## Abstract

Accurate estimation of energy expenditure (EE) in children and adolescents is required for a better understanding of physiological, behavioral, and environmental factors affecting energy balance. Cross-sectional time series (CSTS) models, which account for correlation structure of repeated observations on the same individual, may be advantageous for prediction of EE. CSTS models for prediction of minute-by-minute EE and, hence, total EE (TEE) from heart rate (HR), physical activity (PA) measured by accelerometry, and observable subject variables were developed in 109 children and adolescents by use of Actiheart and 24-h room respiration calorimetry. CSTS models based on HR, PA, time-invariant covariates, and interactions were developed. These dynamic models involve lagged and lead values of HR and lagged values of PA for better description of the series of minute-by-minute EE. CSTS models with random intercepts and random slopes were investigated. For comparison, likelihood ratio tests were used. Log likelihood increased substantially when random slopes for HR and PA were added. The population-specific model uses HR and 1- and 2-min lagged and lead values of HR, HR^{2}, and PA and 1- and 2-min lagged values of PA, PA^{2}, age, age^{2}, sex, weight, height, minimum HR, sitting HR, HR × height, HR × weight, HR × age, PA × weight, and PA × sex interactions (*P* < 0.001). Prediction error for TEE was 0.9 ± 10.3% (mean ± SD). Errors were not correlated with age, weight, height, or body mass index. CSTS modeling provides a useful predictive model for EE and, hence, TEE in children and adolescents on the basis of HR and PA and other observable explanatory subject characteristics of age, sex, weight, and height.

- accelerometers
- heart rate monitoring
- calorimetry
- energy cost of activity
- basal metabolic rate
- sleeping metabolic rate

accurate estimation of energy expenditure (EE) in free-living children and adolescents is integral to an understanding of the physiological, behavioral, and environmental factors affecting energy balance and, thus, normal growth and development. Several methods, including direct calorimetry, indirect calorimetry, and noncalorimetric methods, can be used to estimate EE (24). Because of the confinement and intrusiveness of calorimetry and the cost of the stable-isotope doubly labeled water method, alternative noncalorimetric methods have been developed to estimate EE under field conditions based on heart rate (HR) and physical activity (PA) measured by accelerometry.

The potential for prediction of EE from HR was recognized several decades ago. In 1950, Berggren and Christensen (1) proposed HR as an index of metabolic rate on the basis of the nearly linear relationship between HR and EE during submaximal muscular work. Several limitations of HR monitoring, some biologically inherent and others technological, were soon recognized. Although metabolic demand is a major contributor to the circulatory load, the cardiovascular system supports many other functions, such as heat exchange. The cardiovascular system responds to demand through alterations in blood pressure, stroke volume, and blood volume, as well as HR. In particular, HR does not discriminate well at low levels of EE (17). Because subject characteristics such as age, sex, body posture, and fitness level affect the slope and intercept of a linear HR-EE relationship, the method requires individual laboratory calibration of the HR-EE relationship (1).

Recent technological advances in integrated circuitry and memory capacity directed research efforts toward use of accelerometry to predict EE. Although moderate-to-high correlations between accelerometer counts and EE have been reported for groups of children, individual prediction errors for EE were unacceptably high (7, 14, 15, 18, 19, 29, 30, 33, 36, 37). Limitations inherent to all accelerometers for estimation of EE remain. Static work and movement against external forces are not accurately captured by accelerometers. To overcome these limitations, a number of investigators have combined HR and PA to improve estimates of free-living total energy expenditure (TEE) (4, 5, 8, 10, 15, 26, 31, 34–36). In general, the combination of HR and PA provided more precise estimates of TEE than either used independently, but models relied on individual laboratory calibration of the HR-EE relationship. Simplified calibration procedures to capture between-subject variance in the relationships between EE, HR, and PA have been proposed (6). Other investigators have considered random intercept models to account for individual variation (6).

Besides new technologies and the addition of other physiological data, more sophisticated analytic approaches, such as quadratic discriminant analysis and hidden Markov modeling (28), a two-regression model based on data variability (9, 11), and artificial neural network modeling (32), have been taken to predict EE from accelerometry. Here, we introduce another analytical approach, cross-sectional time series (CSTS) modeling, for the prediction of EE with use of a new technological device.

The usual methods for predicting minute-by-minute EE from HR and PA are linear or polynomial regression models that do not take into account the interdependent nature of the measurements. Since successive observations are usually serially correlated and the variance of repeated measurements is not usually constant over the duration of the study, strong assumptions of regression methods, such as independent and identically distributed residuals, may be violated and lead to biased estimates. Minute-by-minute EE are repeated measurements on the same individual over time, and we expect some dependence between points measured close in time. Therefore, statistical analysis of repeated-measures data must address the issue of correlation between measures on the same individual. The correlation between observations is an advantage for the prediction of EE. CSTS is a parametric method based on regression and time series analysis to model a collection of correlated data. CSTS analysis provides a body of techniques for analysis of the dynamics of the dependent structure of repeated observations over time (12, 16).

The specific aim of the present study is to formulate and apply CSTS modeling to predict minute-by-minute EE and, hence, TEE from HR, PA, and observable subject variables. Our study is motivated by the need to construct models that accurately predict the TEE of individuals and better describe a collection of series of minute-by-minute EE, with the correlation structure of the data taken into account. These CSTS models will be useful in the study of the dynamics of minute-by-minute EE and the factors that influence them.

## MATERIALS AND METHODS

### Study Design

CSTS models for the prediction of minute-by-minute EE and, hence, TEE from HR and PA were developed and tested in 109 children and adolescents by use of room respiration calorimetry and HR and PA monitoring. Subject inclusion criteria stipulated that the children had to be 5–18 yr of age, healthy, and free from any medical condition that would limit participation in physical activity or exercise. For development of robust prediction equations, the sample was balanced with respect to sex and weight status, with the intent to recruit 50% overweight subjects, defined as having a body mass index (BMI) ≥95th Center for Disease Control and Prevention percentile (21).

The study design entailed a 24-h stay in a room respiration calorimeter for simultaneous measurements of EE, HR, and PA. The Institutional Review Board for Human Subject Research for Baylor College of Medicine and Affiliated Hospitals approved the protocol. All parents gave informed consent, and the children gave informed assent to participate in the study.

### Room Respiration Calorimetry

O_{2} consumption (V̇o_{2}) and CO_{2} production (V̇co_{2}) were measured continuously in one of our two 30-m^{3} room calorimeters for 24 h. Our calorimeters were designed to achieve a minimum response time of 2 min. The calorimeter construction, instrumentation, engineering, and data reprocessing to achieve this goal are thoroughly described in our earlier publication (27). Briefly, V̇o_{2} and V̇co_{2} are measured with paramagnetic O_{2} (Oxymat 6) and nondispersive infrared CO_{2} (Ultramat 6) gas analyzers (Siemens, Karlsruhe, Germany) and thermal mass flow controllers (models 740 and 840, Sierra Instruments, Monterey, CA). Our 1-min sampling interval represents the average of ten 6-s intervals. Errors from 24-h infusions of N_{2} and CO_{2} were −0.34 ± 1.24% for V̇o_{2} and 0.11 ± 0.98% for V̇co_{2}. EE was computed using the de Weir equation (38). In addition, HR was recorded by telemetry in the room calorimeter (model DS-3000, Fukuda Denshi, Tokyo, Japan). V̇o_{2}, V̇co_{2}, EE, and HR were averaged at 1-min intervals.

All subjects completed a series of physical activities under supervision in the same order between 9 AM and 5 PM, with “free time” and meal time between measurements. Specific measurements obtained in the room calorimeter included working on a computer for 20 min while sitting in a chair; playing PlayStation 2 games for 20 min in a sitting position; walking on the treadmill (model C942, Precor, Woodinville, WA) at 2.5 miles/h (mph) for 15 min; walking on the treadmill at 3.1 or 3.7 mph set according to age, capability, and safety of the children for 15 min; slow jogging on the treadmill at 3.1–4.3 mph set according to age, capability, and safety of the children for 15 min; playing EyeToy in the standing position for 15 min; performing aerobic exercises, as demonstrated on a videotape, for 15 min; watching a movie while sitting in a lounge chair for 90 min; walking on the treadmill at 1.8 mph for 10 min; jogging/running on the treadmill at 3.7–6.2 mph set according to age, capability, and safety of the children for 15 min; cool-down after jogging/running while sitting for 20 min; resting in a lounge chair while watching television for 20 min; sitting and assembling a floor puzzle for 15 min; dancing to music for 15 min; basal metabolic rate (BMR) measured under thermoneutral conditions (22–24°C) upon awakening after a 12-h fast for 30 min. Except for the BMR measurement, all activities were performed in the fed state. The children were given breakfast at 8:30 AM, lunch at 12 PM, a snack at 2:45 PM, and dinner at 5:30 PM.

Prediction models for rates of EE (kcal/min) are presented for the 24-h period (TEE), the awake period (awake EE), and the sleep period (sleep EE). In addition, a CSTS model was developed for the prediction of activity EE (AEE) during the awake period, defined as AEE = EE − BMR − 0.1 EE, with the assumption that the thermic effect of food is equal to 10% of TEE.

### HR and PA Monitoring

HR and PA were measured by Actiheart (Mini Mitter, Respironics, Bend, OR), a compact (7 mm thick, 33 mm diameter, 10 g total weight) ambulatory device equipped with an omnidirectional accelerometer and ECG signal processor. The accelerometer contains a piezoelectric transducer, a crystalline material that generates a surface voltage in response to a change in motion. The piezoelectric transducer used in Actiheart is of the bimorph element type, the primary response of which is achieved when the motion is parallel to a preferred axis of the crystal. Thus the piezoelectric sensor in Actiheart may be referred to as an anisotropic sensor; the motion sensitivity of this sensor is greatest along the preferred axis (vertical, when worn as prescribed), but it will produce lesser signals when the motion is perpendicular to the preferred axis. The 128-kb memory capacity of allows data storage for 11 days at the 15-s epoch setting. The piezoelectric element (frequency range 1–7 Hz) generates a transient charge when exposed to time-varying acceleration. This produces a voltage signal, which is converted to a binary signal, resulting in 256 distinct levels of acceleration. The dynamic range of the accelerometer is ±25 m/s^{2} (±2.5 *G*) and its sensitivity per bit is approximately one count per 0.025 *G* or one count per 0.23 m/s^{2}. The binary signal is summed over a 15-s epoch. Actiheart digitizes the ECG signal and calculates the HR from the interbeat interval (IBI). It detects the QRS complex by identifying the location of the R-wave line of steepest descent. The logger firmware applies a digital threshold for this differential value that compensates for variation in the signal level due to physiological changes, noise, and physical movement. Sixteen consecutive IBIs are measured, and the average of the 16 IBSs is calculated. Any of the IBI values that are >37.5% of the average are identified and discarded. The average IBI is recalculated, and its inverse is multiplied by 60 to obtain the HR.

Before each test, the calibration of the Actiheart device is confirmed using the Motion Performance Verification System (Mini Mitter). Since the accelerometer is designed to detect vertical movement with the subject in the upright position, the positioning of the Actiheart on the chest is important for maximum accuracy. The main sensor is attached left of the sternum, and the lead is parallel along the midclavicular line at the level of the third intercostal space (upper position) or just below the left breast (lower position) using electrodes (Skintact Premier, Leonhard Lang, Innsbruck, Austria). Variability of PA measurements within and between Actiheart monitors was tested using repeated measurements on the Motion Performance Verification System. Intramonitor coefficient of variation (CV) was 1.3%, and intermonitor CV was 4.3%. HR measurements of the Actiheart monitors were in perfect agreement with an ECG simulator (Dale Technology, Thornwood, NY) operating at rates of 60 and 120 beats/min. Agreement between HR independently measured by the Fukuda room calorimeter and Actiheart was 0.2 ± 0.1% for all subjects.

### Statistical Analysis

Values are means ± SD. Descriptive statistics and CSTS modeling were performed using STATA (release 8.2, Stata, College Station, TX) and SPSS (release 11.50, SPSS, Chicago, IL). HR and PA data acquisition by Actiheart is set at 15-s intervals. Actiheart data were collapsed into 60-s intervals and aligned with the minute-by-minute EE data. HR data were filtered with an upper cutoff of 240 beats/min and a lower cutoff set at 10 beats/min below the subject's average sleeping HR. Minute-by-minute EE are summed for discrete PAs and 24-h periods. HR during awake and sleep periods were calculated as the average HR for the entire times during which the subject was awake and asleep. Minimal HR was determined to be the lowest 20-min average of HR during sleep. Sitting HR was the average HR for a 20-min period while the subject was watching television in the upright seated position.

### CSTS Model

To formulate and apply CSTS-type models to predict the minute-by-minute EE based on HR, PA, and other potential covariates, let *y*_{ij} denote the minute-by-minute EE measures on the *i*th individual at consecutive time points *j*. We consider the following CSTS or mixed regression model with random intercepts and random slopes (1) where β is a vector of regression coefficients associated with the covariates *x*_{ij} (e.g., HR and PA) and contains population-specific parameters describing average trends, *b*_{i} are independent vectors of random effects associated with covariates *z*_{ij} and contain subject-specific parameters describing how the response of the *i*th individual deviates from the mean response over time, and ε_{ij} is the random noise for the *i*th individual at time *j*. We assume that *b*_{i} and ε_{ij} are mutually independent. The term *x*_{ij}β comprises the fixed-effects portion of the model and the term *z*_{ij}*b*_{i} comprises the random effects, and *z*_{ij} is a subset of *x*_{ij}, that is, some subset of the regression parameters that varies randomly from one child to another, accounting for sources of heterogeneity in the population. In *model 1*, *z*_{ij}*b*_{i} + ε_{ij} imposes a correlated error structure. It is clear that different types of CSTS models can be defined by different specifications of the correlation structure. The distinguishing feature of *model 1* is that the regression coefficients may differ across individuals; in particular, if *x*_{ij} = *z*_{ij}, then each individual has his or her own regression parameters. Because the data are inherently unbalanced, accounting for the covariance among the repeated observations of EE on the same individual using CSTS with random effects is more appropriate. Also, because CSTS models can explicitly distinguish between fixed and random effects, they allow the analysis of between- and within-subject sources of variation. A key idea of the application of CSTS is that, by pooling information from a number of time series of minute-by-minute EE, we can obtain a parsimonious model as well as more accurate estimates of the parameters of the model.

In a submodel of *model 1*, it is assumed that *z*_{ij}*b*_{i} = *b*_{i}, that is where *b*_{i} and ε_{ij} are independent and identically distributed normal random variables with mean zero. This model with random intercepts and fixed slope describes the mean response trajectory over time for any individual as well as the mean profile in the population, with the assumption of the same correlation between any pair of measurements (i.e., compound symmetry). Here the subject-specific intercept *b*_{i} reflects individual heterogeneity and induces correlation among the measurements within each individual EE series. *Submodel 1* provides a parsimonious model compared with *model 1* if it captures the subject-specific variations in the data.

Although we are primarily interested in estimating the population parameter β, we have also calculated estimates for random effects, since they reflect the extent to which the subject-specific profiles deviate from the overall average profile. The assumption of normality was tested by examination of the histograms for the random coefficients.

Application of a CSTS model requires the specification of the mean process, together with a model for the covariance structure of the responses on each subject. To develop an appropriate CSTS model, we followed an approach similar to recommendations of Diggle (12). The first step is selection of an initial set of fixed effects in sufficient generality to specify the mean structure. The second step is selection of random effects and intraindividual variation to specify a model for the covariance structure. A set of sample autocorrelation and partial autocorrelation coefficients for the estimated disturbance series was evaluated. Akaike's information criterion and Bayesian information criterion were used to compare the models with different covariance structures.

### Development of the CSTS Model

To develop an appropriate CSTS model, we selected an initial set of fixed effects, including linear and quadratic terms of HR, PA, and their interactions, in sufficient generality to specify the mean structure. Next, we selected random effects (HR and PA with unstructured variance-covariance matrix) and intraindividual variation to specify a model for the covariance structure. We investigated a set of sample autocorrelation and partial autocorrelation coefficients for the estimated disturbance series. Three categories of predictor variables were incorporated into the CSTS models. The first category consists of the time-varying variables: HR, HR^{2}, 1- and 2-min lagged values of HR (HR-lag1 and HR-lag2), 1- and 2-min lead values of HR (HR-lead1 and HR-lead2), PA, PA^{2}, and 1- and 2-min lagged values of PA (PA-lag1 and PA-lag2). HR^{3} was evaluated but did not add significantly to the CSTS model. The second category consists of subject-specific characteristics such as sex and ethnicity or time-invariant variables during the study period, such as age, height, weight, minimal HR, and sitting HR. The third category consists of appropriate interaction terms between HR, PA, and other variables.

### Evaluation of the CSTS Model

We analyzed concordance between the observed and predicted TEE. Concordance between two methods indicates the extent to which measurements made by one of the methods can serve as a surrogate for the other. Two variables are in perfect concordance if the points in the scatterplot of the two variables lie along the line of equality (the concordance line), which is the 45° line through the origin, but two variables have perfect correlation if the points lie along any straight line. For a pair of measurements (*X*_{i}, *Y*_{i}) on a subject, the obvious measure of agreement is simply the difference between them, and if the difference is “sufficiently small,” then they can be used interchangeably. Bland and Altman (2) recommend a graphical method to assess concordance or agreement between two variables. Their plot is based on the fact that, for a pair of measurements, their agreement is simply the difference between them. If this difference is sufficiently small, which is determined by substantive criteria, rather than a purely statistical one, then there is an agreement or concordance. This idea can also be applied to measurement of prediction errors. Bland and Altman propose to plot the difference vs. the “true” value or mean of the two methods and the “limits of agreement,” which are two horizontal lines indicating mean ± 2SD of intraindividual differences. In our application, the observed TEE measured by calorimetry was taken as the true value and used on the *x*-axis.

Although the Bland-Altman (2) diagnostic plot of the difference vs. the mean can provide insight into the measurement differences between two methods, it does not provide a single measure of agreement. Krippendorff (20) and Lin (22, 23) considered the concordance correlation coefficient (CCC), which is appropriate for measuring agreement when the data are measured on a continuous scale. The CCC consists of a precision component, the Pearson correlation coefficient, which measures how closely observations lie on the line fit to the data, and an accuracy component, which measures how closely the fitted line deviates from the 45° line through the origin. Therefore, when the concordance correlation is high, we can be more confident about the similarity of the two methods.

## RESULTS

### Subject Characteristics

The 61 boys and 48 girls who participated in the study are described in Table 1. Mean age of the children was 12.3 ± 3.5 yr, with a range evenly distributed across 5–18 yr of age. Forty-one percent of the boys and 46% of the girls were classified as overweight.

### Observed EE, HR, and PA

The mean values for EE, HR, and PA for the entire 24-h, awake, and sleep periods are summarized in Table 2. The mean PA level, defined as the ratio of TEE to BMR, was 1.54 ± 0.13, indicating that a low-to-moderate PA level was achieved within the room calorimeter. As potential indicators of an individual's fitness level, sleep HR, minimal HR, and sitting HR were extracted from individual 24-h files. An example of one subject's pattern of HR, PA (Fig. 1*A*), and measured and predicted EE (Fig. 1*B*) is illustrated.

### CSTS Modeling

The CSTS model is based on time-varying variables, i.e., HR and PA, subject-specific variables, and appropriate interaction terms between HR, PA, and other variables. Significant covariates included weight, height, minimal HR, and sitting HR. Although age, age^{2}, and sex were not significant in all models, they were retained as basic descriptors of children. Significant interactions between HR or PA and weight, height, sex, or age were specific to each model.

Inclusion of the time-dependent variables, HR and PA, and the lagged covariates, lagged HR and PA, and lead values of HR substantially improves the prediction of EE. Subject characteristics of age, age^{2}, sex, weight, and height account for 86% of the variance in TEE. However, we are not only interested in predicting TEE; our goal is to predict minute-to-minute EE, from which we sum values to arrive at TEE, or the shorter awake or sleep period. The subject characteristics cannot explain minute-to-minute intraindividual variation in EE, since they are constants. The contribution of HR and PA to the prediction of EE was evaluated by CSTS analysis (Table 3) and with subject characteristics (Table 3). The CSTS model based only on the HR and PA variables explained 72% of the variability in minute-by-minute EE. This CSTS model tracks EE reasonably well (Fig. 2); however, the prediction errors are unacceptably high (8.7 ± 30.9%). Inclusion of the subject characteristics anchors the EE predictions, and PA and HR capture the variations in EE. Together, subject characteristics, PA, and HR in our CSTS model explained 90% of the variability in minute-to-minute EE. The *R*^{2} goodness-of-fit criterion for the minute-by-minute PA and HR + subject characteristics model is stronger than the *R*^{2} for the regression model predicting EE from subject characteristics.

Initially, we fitted *submodel 1* to the collection of EE series; later, we added random slopes for HR and PA. Using the likelihood ratio test, we chose the model with random slopes. With the assumption that HR and PA vary randomly, this would induce covariances among the repeated observations of EE that are functions of the HR and PA of the individual. The likelihood ratio test comparing the random intercepts model (i.e., *submodel 1*) with ordinary regression is highly significant (*P* = 0.001), and the likelihood ratio test comparing the random intercepts and random slopes with random intercepts model is also highly significant (*P* = 0.001). The log likelihood increased substantially when we added random slopes for HR and PA, providing evidence that the random slopes are needed.

### Evaluation of the CSTS Model

#### TEE.

As expected, the subject-specific CSTS model resulted in nearly perfect concordance between the observed and predicted TEE. The CCC was almost 1.0, and the mean prediction error for TEE was 0.0 ± 0.1%. By application of 1,440 min of EE to HR and PA data, an individual's coefficients in the CSTS *model 1* were defined with a high level of accuracy. Because our specific aim is to develop a laboratory calibration-free prediction equation of TEE, we proceeded with the population-specific mean response, averaged over the distribution of the subject-specific effects. The degree of concordance between the measured and predicted TEE for the population-specific CSTS model (i.e., predicted TEE based on the vector of fixed effects obtained from *model 1*) is illustrated in Fig. 3.

The final population-specific CSTS models for the prediction of EE from HR and PA for estimates of the fixed effects are presented in Table 3. The random-effects parameters were 0.006 ± 0.0004 (SE) [95% confidence interval (CI) = 0.005–0.007] for the HR SD, 0.0006 ± 0.00004 (95% CI = 0.0006–0.0007) for the PA SD, and −0.55 ± 0.07 (95% CI = −0.67 to −0.40) for the correlation between HR and PA. When these variance components are compared with their standard errors, there is strong evidence to support their retention in the model. We have assumed that the variance-covariance matrix of *b̂* is unstructured, which provides a more flexible model. We examined the underlying distribution of random effects via histograms and normal Q-Q plots, and normality assumptions of the model were satisfied, as illustrated by the histograms of the best linear unbiased prediction for random intercepts and slopes (HR) in *model 1* (Fig. 4). Similar results were seen for the random slopes for PA. In addition, the normal Q-Q plots were close to a straight line, suggesting that the distributional assumption is reasonable.

The Bland-Altman plot indicates that the predicted values of TEE are in good agreement with the observed TEE and that there is no bias with increasing TEE (Fig. 5). Sixty-eight percent of the predicted TEE values are within 10% of the observed TEE. The CCC for TEE was 0.94. The prediction error for TEE was 11.0 ± 205 kcal or 0.9 ± 10.3% (Table 4). Prediction errors were not correlated with sex, age, weight, height, or BMI. The prediction errors for TEE in the nonoverweight and overweight boys and girls are presented by age tertiles (Fig. 6).

### Evaluation of CSTS Model

#### Components of TEE.

The prediction errors for EE of discrete physical activities were as follows: 5.3% for BMR, 6.5% for watching a movie, −2.6% for working on the computer, 0.2% for playing PlayStation, −18.9% for assembling a floor puzzle, 6.5% for aerobics, 10.3% for dancing, 0.7% for playing EyeToy, −11.0 to 0.8% for walking, −2.4 to 2.0% for jogging, and 16.9% for cool-down.

Applying the 24-h CSTS model, we found that the prediction errors were greater during the sleep (6.1 ± 15.0%) than during the awake (−0.5 ± 9.9%) period. Therefore, we explored the development of CSTS models based on the awake and sleep periods separately. The parameters and prediction errors for these models are presented in Tables 3 and 4, respectively. Predicting TEE from the combined results of the separate models improved the prediction errors for TEE slightly, to 8.7 ± 186.3 kcal or 0.8 ± 9.2%. Seventy-two percent of the predicted TEE values are within 10% of the observed TEE for the combined model. Use of the sleep model improved the prediction error for sleep EE alone to 1.5 ± 8.7%. Applying CSTS analysis, we developed a model for the prediction of AEE from subject characteristics, HR, and PA. AEE closely tracked EE; AEE is an affine transformation of EE during the awake period (Fig. 7). The parameters and prediction errors are presented in Tables 3 and 4. The prediction error for AEE was 0.4 ± 108 kcal or 1.1 ± 16.0%.

## DISCUSSION

Application of CSTS modeling resulted in an accurate population-specific model for the prediction of minute-by-minute EE and, hence, TEE from HR and PA. This represents a significant advancement in field methodology for the estimation of TEE, since its application is laboratory calibration free and prediction errors are acceptable at the level of the individual.

The combination of HR and PA to estimate free-living EE is not new (4, 5, 8, 10, 15, 26, 31, 34–36). Our research group utilized PA to assign HR to one of two linear regressions relating HR to EE (26). In that adult study, the mean prediction error of the HR + PA model was −3.4 ± 4.5% (26). In a subsequent study, we applied the combined HR + PA model to the prediction of TEE in 20 children (36). Again, the combined HR + PA model yielded the lowest errors for TEE (−2.9 ± 5.1%). A similar approach was taken in eight adults (31); minute-by-minute HR was converted to EE using individual calibration curves, with the motion data discriminating between periods of inactivity and activity. The percent error of the HR + PA model from calorimeter measures was 0% (−22 to 19%). In another approach, CSA accelerometers were used to discriminate between arm and leg movement, and EE was predicted from the corresponding arm or leg HR-EE regression equation (34, 35). The HR + PA model accounted for 81% of the variance in EE. In the aforementioned studies, individual laboratory calibration of the relationship between HR and EE was required.

Brage et al. (5) tested whether HR + PA was sufficiently precise to preclude the need for individual calibration. On the basis of calibration equations using CSA accelerometers and Polar HR monitoring during treadmill walking and running in 12 men, mean errors for AEE during 24-h room calorimetry were −4.4 ± 29% and 3.5 ± 20.1% for individual and group calibration, respectively. Actiheart HR + PA models were evaluated for the assessment of EE in adults (10) and children (8). In adults, the mean error for the Actiheart HR + PA algorithm was 0.02 kJ·kg^{−1}·min^{−1} (−0.17 to 0.22 kJ·kg^{−1}·min^{−1}) during 18 structured activities, suggesting that the algorithm is acceptable for groups, but not for individuals. In children, Actiheart HR + PA accounted for 86% of the variance in AEE during treadmill walking and running, without systematic bias.

In general, HR + PA overcomes the shortcomings of either method alone to predict TEE. HR is not as useful for predicting EE in light activity as it is in moderate-to-vigorous activity. Accelerometers mounted on the hip tend to overestimate sedentary activities and underestimate moderate-to-vigorous activities that involve upper body movement and inclines (15). For low-intensity activities, accelerometry is useful for discriminating HR responses to EE vs. other extraneous factors. HR + PA has proven to be more accurate and precise than either method alone.

The prediction of minute-by-minute EE from HR and PA is becoming increasingly common with the development of small, noninvasive electronic devices such as Actiheart. However, the mathematical modeling of the HR and PA has been limited to regression models that do not take into account the interdependence of EE, HR, and PA over time. As a result, these methods have not exploited all the information in the raw data. The field of CSTS analysis provides a body of techniques for analyzing the dynamics of the dependent structure of observations, i.e., repeated measurements taken from a cross section of subjects (12, 16). In this framework, the key idea of the application is that, by pooling information from a large number of time series, we can obtain more accurate estimates of the parameters, rather than evaluate a single time series. In general, the heterogeneity among the subjects or cross-sectional units is modeled as a random coefficient; that is, individual specific effects are treated as random. CSTS is a parametric approach to model a collection of correlated data, taking into account within-individual changes and between-individual heterogeneity. Nonparametric methods, such as artificial neural network or hidden Markov modeling, are other alternatives for predicting EE (28, 32). Using a classification based on the CV, Crouter et al. (9, 11) considered a two-regression model for predicting EE. These alternative approaches also have improved on simple regression models for the prediction of EE.

In the present study, we introduce CSTS modeling for prediction of minute-by-minute EE and, hence, TEE from HR and PA and other covariates. Our population-specific CSTS model is distinctively applicable to the dynamic nature of EE data and substantially increased the accuracy of prediction of TEE for individuals. Although changes in PA tend to be more abrupt, minute-by-minute changes in HR and EE are more gradual and intercorrelated. CSTS takes advantage of this correlated structure in forecasting changes in EE based on previous and anticipated data. CSTS can also address the temporal dissociation between EE and HR during intermittent PAs (3, 25). In our CSTS model, the 1- and 2-min lagged values of HR account for HR changes lagging behind changes in EE. Dugas et al. (13) demonstrated that inclusion of HR from the previous minute improved the prediction of EE from HR. Our CSTS model was developed to predict TEE or, alternatively, EE during awake and sleep periods from HR and PA. Other Actiheart algorithms have been developed to predict AEE, a value calculated as the difference between TEE and resting EE, from HR above sleeping HR and PA. These authors contend that interindividual variance in HR-EE relationships is removed by utilizing HR above sleeping HR (8, 10). Such an approach may correct for variation in intercepts among individuals, but not for interindividual differences, in HR-EE slopes.

Our final CSTS model for the prediction of EE from HR and PA was a random-intercepts, random-slopes model with subject characteristics and relevant interactions. Evaluation of the CSTS model indicated satisfactory fit to the data. The histograms of the residuals of the intercept and slope coefficients for HR and PA appeared approximately consistent with normality. In addition, the normal Q-Q plots suggested that the distributional assumption was reasonable. The Bland-Altman plot demonstrated good agreement between the observed and predicted TEE and no systematic bias as EE increases.

Although we sought to develop population-specific prediction equations, we evaluated subject-specific prediction equations for TEE. Mean prediction error for TEE was 0.0 ± 0.1%, which represents a significant improvement over other studies that require individual calibration of the HR-EE relationship (15, 26, 31, 34–36). For our population-specific model, the prediction error for TEE was 0.9 ± 10.3%, with no systematic bias by sex, age, or weight status.

Although our primary aim was to predict TEE from minute-by-minute EE, we examined errors in predicting EE during discrete PAs to identify sources of error. The CSTS model tracked the observed EE quite well for most activities, with some exceptions. During the cool-down period following jogging/running, the CSTS model overestimated EE by 17% because of the HR recovery period following exercise. During assembly of a floor puzzle, the model underestimated EE by 19% because of relatively low activity counts in response to upper body movement in the sitting position. For the other activities, which represented lying, sitting, standing, aerobics, dancing, walking, and jogging, the mean prediction errors were ≤11%. We also divided the 24-h period into awake and sleep periods. The overall CSTS model performed less well during the sleep (6.1 ± 15.0%) than during the awake (−0.5 ± 9.9%) period. Because of the low EE during sleep, this error does not contribute substantially to the 24-h prediction error (0.9 ± 10.3%). However, a slight improvement in the prediction of TEE (0.8 ± 9.2%) could be gained by the more complicated process of identifying the sleep and awake periods, applying the separate models for each period, and then combining the two models to obtain TEE. If the EE during sleep is of primary interest, the sleep-specific CSTS model would be more accurate (1.5 ± 8.7%). Our model can be used to produce meaningful predictions of EE in the fed awake state and the fasting sleep state. The CSTS model that we have proposed can be applied for any time period, as long as all the variables in the model are available.

CSTS modeling provides a useful predictive model for minute-by-minute EE and, hence, TEE in children and adolescents on the basis of HR and PA and other observable explanatory subject characteristics of age, sex, weight, and height. The calibration-free population-specific CSTS model represents a significant advancement in field methodology for the prediction of TEE. Future studies will validate this novel CSTS model against the doubly labeled water method in free-living children and adolescents.

## GRANTS

This project was funded by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-74387 and the USDA/ARS under Cooperative Agreement 6250-51000-037.

## Acknowledgments

The authors thank the children who participated in this study and acknowledge the contributions of Mercedes Alejandro for study coordination and the staff of the Metabolic Research Unit for nursing and dietary support.

This work is a publication of the US Department of Agriculture (USDA)/Agricultural Research Service (ARS) Children's Nutrition Research Center, Department of Pediatrics, Baylor College of Medicine and Texas Children's Hospital, Houston, TX.

The contents of this publication do not necessarily reflect the views or policies of the National Institutes of Health or USDA, nor does mention of trade names, commercial products, or organizations imply endorsement by the US Government.

Present address of I. Zakeri: Dept. of Epidemiology and Biostatistics, Drexel University, Philadelphia, PA 19102.

## Footnotes

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- Copyright © 2008 the American Physiological Society