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U.S. Department of Agriculture/ARS Children's Nutrition Research Center, Department of Pediatrics, Baylor College of Medicine, Houston, TX 77030
3 To whom correspondence and reprint requests should be addressed. E-mail: nbutte{at}bcm.tmc.edu.
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ABSTRACT |
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 TOP ABSTRACT SUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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KEY WORDS: • basal energy expenditure • sleeping energy expenditure • maximal energy expenditure
Appropriate normalization or scaling of energy expenditure (EE)4 data for body mass or composition is critical for understanding variability in basal metabolism, energy requirements, and energy cost of physical activities because body mass and composition are the major predictors of EE. Theoretical, physiological, and mathematical arguments that support or dispute different approaches for the normalization of EE for body mass have been made (1–3). Central to any statistical analysis and normalization of EE is the idea of selecting an adequate model. Constant ratio models (i.e., kcal/kg) are commonly used in many applications, but they do not take into account the nonlinearity and thus fail to produce a variable independent of body weight (2,3). Tanner showed that the mathematical bias may lead to spurious conclusions when individuals that vary in body size are compared using the ratio method (4). Several investigators demonstrated that the assumptions of the ratio method are not met for either basal metabolic rate or maximal oxygen consumption (2,5). Linear regression models are an improvement, but can create positive and negative biases. Linear models assume an additive error term which is questionable because rates of EE diverge with an increase in scale. Inspection of EE data vs. body weight reveals a curvilinear relation with a strong linear component; these issues must be taken into account in selecting empirical models to adequately explain the underlying data-generating mechanism. Thus, a considerable amount of flexibility for fitting EE data can be gained by considering a nonlinear model (6). A number of different statistical models were proposed for physiological variables adjusting for certain variables such as weight and age. Cole (7) proposed linear and proportional regression models. Nevil et al. (8) considered power function models and showed that the allometric models fit the data on oxygen consumption somewhat better than the Cole model (7). Recently, Davies and Cole (9) advocated power function models to investigate the adjustment of measures of EE for body weight and body composition. The curvilinear power function model between physiological variables and body mass has had a long tradition in physiology and was shown to be superior to linear models (10).
Although stochastic modeling of metabolic rate has been a topic of interest to biologists and statisticians for decades, its application across childhood and adolescence when body mass and composition are changing dramatically has not been studied extensively (11–13). Human studies focused on scaling basal metabolic rate and maximal oxygen consumption for body weight (8,9,11–15). For intermediate rates of EE, the exponents would be expected to vary depending on the level of exertion and extent to which the activity is weight-bearing (16).
In this paper, we explore allometric modeling for the normalization of EE data for body mass or composition in a large cohort of children, ages 5–19 y, for a wide range of physical activities. One of the strengths of the allometric model, compared with other nonlinear models, is that it ultimately rests on a flexible linear regression model; more importantly, it fit the data set in this study quite well and provided a good framework within which to investigate interactions among variables. The specific aims of this study were as follows. 1) To develop models or indices for scaling EE measurements for differences in body mass for children ages 5–19 y; 2) to develop models or indices for scaling EE measurements for differences in body composition for children ages 5–19 y; and 3) to investigate the effects of age, gender, and weight status on the coefficients or the exponent values of the body mass and composition predictors of EE.
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SUBJECTS AND METHODS |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Children (n = 833), ages 5–19 y, enrolled in the VIVA LA FAMILIA Study were used in the present analysis. As part of this family-based study designed to identify factors contributing to childhood obesity in the Hispanic population, the children underwent EE and body composition measurements. After a tour of the Children's Nutrition Research Center and full explanation of the study, all enrolled children and parents gave written informed consent or assent. The protocol was approved by the Institutional Review Boards for Human Subject Research for Baylor College of Medicine and Affiliated Hospitals and for Southwest Foundation for Biomedical Research.
Anthropometry and body composition
Body weight was measured to the nearest 0.1 kg with a digital balance and height was measured to the nearest 1 mm with a stadiometer. Body composition was determined by dual-energy X-ray absorptiometry (DXA) using a Hologic Delphi-A whole-body scanner (DXA, Delphi-A, Hologic). Total body estimates of fat-free mass (FFM) and fat mass (FM) were obtained by DXA. Total body weight was equal to the sum of FFM and FM.
Room respiration calorimetry
Oxygen consumption (
) and carbon dioxide production (
) were measured continuously in an 18- or 30-m3 room calorimeter for 24 h. The performance of the room respiration calorimeters was described in detail previously (17). Errors from 24-h infusions of N2 and CO2 were –0.34 ± 1.24% for and 0.11 ± 0.98% for (17). All urine was collected during the 24-h calorimetry. Urine samples were acidified with 6 mol/L HCl and refrigerated; urinary volume was measured and nitrogen concentration determined by Kjeldahl digestion (Kjeltec Auto Analyzer 1030, Tecator), followed by a phenol-hypochlorite colorimetric reaction (18). From the 24-h 
and urinary nitrogen excretion, TEE was computed according to Livesey (19). EE for specific activities was calculated using the Weir equation (20). 
EE, and heart rate were averaged at 1-min intervals. Heart rate was recorded by telemetry (DS-3000, Fukuda Denshi) and motion was monitored by a Doppler microwave sensor (D9/50, Microwave Sensors) in the room calorimeter. Except for the BEE measurement, all activities were performed in the fed state. The children were given breakfast at 0830, lunch at 1200, and dinner at 1730. No food or drinks other than water were consumed after 1900 h. Discrete measurements obtained during the 24 h in the room calorimeter were defined as follows:
Total energy expenditure (TEE). Total energy expended during 24 h in a room respiration calorimeter was measured following a standardized protocol.
Basal energy expenditure (BEE). BEE was measured under thermoneutral conditions upon awakening after a 12-h fast. The children were asked to remain still, but awake, for 30 min. The children were monitored both visually and by the motion sensor to confirm that they were lying still (<50 activity counts/min) for the entire measurement.
Sleeping energy expenditure (SEE). SEE was measured through nighttime sleep, verified by the motion sensor and heart rate monitoring.
Cycling energy expenditure. EE was measured for 15 min while cycling on a stationary bike at 20 W (light level) and then at an mean of 57 W (range 20–160 W), set at 60% of the child's maximal oxygen consumption (VO2max) (moderate level).
Walking energy expenditure. EE while walking on a treadmill at 4.0 km/h (2.5 mph) at 0% grade (Model Q55, Quinton Instrument) was measured by collecting expired gases with a metabolic measurement cart (Model 2900, SensorMedics).
Maximal energy expenditure (MaxEE). MaxEE was measured by using a ramp protocol on the treadmill (Model Q55, Quinton Instrument). The treadmill protocol involved a constant speed of 4.0 k/h (2.5 mph) at an initial 0% grade for the first 4 min (21). After the first 4 min, the grade was increased in steps of 2.5%/min to a maximum grade of 22.5%, after which the speed was increased by 1 k/h (0.6 mph) each minute. MaxEE was determined using criteria for children, specifically a heart rate >195 bpm or RQ >1.0 at maximal effort (21,22).
Statistical analysis
ACCESS (version 9, Microsoft) was used for database management. STATA (version 8.2) and SPSS (version 12.0 for Windows, SPSS) were used for statistical analyses. Data are summarized as means ± SD. Statistical significance was set at P < 0.05.
Because there are measurements from siblings from each family, the VIVA LA FAMILIA data set can be considered as either a repeated measures, or, alternatively, as a clustered or hierarchical data set in which observations are clustered by families. The data set is also unbalanced because there are different numbers of observations within families. Because observations within a family are not independent, mixed-effects regression models, including generalized estimation equations with exchangeable correlation structure, were used to examine the associations of outcome energy variables and their predictors such as age, gender, weight status, FFM, FM, weight, and height. However, accounting for the clustering effect resulted in very small changes in the coefficients or exponent values of the predictors. Therefore, for simplicity of the presentation here, we outlined and discussed the regression models without clustering effects. For all models, age was in years, gender was coded 0 for boys and 1 for girls, and weight status was coded 0 for nonoverweight and 1 for overweight, defined as 
95th percentile for BMI (23).
Exploratory analysis of the data included scatter plot smoothing to examine the form of the regression function and to highlight the underlying trend in the scatter plots (24,25). In our data set, logarithms of the outcome variables and continuous predictors improved the associations of the variables and heterogeneity in the regression models. Based on the exploratory analysis, we used allometric or power function models to identify appropriate scaling or normalizing parameters for EE variables. More specifically, we considered models of the following form:
 | (1) |
where Y represents the EE variable, 
are the predictors, 
are unknown parameters that have to be estimated, and 
is a multiplicative error. Model 1 can be transformed to a linear model by taking the logarithm and assuming that has log-normal distribution. After tentatively identifying a model, we carried out model validation using residual analysis and goodness-of-fit tests. Data splitting was used for cross validation; however, the final model was based on the entire data set. The graphical methods for testing normality such as Q-Q plots and histograms indicated no violation of the normality of the residuals. In addition, formal tests for normality were conducted using the Shapiro-Wilk test and goodness-of-fit statistics, i.e., the Kolmogrov-Smirnov statistic and the Anderson-Darling statistic. The P-values for the tests were >0.05, indicating consistency with the normality assumption.
An alternative to Model 1 is to use the following additive nonlinear regression model:
 | (2) |
where is assumed to be normally distributed. Nonlinear regression models are usually estimated by least squares, but there is no explicit formula for the estimates as for linear regression; in addition, iterative procedures are required, for which initial values must be supplied. We used S-PLUS function nls, which uses the Gauss-Newton algorithm (linearizing the nonlinear function). We investigated both models, and both models gave similar results, i.e., almost identical estimates of the parameters. However, regression diagnostics and residual analyses, such as the Q-Q plot, Shapiro-Wilk normality test for residuals, and Kolmogrov-Smirnov and Anderson-darling goodness-of-fit tests revealed that Model 1 fits the data somewhat better than Model 2. Model 1 is intrinsically linear, that is, it can be transformed to a model linear in parameters; this is an advantage because standard linear regression techniques can be applied easily. The fit of Model 1 can also serve as a useful starting value for other fitting procedures, and, in fact, we used the estimated parameters in Model 1 as initial values for estimating the parameters in Model 2. In general, Model 1 would also provide more meaningful normalized values when the results are used in other statistical analysis. Thus, our final analyses are based on Model 1, which provides a reasonably good parametric class of models for a complex physiological problem.
In both models, we investigated the effects of age, gender, and weight status on the exponents of weight, height, FFM, and FM. For example, the effects of gender on the exponents can be investigated by the following exponential model:
 | (3) |
which can be linearized by taking natural logarithms of both sides, giving
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The exponential Model 3 was extended to include age, gender, and weight status as well as interactions among the variables. We also investigated the interactions between the classification variables such as gender and weight status and other predictors. Of course, interaction terms were considered mainly on the basis of the physiological reasons. We also investigated the quadratic effect of age, in addition to the linear effect, but it was not significant.
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RESULTS |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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0.5) were significantly affected by age (P < 0.04); however, the age effect was minor, contributing <1% to the adjusted r2 of the regression model. In contrast, gender had a significant effect on the exponents for normalizing TEE, BEE, and SEE by weight (P = 0.001). Weight status also had a significant effect on the exponents for normalizing BEE and SEE by weight (P = 0.001).
Scaling cycling and treadmill walking/running for body weight.
Rates of EE in moderate-to-vigorous physical activities scaled somewhat differently. Weight accounted for 65 and 85% of the variation in cycling and treadmill walking, respectively (Table 3). Cycling at 20 W resulted in an exponent for weight (0.47) close to those for TEE, BEE, and SEE. Higher rates of exertion during cycling, walking at 4.0 k/h (2.5 mph), and maximal treadmill walking/running produced exponents approaching 0.7. Body mass exponents for these activities were affected by weight status (P = 0.04–0.001), but not by gender or age.
Scaling TEE, BEE, SEE for body composition: FFM and FM. FFM (kg) was a dominant predictor of TEE, BEE, and SEE, explaining 75–85% of the variation in EE (Table 4). The exponents for FFM (0.60, 0.55, and 0.60) for the scaling of TEE, BEE, and SEE were not significantly affected by age. There was no association between the allometrically scaled variable BEE (i.e., BEE/FFM0.55) and FFM (r = 0.01, P = 0.76), thus supporting the validity of the allometric modeling in providing a size-independent index. Similar results held for other allometrically scaled variables in this article.
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Scaling cycling and treadmill walking/running for body composition: FFM and FM. FFM (kg) explained 57–80% of the variation in rates of EE for cycling and treadmill walking/running (Table 4). Lower rates of EE during cycling at 20 W and steady-state treadmill walking scaled to 0.5 and 0.8, respectively. Moderate cycling and MaxEE scaled to 1.0 and 0.9, respectively. These mass exponents were affected by weight status (P = 0.04–0.001), but not by gender or age.
The addition of FM to our allometric model made a minor contribution to the variance explained in cycling/walking/running EE, on the order of 2–6%, and in some cases resulted in negative coefficients. Thus, in further analyses, FM was dropped because a simpler model would be more efficient.
Specific scaling exponents for the normalization of EE by weight or FFM are provided in Table 5 for those models in which the interaction between ln weight or ln FFM and gender or weight status was significant. Presented by weight status, the pattern of scaling exponents and the length of their 95% CI, which vary as a function of the EE, are shown in Figure 2.
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DISCUSSION |
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TOPABSTRACTSUBJECTS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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As shown here and by others, a curvilinear power function model between physiological variables and body mass was physiologically and empirically superior to simple linear models (6,10). A curvilinear power function model is intrinsically linear because it can be log-transformed to a model linear in its parameters, which is an advantage for further statistical analyses. The power function model assumes a multiplicative error ratio term, whereas competing models such as the simple ratio and least-squares linear regression assume a constant additive error term.
In general, weight and FFM were comparable predictors of EE, with the exceptions of the physical activities of highest exertion. In these cases, FFM was a superior predictor of moderate cycling EE and MaxEE, accounting for a greater percentage of the variance explained in EE. Inclusion of FM did not improve the prediction of EE based on FFM alone.
For BEE and lower levels of EE, weight scaled to 0.5 and FFM to 0.6. Basal metabolism declines throughout childhood and approaches adult values late in adolescence. The decline in basal metabolism relative to body weight is likely secondary to the differential in growth rates of organs with high metabolic rates (i.e., brain, liver, heart, and kidney) relative to those with lower metabolic rates (i.e., muscle and fat) (26,27) and changes in the metabolic rates of individual organs and tissues (28). For cycling and treadmill walking/running, the weight exponent approached 0.7, even with a weight-bearing activity. Scaling EE for FFM resulted in slightly higher exponents of 0.6 for lower rates of EE and 0.8–1.0 for higher rates of EE.
Nevill (8) suggested that for weight-bearing physical activities, the likely body size denominator will be total body mass. For non-weight–bearing activities, such as cycling and rowing, the body size denominator of a scaled index is likely to be considerably less. In fact, activity-specific allometric exponents ranging from 0.30 for sitting, 0.49 for cycling, 0.80 for walking at 3.2 k/h (2 mph) and 1.05 at 6.4 k/h (4 mph), and 0.50 for 24-h TEE were demonstrated in 42 women using 24-h room calorimetry (16). In our data set of children and adolescents, the exponents also approached 1.0 as the level of EE increased, irrespective of the weight dependence of the activity. For moderate cycling EE, weight scaled to 0.78 and FFM to 0.89. For MaxEE, weight scaled to 0.69 and FFM to 0.90. These values are similar to reports in adults in which VO2max scaled to the two-thirds power for weight (6,12) and 1.0 for FFM (14). In a sample of 50 boys, ages 7–13 y, the exponential relation between VO2max and body weight was 0.88 (29). In another sample of 25 boys, 8–16 y, the exponent was 0.95 (30).
Our analysis suggested that the EE to weight relation for some physical activities differs between boys and girls, and nonoverweight and overweight children. Accounting for body composition in our examination of the EE to FFM did not eliminate the interactions with gender and weight status. At lower levels of exertion, the ß-coefficients for weight or FFM were higher in the overweight than nonoverweight children, indicating a greater contribution of weight or FFM to EE. At the higher levels of EE during cycling and the maximal test, the opposite was seen. Factors such as the composition of FFM, body shape, cross-sectional area of the primary muscles, leg length, efficiency of body movement, motivation, and training all may influence the EE to weight or FFM relations.
This large data set of EE and body composition in Hispanic children permitted us to explore models for the normalization of EE across a broad range of age, body size and composition, and physical activities. To test whether these scaling exponents were applicable to other groups of children, we retrieved BEE and maximal EE data from previous studies on 289 Caucasian, Hispanic, and African-American children and adolescents. The CI for the different ethnic groups overlapped the CI presented in this paper.
In conclusion, appropriate scaling of EE for body weight and composition of children and adolescents varied primarily as a function of the level of EE. In some instances, the exponents for scaling EE by body weight or composition were influenced by gender and weight status, but not by age.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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2 Funded with federal funds from the NIH R01 DK59264 and from U.S. Department of Agriculture/ARS under Cooperative Agreement 58-6250-51000-037. The contents of this publication do not necessarily reflect the views or policies of the U.S. Department of Agriculture, nor does mention of trade names, commercial products, or organizations imply endorsement by the U.S. Government.
4 Abbreviations used: BEE, basal energy expenditure; DXA, dual energy X-ray absorptiometry; EE, energy expenditure; FFM, fat free mass; FM, fat mass; MaxEE, maximal energy expenditure; SEE, sleeping energy expenditure; TEE, total energy expenditure; oxygen consumption; carbon dioxide.
Manuscript received 9 December 2005. Initial review completed 10 January 2006. Revision accepted 20 February 2006.
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