## Abstract

OBJECTIVE. Our goal was to quantify the magnitude of energy imbalance responsible for the increase in body weight among US children during the periods 1988–1994 and 1999–2002.

METHODS. We adopted a counterfactual approach to estimate weight gains in excess of normal growth and the implicit “energy gap”—the daily imbalance between energy intake and expenditure. On the basis of Centers for Disease Control and Prevention growth charts, we constructed weight, height, and BMI percentile distributions for cohorts 2 to 4 and 5 to 7 years of age in the 1988–1994 National Health and Nutrition Examination Survey (*N* = 5000). Under the counterfactual “normal-growth-only” scenario, we assumed that these percentile distributions remained the same as the cohort aged 10 years. Under this assumption, we projected the weight and height distributions for this cohort at 12 to 14 and 15 to 17 years of age on the basis of their baseline weight-for-age and stature-for-age percentiles. We compared these distributions with those for corresponding age groups in the 1999–2002 National Health and Nutrition Examination Survey (*N* = 3091) ∼10 years after the 1988–1994 National Health and Nutrition Examination Survey. We calculated differences between the counterfactual and observed weight distributions and translated this difference into the estimated average energy gap, adjusting for increased total energy expenditure attributable to weight gain. In addition, we estimated the average excess weight accumulated among overweight adolescents in the 1999–2002 National Health and Nutrition Examination Survey, validating our counterfactual assumptions by analyzing longitudinal data from the National Longitudinal Survey of Youth and Bogalusa Heart Study.

RESULTS. Compared with the counterfactual scenario, boys and girls who were aged 2 to 7 in the 1988–1994 National Health and Nutrition Examination Survey gained, on average, an excess of 0.43 kg/year over the 10-year period. Assuming that 3500 kcal leads to an average of 1-lb weight gain as fat, our results suggest that a reduction in the energy gap of 110–165 kcal/day could have prevented this increase. Among overweight adolescents aged 12 to 17 in 1999–2002, results indicate an average energy imbalance ranging from 678 to 1017 kcal/day because of an excess of 26.5 kg accumulated over 10 years.

CONCLUSIONS. Quantifying the energy imbalance responsible for recent changes in weight distribution among children can provide salient targets for population intervention. Consistent behavioral changes averaging 110 to 165 kcal/day may be sufficient to counterbalance the energy gap. Changes in excess dietary intake (eg, eliminating one sugar-sweetened beverage at 150 kcal per can) may be easier to attain than increases in physical activity levels (eg, a 30-kg boy replacing sitting for 1.9 hours with 1.9 hours walking for an extra 150 kcal). Youth at higher levels of weight gain will likely need changes in multiple behaviors and environments to close the energy gap.

The prevalence of overweight children and youth is rapidly rising in the United States^{1,2} as well as many other regions in the world.^{3,4} Over the past 3 decades, the prevalence of overweight has doubled among preschool-aged children and adolescents, and the prevalence increased threefold among children 6 to 11 years of age.^{2} Approximately 9 million children over the age of 6 are considered overweight, defined as BMI ≥95th percentile on the Centers for Disease Control and Prevention (CDC) growth charts.^{5} Ethnic disparities in overweight are growing as rates increase faster among black and Hispanic children.^{6,7} Moreover, adolescents living in lower-income households are more likely to be overweight compared with youth living in higher-income households.^{8}

Many epidemiologic studies document the likely role of excess adiposity in increasing mortality and morbidity,^{9–12} including cardiovascular diseases,^{13} cancer,^{14,15} and reduced functional status.^{16} An estimated 60% of overweight children between the ages of 5 and 10 have already developed at least 1 cardiovascular disease risk factor such as hyperinsulinemia, hypertension, and hyperlipidemia, and 25% have ≥2 risk factors.^{17} The incidence of type 2 diabetes, until recently thought to be almost exclusively an adult-onset disease, has dramatically increased among youth.^{18,19} Moreover, overweight children and adolescents are more likely than their peers to experience negative social and psychological consequences such as marginalization, low self-esteem, stigmatization, and discrimination.^{20–23}

An analysis by Hill et al^{24} estimated body weight changes in US adults between 20 and 40 years of age during the period of 1992–2000 and quantified the magnitude of energy imbalance that could explain such changes. They asserted that “affecting energy balance by 100 kilocalories [420 kJ] per day could prevent weight gain in the majority of the population.” Their findings provide salient and tangible targets for implementing population preventive interventions. However, to provide similar estimates in the youth population requires methodologic adjustments. Throughout adulthood, a positive shift in weight distribution can be reasonably seen as the result of a positive “energy gap” (energy intake exceeding expenditure). The major challenge in estimating energy imbalance in childhood involves the fact that excess energy beyond daily activities is essential for growth. Unlike adults, all children should grow taller and heavier as they age. The issue is how much weight gain would be considered excessive beyond normal growth. Hence, we propose a counterfactual framework to project changes in the weight distribution of a hypothetical population if they only gain weight commensurate with their gain in height. This hypothetical scenario represents “what would have been” if the US youth had not experienced excess weight gain beyond normal growth. We then quantify the population-level deviation of reality from this counterfactual scenario to estimate the average excess weight gain and corresponding energy imbalance for US children during the period of 1988–1994 to 1999–2002. We also aim to address one potential criticism of the Hill et al approach: their calculations do not address the recognized fact that as individuals gain weight, they will also increase energy expenditure because of this extra weight.^{25,26}

## METHODS

### Descriptive Analysis

We compared data from children (2–11 years of age) and adolescents (12–17 years of age) in 2 nationally representative population studies: the National Health and Nutrition Examination Survey (NHANES) III (1988–1994) and NHANES 1999–2002. The NHANES is an ongoing series of nationwide surveys and clinical examinations conducted by the National Center for Health Statistics. The surveys use a multistage, clustered, probability sampling strategy to select households and individuals to provide national estimates representative of the civilian noninstitutionalized US population. Beginning in 1999, the NHANES began collecting data every year. This analysis is based on the first 4 years of the continuous NHANES data collection (1999–2002) and NHANES III (1988–1994). Weight and height are measured in mobile clinic settings by using standard NHANES protocols. A complete description of data-collection procedures and analytic guidelines are available on the National Center for Health Statistics Web site (www.cdc.gov/nchs/nhanes.htm).

For all NHANES subjects with complete height and weight measurements, we calculated weight-for-age, stature-for-age, and BMI-for-age percentiles as a function of gender on the basis of the revised CDC growth charts.^{5,27} Because body composition changes during growth, the distribution of these percentiles, both epidemiologically and clinically, have been widely used to monitor changes in relative body size across gender and age groups. Instead of defining a range of reasonable values for each age, we excluded subjects with BMI more extreme than the 0.5th or 99.5th BMI-for-age percentiles.

### Synthetic Cohorts and Counterfactual Weight Trajectory

We created a “synthetic” cohort of children by linking subsets of children between the 2 cross-sectional studies: children 2 to 7 years old in the NHANES III (born between 1981 and 1992) who approximately represent the same birth cohorts as adolescents 12 to 17 years old in the NHANES 1999–2002 (born between 1982 and 1990). The midyear of the NHANES III is 1991, and the midyear of the NHANES 1999–2002 is 2001, so we have 2 “snapshots” of this synthetic cohort ∼10 years apart. Although the cross-sectional nature of the surveys precludes the exact matching of birth cohorts, we aim to capitalize on the large sample size and wide geographical coverage in making nationally representative inferences.

We used a counterfactual scenario as the idealized trajectory of weight at the population level. Between 1988–1994 and 1999–2002, the average height of US children and adolescents did not increase significantly (Table 1). Therefore, when we converted the height of our synthetic cohort of children at baseline (1988–1994) into a stature-for-age percentile using the CDC growth charts, we assumed that the distribution remained stable over time.

Assuming stability in height distributions, we hypothesized that the distribution of weight-for-age percentile in a cohort of children with perfect energy balance (ie, weight gain only in proportion to height gain) should also remain the same as they age. This projection of a weight-distribution shift for the synthetic cohort in the NHANES III 10 years later represents an ideal growth trajectory into adolescence that, although heavier in actual body weight, maintains an identical weight-for-age percentile distribution. The deviation from this counterfactual “normal-growth-only” trajectory of weight distribution of the actual weight distribution among the synthetic cohort in the NHANES 1999–2002 therefore provides us with an estimate of the distribution of excess weight gained during the 10-year period.

Operationally, for each sampled child in the NHANES III synthetic cohort, we calculated the expected body weight for an adolescent 10 years older and of the same weight-for-age percentile. This mapping from a given percentile to corresponding age- and gender-specific weight is accomplished by reversing the statistical procedure that produced percentiles in the construction of the CDC growth charts (Appendix 1).^{28,29} Finally, we calculated the difference between the counterfactual distribution and the actual weight distribution in the NHANES 1999–2002 as the area enclosed between the 2 cumulative weight-distribution curves (Fig A1).

### Translating Excess Weight Gain to Daily Energy Gap

To estimate the excess energy responsible for fueling the average excess weight gains observed in the NHANES, we assumed that the accumulation of excess weight is linear over the years and that each pound of excess weight gain as fat is associated with 3500 kcal stored.^{24}

In addition to the assumptions described above, similar to Hill et al,^{24} we assumed 63% efficiency of the conversion of this imbalance of energy intake over expenditure into weight gain, the average efficiency of energy storage from composite diet.^{30–32} We further adjusted for the increase in average energy requirements associated with weight gain. This adjustment aims to reflect the fact that, at a given level of physical activity, greater energy intake is required to move a higher body weight.^{31} Hence, to sustain a continuing excess-weight trajectory, an energy surplus needs to not only deposit fat mass (as accounted for in the Hill et al approach) but also offset the margin created by elevated energy expenditure.^{25} We modeled this time-dependent dynamic process using a differential equation to approximate the magnitude of this margin (detailed in Appendix 2) as a linear function of both the realized excess weight gain and average daily physical activity level (PAL).^{26}

We provide 2 sets of adjusted energy-gap estimates: the primary estimates are based on the average age-specific PAL implied by the Food and Agriculture Organization (FAO)/World Health Organization (WHO) energy-requirement recommendations for youth 2 to 17 years of age (“average physical activity”)^{26}; the supplemental estimates assume “light physical activity.”^{33}

### Excess Weight Gained Among Overweight Adolescents

A question of particular concern for pediatricians is the magnitude of the energy gap that leads to overweight status (BMI ≥95th percentile) among adolescents. We validated our counterfactual assumptions using 2 longitudinal samples to study the actual shifts in body size among children who became overweight adolescents. We compared longitudinal changes in weight and BMI in contrast to stature-for-age percentile changes in 2 cohorts: the National Longitudinal Study of Youth (NLSY) and the Bogalusa Heart Study; detailed information on these cohorts^{34–36} has been published elsewhere. Briefly, data were collected every 2 years since 1986 on children born to a nationally representative sample of women enrolled in the NLSY in 1979, from a child’s birth through age 14. To be consistent with the time span of our NHANES analyses, we analyzed the subset of NLSY children who were 2 to 4 years of age in 1990 and had a BMI of ≥95th percentile in 2000 when aged 12 to 14 years (*n* = 39), and all measurements of weight and height were obtained by the interviewers. The initial Bogalusa Heart Study population consists of all children and young adults living in ward 4 of Washington Parish, Louisiana, which includes the city of Bogalusa. We studied the subset of children from the Bogalusa cohort who were 5 to 7 years of age in 1981–1982 and were considered overweight in 1992–1993 when aged 16 to 18 (*n* = 40).

The results can indicate the validity of using the simple counterfactual: “what if” their counterfactual weight-for-age percentile would have equaled their height-for-age percentile. We compared these estimates across the NLSY and Bogalusa samples and applied the results to the NHANES data.

### Statistical Analysis

All statistical procedures were conducted in SAS 9.1 (SAS Institute, Cary, NC) and SUDAAN 9.0.1 (Research Triangle Institute, Research Triangle Park, NC) software. Age standardization of overweight prevalence and mean height and weight in Table 1 used direct standardization based on the 2000 US census.^{37,38} All statistics based on the NHANES and the NLSY data were weighted to adjust for unequal probabilities of sampling. Variance estimates from the NHANES analyses were adjusted for the stratified and clustered structure of the national sample using the robust variance estimation method.^{39}

## RESULTS

### NHANES Cross-sectional Comparisons

Samples included 6964 children and 2229 adolescents from the NHANES III (1988–1994) and 3260 children and 3091 adolescents from the NHANES 1999–2002. Sample size and the age-standardized prevalence of overweight (BMI ≥ 95th percentile) according to demographic subgroups are summarized in Table 1. As reported previously,^{2,40} between 1988–1994 and 1999–2002, the overall age-standardized prevalence of overweight significantly increased in children (8% vs 11%; *P* = .006) and adolescents (10% vs 13%; *P* = .002) during the 10-year period. Increases occurred among both boys and girls. Among race/ethnicity groups, Mexican American children and adolescents have the highest prevalence of overweight, followed by non-Hispanic black children and adolescents. In addition, although children from lower-income families had a slightly lower prevalence of overweight compared with their higher-income counterparts in the NHANES III, this was reversed in 1999-2002. Adolescents from lower-income families had higher overweight prevalence in both periods, although overweight adolescents from higher-income families increased by 50%. The mean height in all groups of youth remained virtually unchanged during the 2 time periods (Table 1).

### Synthetic Cohorts and the Counterfactual Growth Trajectory

Figure 1 illustrates the baseline (age 2–7, NHANES III), counterfactual trajectory (age 12–17), and the actual (age 12–17, NHANES 1999–2002) BMI, weight, and height cumulative frequency distributions for the synthetic cohort (left panels). As expected, the baseline curves are to the left of the others, because they represent the body-size distributions of children much younger in age. Note that for height distributions, the counterfactual trajectory largely overlaps the actual curves in 1999–2002. This close approximation can also be demonstrated on the mean-difference plots (right panels). In these plots, the deviation of the actual distributions from the counterfactual scenario of corresponding percentiles (y-axis) is graphed against the mean of the same percentiles (x-axis) (see Appendix 3). The deviance of height distributions scatters around zero, mirroring the overlapping height-distribution curves and providing strong evidence for our assumption of constant stature-for-age percentile distribution during growth. As a result, the shifts in BMI and weight distributions beyond expectation under normal growth represent excess weight gain during these 10 years. The mean-difference plots indicate that deviation from the normal growth trajectory tends to be more prominent at higher percentiles.

We calculated the areas between the 2 curves to estimate the average excess weight gain in kilograms among this synthetic cohort during the 10-year period (see Fig A1 for details). The resulting overall and subpopulation estimates of excess weight gain and corresponding mean energy gap are summarized in Table 2. On the basis of the synthetic cohorts from the NHANES, we estimate that US children between 2 and 7 years of age gained an excess of 4.3 kg beyond normal growth during the 10-year period. Averaged over the 10-year period,^{24} we calculated that a 0.43-kg annual excess weight gain is fueled by an average of 131 kcal of excess energy intake over expenditure on a daily basis. Because of our assumption of uniform excess weight gain, this energy gap changes over time as the children grow, ranging from 14 kcal/day in the beginning of the first year to 247 kcal/day toward the end of the tenth year.

Corresponding estimates indicate a greater energy gap among children who were younger (age 2–4) than who were older (age 5–7) at baseline, children from lower-income families than from higher-income families, and among Mexican American and black compared with non-Hispanic white children. However, estimates of the average daily energy gap for all groups are no larger than 165 kcal/day. If we vary the energy efficiency assumption from 50%, as assumed by Hill et al,^{24} to an upper bound of 75%,^{30–32} our energy-gap estimate of 131 kcal/day would range from 165 to 110 kcal/day. Furthermore, when we assume a lower age-specific activity level throughout the 10-year period instead of the average level according to the WHO/FAO report,^{26} the energy-gap estimates decrease by ∼13% (to 113 kcal/day, as indicated in Table 2). Finally, we follow the recommendations from the WHO/FAO report to use basal metabolic rate (BMR) prediction equations that assume linear increase in energy expenditure by weight gain. Had we alternatively used the nonlinear equations according to existing evidence suggesting a possible nonlinear relationship,^{26,32,41} we would have obtained slightly higher energy-gap estimates (∼13 kcal higher for a typical boy in the NHANES who gained 4.5 kg of excess weight).

### Excess Weight Gain and Energy Gap Among Overweight Adolescents

When weight and height changes were traced for each overweight NLSY adolescent aged 12–14 years at follow-up (*n* = 39), their average stature-for-age percentile modestly decreased from 66 in 1990 to 61 in 2000, and there were large increases in mean BMI-for-age (58 to 98) and weight-for-age percentiles (67 to 97). Similar results were observed among the Bogalusa Heart Study adolescents aged 16–18 years who were overweight at follow-up in 1992–1993 (*n* = 40). Over the ∼11 years’ time, their mean stature-for-age percentile decreased (63 to 58), and there were large increases in mean BMI-for-age (73 to 98) and weight-for-age percentiles (71 to 97). Compared with these 2 cohorts, the overweight adolescents (*n* = 299 aged 12–14 and *n* = 232 aged 15–17) in the NHANES 1999–2002 showed similar distribution of weight, BMI, and height for their age. Among the 12- to 14-year-olds, weight-for-age percentiles averaged 97, whereas height-for-age percentiles averaged to 61 (the same as for those in the NLSY). Similarly, the 15- to 17-year-old overweight NHANES subjects had a mean weight-for-age percentile of 97, in comparison to their mean stature-for-age percentile of 53.

To quantify this disconnection between the growth in weight and height among the overweight NHANES adolescents, we calculated the difference between their actual weight and a hypothetical “ideal weight,” which we defined as a weight-for-age percentile equaling their stature-for-age percentile. Under this counterfactual scenario, we estimated that the overweight 12- to 14-year-old NHANES adolescents were, on average, 24 ± 8 kg (median: 26 kg) above their idealized weight. The corresponding NLSY estimate in this analytic scenario would be 30 ± 16 kg (median: 26 kg). As for the older adolescents, we calculate that the overweight 15- to 17-year-old NHANES adolescents accumulated an average of 30 ± 9 kg (median: 29 kg) in excess. The same calculation applied to the Bogalusa Heart Study sample resulted in an average of 31 ± 11 kg (median: 30 kg) of excess gain.

The combined estimate of average excess weight gained among all overweight NHANES adolescents aged 12 to 17 years is 26.5 ± 9.7 kg (median: 26 kg). Following the same calculation that produces 110- to 165-kcal energy-gap estimates for all US adolescents, this average excess weight accumulated over 10 years’ time would indicate a daily imbalance of 678 to 1017 kcal.

## DISCUSSION

In its 2005 report “*Preventing Childhood Obesity: Health in the Balance*,” the Institute of Medicine recommended reporting changes in the mean BMI and the shape of the whole BMI distribution as the benchmarking metric for evaluation of clinical guidelines and pubic health campaigns.^{42} We believe that estimates of energy imbalance underlying the observed excess shifts in weight distribution can be informative in facilitating communications in clinical, community, and policy settings.

We used a counterfactual approach, which acknowledges the natural changes in BMI distribution associated with growth, to calculate excess weight gains experienced by children in the United States. Between 1988–1994 and 1999–2002, we estimate that the cohort of children between ages 2 and 7 at baseline gained an excess of 4.3 kg over 10 year’s time. This translates to an average energy gap of 110 to 165 kcal/day. Compared with the analysis of adults by Hill et al,^{24} our estimates are relatively larger, because we take into account the increased energy expenditure resulting from excess weight. Had we not taken into account the excess energy required to maintain the extra weight, the estimated energy gap would have been much smaller (14 kcal/day).

Our findings are limited by the validity of our assumptions, including our estimates of BMR based on body weight, age, and gender. We calculated the energy gap for 2 levels of PAL: both the average and light levels noted in the WHO/FAO report.^{26} There is some evidence that US children may be in the light range.^{33} In addition, our assumption of a 63% efficiency of energy deposition from dietary intake represents estimation. Experimental studies have shown variations of this efficiency between fat and fat-free mass, by dietary content, and between individuals.^{30,43,44} It is also important to note that we assume that the calculated energy gap pertains to excess gain of weight and not to weight loss. Finally, we made inference from a counterfactual framework, which relies on model assumptions instead of controlled experiments. The validity of our estimates depends on the linkage of cohorts from cross-sectional data, which is subject to sampling variability as well as changes in population demographics resulting from immigration or childhood morbidity/mortality. We also stress that the interpretation of our results should be at the population level, not for individual children.

These results suggest that the behavioral modifications required for preventing excess weight accumulation in the US pediatric population are of manageable scale for most children. The accumulation of small lifestyle or environmentally induced changes in diet and physical activity could make significant differences. More importantly, because we show that an increasingly large energy excess must be sustained to continue a trajectory of excess weight gain, early prevention may be crucial. A typical child in the NHANES synthetic cohort aged 2 to 7 years at baseline gains an excess 0.43 kg every year for a decade, accumulating a total excess of 4.3 kg. The excess energy needed to produce this excess weight gain increases from ∼40 kcal when the child is only 0.5 kg in excess, to 120 kcal when 2 kg in excess, and finally up to 230 kcal when 4 kg in excess to normal growth (see Fig A2). A possible implication of this scenario is that early recognition of behavioral risks and intervention could be more effective than attempting changes in habits and environments after a weight-gaining pattern has been sustained for years.

A wide range of behavioral and environmental strategies can be imagined. For example, one strategy would be to increase the daily PAL of children. Using the same approach with BMR and PAL as detailed in Appendix 3, we calculate that a typical 9-year-old boy weighing 30 kg could burn an extra 150 kcal every day by replacing 1.9 hours of sitting with 1.9 hours of walking. Increasing physical education classes for the same child from once a week to 3 times a week can mean a difference of 240 kcal/week (assuming that children are active half of the time in class: 30/60 min in class, with activity intensity equivalent to playing volleyball instead of sitting in class).

Another strategy might be reducing portion size or reducing intake of certain foods. One source of calories associated with overweight in both observational and experimental studies is sugar-sweetened beverages.^{45,46} Studies suggest that calories from these beverages were often not offset by reduction of intake elsewhere.^{47,48} Thus, if this same child replaced one 12-oz can of sugar-sweetened beverage per day with water, this change could mean a difference of 150 kcal/day. In addition, eating at fast food restaurants has been associated with an additional 126 kcal/day,^{49} indicating another intervention target.

Reductions in television-viewing time could be another important strategy. Multiple observational and experimental studies link excess television viewing to increased overweight.^{50–53} If the same 30-kg boy replaces 1 hour of television-watching time with 1 hour of slow walking, the difference in energy expenditure would be ∼55 kcal. However, both observational and experimental studies document substantial effects of television-viewing time not only on physical activity but also on dietary intake.^{54–56} Each hour per day decrease in television viewing has been associated with a reduction of total energy intake of 167 kcal/day (95% confidence limits: 136, 197).^{56} In experimental studies, Epstein et al^{54,55} found even larger effects. These results suggest that the role of television viewing on energy imbalance from both promoting sedentary behavior and encouraging dietary intake can be substantial.

Previous research has documented that shifts in BMI distribution in children over the past decades are characterized by a disproportionately elongating upper tail.^{6,57} The largest gains in overweight have thus occurred among those already overweight or at risk of overweight. The health risks associated with elevated BMI are also higher among this group: elevated blood pressure and insulin level are twice as common in children above the 97th percentile as in children between the 95th and 97th BMI percentiles.^{17} Preventing more children from becoming overweight, therefore, is crucial. Our results for youth who were overweight at ages 12 to 17 in the NHANES 1999–2002 indicate that they had accumulated an average of 26.5 kg in excess to normal growth, 5 to 6 times higher than the population average. The corresponding energy gap mounted to 678 to 1017 kcal/day. It is likely that focusing on single risk factors will not be enough; eliminating the consistent energy imbalance responsible for this level of excess weight will clearly require changes in multiple environmental and behavioral factors that contribute to sedentary lifestyles and excess dietary intake.

To date, population-based prevention approaches are believed to hold the key for reversing the forecast of this “epidemic.”^{42} Substantiating these initiatives requires an evidenced-based framework to ensure that goals and recommendations are consistent with biological, psychosocial, and environmental knowledge. A comprehensive strategy, possibly similar to or even more extensive than the concerted interventions in tobacco control, is a critical need. Pediatricians stand at a pivotal position in fostering such effort.^{58} In addition to their close watch on the etiology and complications associated with already-overweight children, pediatricians and pediatric nurse practitioners are expected to play important roles in assisting families and communities initiate environmental and individual changes to halt the overweight epidemic.

## APPENDIX 1: STATISTICAL REVERSAL OF CDC GROWTH-CHART MACROS

The purpose of reversing the statistical program (“macro”) provided by the CDC in their construction of growth charts is to automate the process of looking up gender- and age-specific values of BMI, weight, or height for a given percentile.

### The Original SAS Macro

The original SAS program provided on the CDC Web site allows researchers to generate indices of anthropometric status (in percentiles) for his or her own target population from birth to 20 years of age on the basis of their subjects’ gender, age (in months), and body measures. For example, with a 5-year-old boy who weighs 41.5 lb and is 43 in tall, the program calculates that he is at the 50th BMI-for-age percentile. This procedure follows the mathematical algorithm based on the modified LMS method.^{28,29} The program can be downloaded at www.cdc.gov/nccdphp/dnpa/growthcharts/sas.htm.

The development of 2000 growth charts involves estimating 3 parameters from nationally representative data using the LMS technique: the median (M), the generalized coefficient of variation (S), and the power in the Box-Cox transformation (L). The L reflects the degree of skewness. The detail of this procedure and the subsequent smoothing process is outside the scope of this article; however, detailed information can be obtained from the CDC at www.cdc.gov/nchs/data/series/sr_11/sr11_246.pdf.

For a given body measure (eg, BMI), the process can be summarized as follows:

Identify outlier values and assign “missing” to the percentile output.

Look up L, M, and S values specific to the subject’s age (in months) from the standard distributions that the CDC constructed (let’s call them L*, M*, and S* to denote them specific to the subject’s age and gender).

Find his or her

*z*score (*z**) and corresponding percentile (PCT*) by mapping it on the standard distribution (defined by L*, M*, and S*). Mathematically, the “mapping” entails the following: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{If\ L}{\ast}\ {>}{-}0.01\ \mathrm{and\ L}{\ast}\ {<}0.01),\] \end{document}(1)\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{then}\ z{\ast}{=}\mathrm{log}(\mathrm{BMI}/\mathrm{M}{\ast})/\mathrm{S}{\ast}.\] \end{document}(2)\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{Otherwise},\ z{\ast}{=}\frac{(\mathrm{BMI}/\mathrm{M}{\ast})^{\mathrm{L}{\ast}}{-}1}{\mathrm{L}{\ast}{\times}\mathrm{S}{\ast}}\ (\mathrm{BMI},\ \mathrm{M}{\ast},\ \mathrm{S}{\ast},\ \mathrm{and\ L}{\ast}\ \mathrm{given}).\] \end{document}(3)

The statistical relation between percentile and *z* score is:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{PCT}{\ast}{=}\mathrm{PROBNORM}(z{\ast}){\times}100,\] \end{document}(4) where PROBNORM(*z**) is a function that returns the probability that an observation from the standard normal distribution is ≤*z**. In other words,
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{PROBNORM}(z{\ast}){=}\frac{1}{2{\pi}}\ {{\int}_{{-}{\infty}}^{z}}\mathrm{exp}({-}\frac{1}{2}v^{2})dv\ \mathrm{for\ any\ v}{\sim}\mathrm{norm}(0,1).\] \end{document}(5)

### Reversed Macro

In constructing our counterfactual projection of distributions, we need to repetitively calculate the corresponding BMI, weight, and height values for the hypothetical future for each NHANES III child. For example, for a sampled child in NHANES III who is a 3-year-old girl who is at the 50th weight-for-age percentile, we calculate the corresponding weight (in kg) for the 50th weight-for-age percentile for 13-year-old girls. In other words, we need a statistical algorithm that serves the opposite of the function provided by the original macro on the CDC Web site. A straightforward reverse of the original mathematical algorithm is: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{If\ L}{\ast}\ {>}{-}0.01\ \mathrm{and\ L}{\ast}\ {<}0.01),\] \end{document}(6)\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{then\ BMI}{^\prime}{=}\mathrm{M}{\ast}{\times}\mathrm{exp}(\mathrm{S}{\ast}{\times}z{\ast})\] \end{document}(7)\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{Otherwise\ BMI}{^\prime}{=}\mathrm{M}{\ast}{\times}(\mathrm{I}{+}\mathrm{L}{\ast}{\times}\mathrm{S}{\ast}{\times}z{\ast})^{\mathrm{I}/\mathrm{L}{\ast}}\ (z{\ast},\ \mathrm{M}{\ast},\ \mathrm{S}{\ast},\ \mathrm{L}{\ast}\ \mathrm{given}).\] \end{document}(8)

## APPENDIX 2: ADJUSTMENT FOR INCREASE IN TOTAL ENERGY EXPENDITURE AFTER WEIGHT GAIN

We assume the excess weight gain is linearly accumulated over 10 years (eg, an average child gains 0.43 kg/year for a total of 4.3 kg in excess weight). Thus,
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{daily\ weight\ gain}{=}\mathrm{total\ excess}/(10\ {\times}\ 365),\] \end{document}(9) where total excess = 4.3 kg in the above example. Excess weight accumulated up to any given time point *t* can be estimated as:
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{excess\ weight}(t){=}t{\times}\mathrm{daily\ weight\ gain}.\] \end{document}(10)

Assuming no change in PAL, this excess weight leads to an elevated rate of energy expenditure. The energy surplus (on day *t*) to sustain a continuing weight gain is the sum of (1) joules required to offset excess energy expenditure from already-gained weight up to time *t*, and (2) energy stored to further gain weight (by the amount of daily weight gain.) We used a differential equation as follows to estimate this time-varying quantity, the daily gap (daily energy surplus):
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\mathrm{daily\ gap}(t){=}\ \frac{1}{f}{\times}{[}\mathrm{c}{\times}\mathrm{daily\ weight\ gain}{+}k{\times}\mathrm{excess\ weight}(t){]}.\] \end{document}

The parameter *f* reflects our assumption that the efficiency of energy storage is 63%. Multiplier *c* represents the conversion factor of excess energy to weight gain. We assume the same value as did Hill et al^{24} (ie, 3500 kcal stored equals 1 lb of weight gain as fat; hence, in standard units, *c* = 3500 × 2.205 = 7717.5 kcal/kg). Multiplier *k* represents the average additional daily energy expenditure associated with carrying each extra kilogram of excess body weight (as opposed to carrying no excess weight as the counterfactual normal growth cohort). We adopted the most widely used Schofield equations^{59} that assume a linear relationship; however, other sources have suggested a nonlinear relationship between body weight and total energy expenditure or BMR.^{26,32,41}

We derive 2 values of *k* assuming different PALs. We assume that total energy expenditure is proportional to body weight, and there is no change in PAL associated with gains in excess weight, although at least 1 study has found a small difference (eg, 5% lower) in PAL among overweight youth.^{60} We calculate age- and gender-specific products of (1) the incremental BMR (kcal/day) per kg higher body weight and (2) PAL.

Quantity (1) is estimated from the BMR prediction equations developed by Schofield^{59} (FAO/WHO report, Tables 4.2 and 4.3).^{26} The slope coefficients of these equations (eg, 22.706 for boys between 3 and 10 years of age) represent this incremental increase; for example, a 20-kg boy expends 22.706 kcal/day more on basal metabolism than a 19-kg boy of the same age. As for PAL, the FAO/WHO report provided age- and gender-specific estimates for both average and light activity levels. We impute a number of light PAL estimates in some younger age groups using the same average-to-light PAL ratios in other ages. Finally, we average these age- and gender-specific products of quantities 1 and 2 to derive our overall *k* = 34.14, assuming an average level of PAL. Assuming the lighter PAL value leads to *k* = 29.11. Fig A2 shows the resulting time course of daily gap(*t*). The main text reports the average value of daily gap(*t*) over the entire 10-year period for different subpopulations.

Intensity and implications on energy expenditure of various physical activities (measured as multiples of BMR, equivalent to PAL) are taken from Annex 5 of the WHO/FAO report.^{26}

## APPENDIX 3: MEAN-DIFFERENCE PLOTS AND THEIR INTERPRETATIONS

Let *x* represent a continuous random variable and *f*(*x*) represent the probability density function of *x*; then,
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ {{\int}_{{-}{\infty}}^{{\infty}}}\ f(x)dx{=}1,\] \end{document}(12) and the cumulative distribution function (or distribution function)
\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ F(x){=}\ {{\int}_{{-}{\infty}}^{x}}\ f(x)dx{=}P(X{\leq}x).\] \end{document}(13)

One way of quantifying the shift of a cumulative distribution is by the area enclosed by the “before” and “after” curves. To conceptually link the graphical presentations of distribution shifts, the Fig A3 contains 3 generic cases, each presented in 3 different ways. The left panels illustrate the frequency distribution of the population in 2 time periods, for example. The middle panels show the corresponding cumulative density distribution. Finally, the right panels show the mean-difference plots.^{57} Each point on the mean-difference plot represents the difference against the mean for the corresponding percentile of the 2 distributions.

For example, if the 20th percentile of distribution *A* is 18.5 and the 20th percentile of distribution *B* is 22.5, then one will plot 4.0 (22.5 − 18.5) on the y-axis against 20.5 [(22.5 + 18.5)/2] on the x-axis.

The mean-difference plot is informative in identifying a shift that is by a greater amount at one end of the distribution (case 3), such as in the case of weight shift in children. The plot will show a pattern of bigger differences at higher percentiles.

## Acknowledgments

This study was supported by Robert Wood Johnson Foundation grant 052194. The sponsor reviewed the manuscript but had no role in the design and conduct of the study, interpretation of the data, or preparation of the manuscript.

We thank Drs William Dietz, Claude Bouchard, and Boyd Swinburn for valuable comments. We are also in debt to Drs Gerald Berenson and David Freedman for providing access and analytical assistance with the Bogalusa Heart Study data.

## Footnotes

- Accepted June 23, 2006.
- Address correspondence to Y. Claire Wang, MD, ScD, Department of Health Policy and Management, Harvard School of Public Health, 718 Huntington Ave, 2nd Floor, Boston, MA 02115. E-mail: ywang{at}hsph.harvard.edu
The authors have indicated they have no financial relationships relevant to this article to disclose.

Earlier versions of this manuscript were presented at the Pennington Scientific Symposium; December 4–6, 2005; Baton Rouge, LA; and the 27th annual meeting of the Society for Medical Decision Making; October 21–24, 2005; San Francisco, CA.

## REFERENCES

- Copyright © 2006 by the American Academy of Pediatrics