The multilevel model has the general form
is a vector of longitudinal responses for all subjects (e.g., systolic blood pressure [SBP] or total cholesterol);
X is the design matrix;
β is a vector of unknown regression coefficients;
Z is a design matrix for between-subject variations;
u is a vector of random deviations between subjects;
and ε is a vector of within-subjects random errors.
The term Xβ is the fixed part of the model and describes the mean response as a function of age and other covariates; Zu and ε constitute the random part of the model. In the Project HeartBeat! analyses, Xβ describes the overall mean trajectory of the response variable, Zu describes the inter-individual variation (Level 2), and ε describes the intra-individual variation among various repeated measurements (Level 1). In the hierarchic design, Level 1 corresponds to repeated measurements of the response variables within subjects, and Level 2 corresponds to the individual subjects.
It was assumed that the random deviations followed multivariate normal distributions, with u~N(0, Ω2) at Level 2 and, independently, ε~N(0, σε2I) at Level 1. For each model, a residual analysis was carried out to detect patterns inconsistent with these assumptions. No major problems were found in this regard, so the assumptions were presumed to be at least approximately satisfied. For large samples (in excess of 5500 for this study), the distribution of the estimates of the fixed coefficients (the β's in the model above) tends to be normal, so the Wald tests, based on the ratio of the estimated parameter to its estimated SE, were used to test the significance of the parameters.
The models used in this study contained terms in the design matrix X
expressing the dependence of the outcome variable on a linear combination of predictors, such as race; gender; the linear, quadratic, and cubic terms in age; and the interactions of these terms. The quadratic and cubic terms in age were necessary to describe the nonlinear trajectories from ages 8 to 18 years.1,2
Age to the nearest day was calculated for each child at each occasion of measurement. These ages were then “centered” by subtracting 12 years (the approximate mean) from the age before fitting the model. Maximum likelihood estimates of the model parameters and the variances and covariances were calculated for each model fitted. From the estimated coefficients, the age- and covariate-adjusted population average over the entire age span were calculated. Because the analysis was intended primarily to describe the trajectory of the response rather than to find the most succinct model, in most instances all the terms were kept in the model when calculating predicted values.
Multilevel models using race (R), gender (G), and age (A) with interaction terms RG, RA, GA, RGA, A2, and A3 as predictor variables were found to fit the data well because these models allow different nonlinear growth curves for each of the four race–gender categories. This analysis allowed an adequate description for both the individual trajectories and the population averages.
All statistical tests were carried out at the 5% level, and all CIs were reported at the 95% level. Because the principal goal of the analysis was description of growth patterns rather than formal hypothesis testing, no adjustment was made for multiple testing.
An example of the typical use of multilevel models in the statistical analysis of the Project HeartBeat! (1991–1995) data is given by constructing a model of SBP as a function of race, gender, and age. Graphic illustrations and interpretations are emphasized in this example. A subset of 358 subjects with a total of 3152 SBP measurements was selected for the example (2008). The number of subjects (observations) for each race–gender subgroup were: 167 (1461) nonblack boys, 150 (1358) nonblack girls, 29 (227) black boys, and 12 (106) black girls.
As a first step in the analysis, the data for SBP were plotted against age for each of the four race–gender groups as shown in . In these plots, measurements for the 358 individual subjects are connected with line segments. These plots were useful in visualizing the overall patterns in the data and anticipating the analytic steps to follow. The upward trend in SBP was apparent in each group, and the vertical scatter of the data points reflects the combined variability of SBP within and between subjects. A simple two-level regression model was fitted to these data. The model was of the form
is the i
th measurement of SBP for the j
Systolic blood pressure (SBP) versus age for four race–gender groups, Project HeartBeat! (1991–1995); for a subsample of 3152 observations on 358 subjects; SBP readings are for individuals, joined by line segments
R and G are indicator variables for race and gender (with interactions);
A is the age at measurement minus 12 years (with quadratic and cubic terms);
β's represent unknown regression coefficients;
u0j is the random deviation in intercept and u1j is the random deviation in the linear term for the jth subject;
εij is the random deviation in the ith measurement for the jth subject;
u0j and u1j~N2(0, Ω2), where Ω2 is a 2 × 2 variance–covariance matrix; and
εij~N1(0, σε2), independently from u0j and u1j.
The first line of the model is the “fixed part” and is similar to a standard multiple regression model. The second line is the “random part” expressing random deviations at the subject level through the u
terms, and at the measurement level through the ε
term. An iterative method was used to find the maximum likelihood estimates of the regression coefficients β
and the variances and covariances Ω2
along with their SEs.6
The results are presented in .
Estimated coefficients and p-values of a two-level model for systolic blood pressure; Project HeartBeat! (1991–1995)a
The p-values are determined by the Wald test, which compares the ratio of the coefficient estimate to its SE with the standard normal distribution. By this test, none of the terms involving R (race) is statistically different from zero at the 0.05 level, and all could be omitted from further consideration, if desired. The predicted values from the model (with all fixed terms retained), with 95% CIs, are presented in .
Predicted values with 95% CIs for systolic blood pressure (SBP) vs age in four race–gender groups, Project HeartBeat! (1991–1995), for a subsample of 358 subjects; model coefficients given in
The trends anticipated from were confirmed by the fitted model. Note that the width of the CIs reflect the precision with which the population average has been estimated. The difference in precision for the curves of the black and nonblack children is attributed mainly to the differing sample sizes.
Standard residual analysis techniques were used to assess the tenability of the distributional assumptions. The plot of standardized within-subject residuals, εij, against age in exhibited no obvious trends with age, suggesting that the Level-1 variance is constant with respect to age. The three age cohorts can be faintly discerned in this plot by noting the greater density of observations around the ages of overlap between cohorts, 11–12 and 14–15 years. The standardized within-subject residuals were also plotted against their normal scores, and the resulting nearly straight line is consistent with the assumption that εij ~N1(0, σε2).
Scatter plot and normal-scores plot of within-subject residuals for systolic blood pressure (SBP) versus age, Project HeartBeat! (1991–1995), for a subsample of 3152 observations on 358 subjects
It was assumed that the distribution of the between-subject residuals u0j for intercept and u1j for age has the bivariate normal distribution N2(0, Ω2), where Ω2 denotes the 2 × 2 variance–covariance matrix. The estimates of these residuals are plotted in . Because the normal score plots appeared quite linear, there was no apparent contradiction to the assumption of normality.
Scatter plot and normal-scores plot of between-subject residuals for systolic blood pressure (SBP) versus age, Project HeartBeat! (1991–1995), for a subsample of 358 subjects
The deviations u0j and u1j were added to the population predictions of , and the result is the subject-specific predictions plotted in . In addition to the overall trends, the pattern of the variability of the subject-specific trajectories about the population M was seen, corresponding to the subject-level random structure of the model.
Subject-specific predictions of systolic blood pressure (SBP) versus age for four race–gender groups, Project HeartBeat! (1991–1995), for a subsample of 358 subjects
In summary, the cubic polynomial model described in the example appeared to provide a useful description of the SBP. Splines or other types of analyses could also have been used, but the cubic polynomial was chosen for its simplicity and its adequacy in consistently describing several different outcomes such as anthropometric measurements, blood pressures, and blood lipids.
In an accelerated longitudinal study design with overlapping cohorts, it is assumed that each cohort is a random sample, differing only in age, from the same underlying population. Under this assumption, data from the various cohorts can be combined to estimate growth curves describing patterns of development of CVD risk factors and other parameters of interest.
To determine if this assumption was reasonable, several of the anthropometric study outcomes, including stature, weight, BMI, fat-free mass (FFM), and percent body fat (PBF), were compared between Cohorts 1 and 2 at their period of overlap (11–12 years) and between Cohorts 2 and 3 at their period of overlap (14–15 years). Regression models adjusting for age and cohort were fitted within each of the four race–gender groups and were used to make these comparisons.
To compare Cohorts 1 and 2, a multilevel model of the form
was fitted to the data obtained from those aged between 11 and 12 years for each race–gender group. In this model, yij
denotes the outcome variable measured on the i
th occasion for the j
th child. The variable age
is centered at 11.5 years; Cohort2
is an indicator variable for Cohort 2; age*Cohort2
is the interaction term; uj
is a subject-level random term; and εij
is a measurement-level random term. In this simple multilevel model, the coefficient β0
is the intercept, and β1
is the slope for children in Cohort 1. For Cohort 2, the intercept is β0+β2
, and the slope is β1+β3
. The difference in intercepts is β2
, and the difference in slopes is β3
; thus the comparisons can be carried out by testing the hypotheses that β2
=0 and β3
=0. A similar model was fitted for comparisons of Cohorts 2 and 3.
Models for stature, weight, BMI, FFM, and PBF were fitted separately on age and cohort membership for each of the two overlapping age ranges. For this analysis, age was centered at the midpoints of each overlap period, so the difference in intercepts in Cohorts 1 and 2 was measured at 11.5 years, and the difference between Cohorts 2 and 3 was measured at 14.5 years. Results revealed only one significant difference between cohorts in intercept, and that was between Cohorts 2 and 3 for PBF (p
<0.05) after adjustments for ethnicity, gender, and age (analyses not shown). There were no significant slope differences between cohorts at either point of overlap for the study variables.14
When these five outcomes were examined in the overlap periods, there were no apparent differences in the intercepts or slopes attributable to the cohorts. Thus, it was concluded that the three cohorts could be combined to estimate growth curves describing patterns of development for children aged 8–18 years.
Comparing Project HeartBeat! with U.S. Growth Data
To judge the degree to which the data obtained in Project HeartBeat! were similar to national data, the 5th, 50th, and 95th percentiles for weight, stature, and BMI by age, race, and gender from Project HeartBeat! were compared graphically with the same percentiles for U.S. children from the National Health and Nutrition Examination Survey (NHANES) I15
and NHANES II16
surveys; results are presented in Mueller et al.14
Plots of the longitudinal data confirmed that for most of the anthropometric variables and for almost every age, the distribution of the measurements was skewed upward, that is, toward the larger values. None, however, appeared multimodal except in regions of extremely sparse data. For fairly large samples, mild to moderate skewness will not appreciably affect estimation of the fixed part of the regression model used to describe the overall trajectory of change.6
Weight percentiles displayed graphically for Project HeartBeat! and NHANES exhibited good agreement for both nonblack boys and girls, except that for the boys, the 95th percentile of weight in Project HeartBeat! tended to exceed that for NHANES for those aged >12 years. Among the black children, there was fair agreement between Project HeartBeat! and NHANES percentiles, except for the 95th percentile, where the former was far larger. This difference may be partly attributable to instability resulting from small samples or it may be a reflection of the increased prevalence of obesity in some groups of children. A generally good agreement was seen in percentiles for stature between Project HeartBeat! and NHANES, except that curves for the former tend to be slightly higher. There was fair agreement in BMI for the nonblack children; for nonblack boys, the BMI for Project HeartBeat! tended to be slightly greater than the BMI for NHANES. For both black boys and girls, the 95th percentile for BMI in Project HeartBeat! greatly exceeded that for NHANES. This pattern was consistent with the pattern for weight.
In summary, the Project HeartBeat! and NHANES percentiles for stature were quite similar, whereas the 95th percentiles for weight and BMI in Project HeartBeat! were skewed upward, likely reflecting a secular trend toward increased obesity. Taken together, these curves, with the exceptions noted, provide evidence that the Project HeartBeat! sample is in reasonable conformity with the NHANES patterns and support the validity of inferences from the present study to the wider U.S. population of children, particularly for nonblack children. Inferences to the population of black children should be considered approximate.14