.0 (0.0, 0.1)  0.1 (0.1, 0.4) 
BMI at infancy peak, kg/m^{2}  2128  0.5 (0.6, 0.4)  0.0 (0.2, 0.1)  0.0 (0.1, 0.2)  0.2 (0.0, 0.3)  0.1 (0.0, 0.3)  0.2 (0.2, 0.3)  0.1 (0.1, 0.3) 
Change in BMI, kg/m^{2}  2128  2.2 (2.6, 1.7)  0.1 (0.9, 0.6)  0.3 (0.5, 1.0)  0.7 (0.1, 1.4)  0.2 (0.5, 0.9)  1.4 (1.7, 1.2)  0.3 (1.4, 0.8) 
Velocity, 10^{2} kg/m^{2}/month  2128  35.4 (41.7, 29.1)  0.5 (12.6, 11.5)  7.1 (4.9, 19.1)  12.9 (2.7, 23.2)  3.1 (7.1, 13.2)  21.4 (24.6, 18.2)  7.1 (23.8, 9.5) 
Area under curve (kg/m^{2}months)  2128  1 (0, 3)  0 (3, 3)  2 (5, 1)  1 (2, 3)  2 (1, 4)  3 (2, 3)  3 (1, 7) 
Infancy peak to adiposity rebound  
Age at adiposity rebound, months  2063  2.1 (3.0, 1.1)  3.3 (5.3, 1.3)  1.6 (3.5, 0.3)  0.1 (1.5, 1.6)  0.2 (1.8, 1.4)  0.6 (1.1, 0.1)  1.9 (4.5, 0.8) 
BMI at adiposity rebound, kg/m^{2}  2063  0.1 (0.2, 0.0)  0.2 (0.0, 0.4)  0.0 (0.2, 0.2)  0.2 (0.0, 0.3)  0.2 (0.0, 0.3)  0.3 (0.3, 0.4)  0.3 (0.0, 0.6) 
Age difference, months  2063  2.3 (3.3, 1.3)  3.2 (5.2, 1.2)  1.4 (3.4, 0.5)  0.1 (1.5, 1.7)  0.2 (1.8, 1.4)  0.6 (1.1, 0.1)  2.0 (4.7, 0.7) 
Change in BMI, kg/m^{2}  2063  0.3 (0.3, 0.4)  0.2 (0.0, 0.3)  0.0 (0.1, 0.2)  0.0 (0.1, 0.1)  0.0 (0.1, 0.1)  0.1 (0.0, 0.1)  0.2 (0.0, 0.4) 
Velocity, 10^{2} kg/m^{2}/month  2063  0.6 (0.5, 0.7)  0.1 (0.1, 0.2)  0.1 (0.2, 0.1)  0.0 (0.2, 0.1)  0.0 (0.1, 0.2)  0.1 (0.0, 0.1)  0.2 (0.0, 0.4) 
Area under curve (kg/m^{2}months)  2063  46 (61, 31)  48 (78, 18)  25 (53, 4)  7 (17, 31)  1 (23, 25)  1 (7, 9)  24 (65, 16) 
Adiposity rebound to age 18 years  
Change in BMI, kg/m^{2}  2063  0.3 (0.5, 0.0)  0.9 (0.4, 1.4)  0.4 (0.0, 0.9)  0.1 (0.3, 0.5)  0.1 (0.3, 0.5)  0.1 (0.0, 0.2)  0.4 (0.2, 1.1) 
Velocity, 10^{2} kg/m^{2}/month  2063  0.1 (0.1, 0.0)  0.3 (0.1, 0.4)  0.1 (0.0, 0.3)  0.0 (0.1, 0.1)  0.0 (0.1, 0.1)  0.0 (0.0, 0.1)  0.1 (0.1, 0.3) 
Area under curve (kg/m^{2}months)  2063  6 (55, 42)  162 (65, 258)  70 (23, 163)  32 (47, 110)  44 (34, 121)  70 (45, 94)  115 (15, 246) 
Discussion
Using repeated growth measures from wellchild visits, we fit childhood BMI trajectory from 1 week to 18 years of age and estimated BMI trajectory milestones and related characteristics. The majority of BMI trajectory characteristics were correlated with each other. Some BMI trajectory characteristics, including age and BMI at infancy peak and adiposity rebound, varied substantially by children's sex, race/ethnicity, and zscore of birth weight, but there was little evidence of cohort effects.
BMI trajectory characteristics
We were able to estimate infancy BMI peak and adiposity rebound for most children. To the best of our knowledge, the present study is the first one to propose the periodspecific AUC to characterize childhood BMI trajectory. We think this novel measure can reflect the child's cumulative "exposure" to excessive body weight; and its potential role in predicting later obesity and obesityrelated diseases warrants further research.
One important but unanswered question in BMI trajectory literature is the extent of correlations among BMI trajectory milestones [14]. Our analysis showed that the majority of BMI trajectory characteristics were moderately or strongly correlated with each other. These correlations may be driven by 2 distinct biological forces. First, human growth is an inherently continuous process: the higher BMI is at infancy peak, the higher it will be at adiposity rebound. Second, the force of 'regression to mean' inhibits too extreme growth: the greater the velocity from 1 week to infancy peak, the lower the velocity from infancy peak to adiposity rebound. This multicollinearity can pose a challenge for separating the independent effects of these BMI trajectory characteristics on adult outcomes. However, the magnitude of correlations between BMI trajectory characteristics estimated in our study should be interpreted cautiously, because we did not observe the characteristics directly, but estimated these characteristics from the same fitted BMI trajectory.
In our cohort, boys and girls had different BMI trajectories and bestfitting models. In line with a previous study [14] and CDC 2000 growth charts, we found that girls were older and had lower BMI at infancy peak, and earlier adiposity rebound. These sex differences may be explained by genetics, growth or sexual hormones, diet, or physical activity levels. One of our novel findings is the racial/ethnicdifferences in BMI trajectory characteristics. Compared to their white peers, nonHispanic black children had BMI trajectory profiles that may be associated with higher risk of later obesity, including younger age at adiposity rebound [17], and larger velocity and greater AUC from adiposity rebound to 18 years of age. However, these racial differences should be interpreted with caution, given insufficient control of socioeconomic status other than the type of health insurance. Consistent with the literature [14], we found that birth weight was a strong predictor for most BMI trajectory characteristics. Overall there were no substantial changes in BMI trajectory characteristics with year of birth, after controlling for other sociodemographics and zscore of birth weight. This suggests that childhood BMI trajectory was fairly stable across the analyzed years in our cohort.
Modeling childhood BMI trajectory
Generally, there are two broad types of methods to estimate childhood BMI trajectory milestones: visualization and modeling [29]. Simple visualization was first used in early studies to determine adiposity rebound as the visual nadir or the point with the lowest BMI [30–32]. Although straightforward and convenient, the age at adiposity rebound estimated by simple visualization is quite arbitrary, especially for children with a flat valley around the nadir, and thus subject to large interobserver variation.
Instead, several recent studies [14, 17, 18, 33–36] have used statistical modeling to identify BMI trajectory milestones more objectively. Commonly, researchers select reasonable combinations of polynomial age terms to fit ordinary regression models within each child [17, 18, 35], or mixed effect models [14, 33, 34] among a group of children. Ordinary regression models require many data points for each child; their estimates are unbiased, but are often subject to large variability. In contrast, mixed effect models need fewer data points for each child and yield more stable estimates, although the estimates may be a little biased, especially for those with very few data points. A study comparing simple regression with mixed effect model for the same sample [36] found estimated BMI values at adiposity rebound were similar between them but estimated ages at adiposity rebound differed.
One common limitation of the existing studies [14, 17] is that they only modeled a segment of childhood. Our novel contribution is developing a good parametric model for BMI trajectory throughout childhood, from 1 week to 18 years of age. Alternatively, some researchers use semiparametric modeling [14, 37], such as cubic and linear spline models, to fit childhood BMI trajectory. Cubic spline models are more flexible and thus may fit the data better than our fractional polynomial models, but they require arbitrary decisions on the number and locations of age 'knots', carry the potential for undesirable multiple infancy peaks and adiposity rebound points, and have limited generablizability of their fitted models due to heavy datadependence [38, 39]. Taken together, all current methods have both advantages and disadvantages. Our method can meet the high need of accurate milestone estimates and is flexible for various study populations and data structures, including missing data and nonfixed age of followups; but it requires a large enough sample to build stable mixed effect models and strong statistical skills. We also note that, although the overall bestfitting fractional polynomial function for the total sample is not necessarily optimal for each individual, it is robust and appropriate especially for those children with only a few repeated BMI measures.
Study strengths
Our study has several strengths. First, the large original dataset yielded a large analytic sample that met our strict eligibility criteria. Second, the small individuallevel residual BMI variance supported the applicability of our selected fractional models for most children. Third, our methods can help researchers estimate novel BMI trajectory characteristics conveniently with common statistical software (e.g. SAS, R, and STATA). As a next step, we plan to develop userfriendly software to make our modeling and estimating process more convenient for general researchers and clinicians.
Study limitations
Our study also had several limitations. One limitation is the quality of the clinical weight and height measures, although the use of a written protocol, annual scale calibration, periodic quality assurance, and mathematical correction for error in length measures under 2 years of age likely reduced measurement errors. In addition, we included only a small proportion of the total sample in the final analysis, and this sample seemed to differ from the excluded sample in race/ethnicity and type of health insurance. The overrepresentation of white children in the analytic sample makes our estimated BMI trajectory characteristics and possibly the bestfitting models less generalizable to racial/ethnic minorities. Our study population was from one multisite pediatric practice in eastern Massachusetts. We did not validate our bestfitting models in an external population. Thus our bestfitting models and estimated means and SD for BMI trajectory characteristics may not be generalizable to other populations. But our methods for modeling childhood BMI trajectory and estimating BMI trajectory characteristics can be broadly used in other studies. Therefore, we recommend other researchers first select the bestfitting models for BMI trajectories in their own samples, and then estimate the corresponding BMI trajectory characteristics, rather than use our bestfitting model and estimated coefficients. Finally, our estimated associations between BMI trajectory characteristics and their predictors from multivariable regression models might be biased, as we did not adjust for some important potential confounders, such as parents' weight and height as well as family socioeconomic status (except the type of child health insurance).
Conclusions
Our mixed effect models with fractional polynomial functions fit childhood BMI trajectories well for most children seen at wellchild visits in this sample. Using our method, one can conveniently estimate BMI trajectory milestones and related characteristics with reasonable accuracy. Future research should evaluate the independent and interactive roles of these novel BMI characteristics on later outcomes. Moreover, prenatal and earlylife determinants of these BMI trajectory characteristics also warrant further investigation.
Abbreviations
 BMI:

Body mass index
 CI:

Confidence interval
 SD:

Standard deviation
 CENTURY:

Collecting Electronic Nutrition Trajectory Data Using eRecords of Youth
 HVMA:

Harvard Vanguard Medical Associates
 AUC:

Area under curve.
Declarations
Acknowledgements
This work was supported in part by The Centers for Disease Control and Prevention, the National Center for Chronic Disease Prevention and Health Promotion (NCCDPHP) (Contract No. 2002008M26882). This work is solely the responsibility of the authors and does not represent official views of The CDC. This study was also supported by a grant from the National Center on Minority Health and Health Disparities (MD 003963) and a grant from the National Institutes of Health (K24 HL 68041).
Authors’ Affiliations
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