Approximately 90% of children diagnosed with type 2 diabetes are classified as either overweight or obese (1). Obesity has been linked to various factors, including a chronic imbalance between energy expenditure (EE) and energy intake, environmental exposures, and genetic predispositions. However, the exact contributions of EE to obesity development remain unclear (2). To combat the growing obesity epidemic among children, behavioral researchers increasingly employ targeted, school-based interventions designed to reduce school-day sedentary behaviors among children (3–5). The effects of these interventions on physical activity (PA) are often monitored using wearable devices, such as accelerometers, which collect frequent measures of estimated calorie expenditures or the number of steps taken (6, 7). Wearable devices typically record data at either the second or minute level over multiple days to monitor PA intensity. Often, these measures are used to estimate the metabolic equivalent of tasks, which can be used to derive the amount of time spent performing sedentary, light, moderate, or vigorous PA (8). Alternatively, the data collected over time by wearable devices can be represented by scalar-valued summary numbers such as mean EE or total EE, or by curves (9–12). When data are presented as curves, functional data analysis, which treats curves as the unit of statistical analysis, is a modeling strategy (13–15). Functional data analysis applies data reductions techniques to the curves and subsequently uses regression approaches for statistical modeling. The data reductions can simply consist in summarizing the data from minute-level observations into hourly mean EE values (11). Furthermore, more complex statistical data reduction techniques, such as functional principal components analyses or polynomial basis expansions, have also been used to for approximating the mean of the curves data have also been used (9, 10, 13, 16–19). Polynomial basis expansions approximate curves by describing their shapes using a few key features, summarizing the information contained within curves into basis functions that adequately capture patterns. Unlike summary statistics, such as the mean, which account for only one source of variation in the data, each basis function accounts for a different source of variation (10). Parametric regression approaches, such as nonlinear or polynomial mixed-effects models, have been considered in functional data settings to parametrically model the effects of curves on an outcome (20–22); however, these approaches are limited by the requirement for strong assumptions regarding the shape of the curve. Thus, semiparametric and nonparametric approaches, which provide more flexibility for fitting curves to data by not requiring a specific parametric form, are standard approaches for analyzing functional data (23–25). Additionally, the ability of these approaches to easily accommodate the high dimensionality of functional data is desirable (13, 14).

In children, overweight and obesity are defined according to age-and sex-adjusted body mass index (BMI) values in the upper ranges (26, 27). However, most studies assessing the impacts of behavioral interventions on BMI rely on traditional linear regression models, which are designed to assess the effects of intervention among children within the normal BMI range and have limited ability to assess the effects of interventions among children classified as overweight or obese. Thus, statistical approaches that permit the evaluation of covariate effects across the entire BMI distribution are preferable when assessing the effects of interventions among children classified as overweight or obese (28). Quantile regression is a statistical technique used to estimate the effects of predictors on quantile functions of a response variable (28–33), such as the median (50th), 85th, or 95th quantiles. Quantile regression is advantageous compared with linear regression because quantile regression does not require the regression residuals to be normally distributed. Using classical mean regression models, such as linear regression, to model BMI as an outcome can provide incomplete information regarding BMI values that lie beyond the mean value, such as values within the distribution tails. Additionally, covariates such as PA and age may have differential impacts on different quantile levels. Therefore, statistical approaches that allow covariate effects to be assessed across the full spectrum of quantile functions are preferable when using BMI as an outcome in obesity studies (28, 29).

Different approaches have been used for assessing the relationship between device-based measures of PA patterns and BMI. Wendel et al. (34) recently used classical linear regression to analyze the effects of introducing stand-biased desks in school on the average change in BMI. Their results indicated that compared with using conventional desks, using stand-biased desks significantly reduced the average change in BMI (p = 0.04). However, analyzing the average change in BMI does not allow for the assessment of how standing desk use affects values above or below the average BMI change. Benden et al. (3) reported that children who used stand-biased desks had a significantly higher mean EE (estimate [Est.] = 0.16, standard error [SE] = 0.04, p < 0.001) than students who used traditional desks. Benden et al. used a hierarchical LMEM to assess the impacts of standing desk use on average SDEE (as a measure of PA); however, this approach is limited to assessing the impacts of standing desk use for those children with “average” SDEE values and cannot assess impacts for those children with SDEE values above or below the average. Additionally, the hierarchical LMEM employed did not model SDEE data as curves.

Trinh et al. (35) studied the effects of PA patterns at baseline on the 3-year change in BMI among elementary school–aged children in Australia and found little evidence to indicate that baseline PA patterns were predictive of future obesity risk when applying classical regression methods that treated objective measures of PA as a summary statistic (average step count per minute) (35). However, using summary statistics to describe PA intensity does not account for potential diurnal PA patterns (9, 11, 12, 36, 37). Approaches that allow assessments of diurnal patterns in PA have also been considered in the literature. For example, Tekwe et al. (9) used functional principal components methods and scalar-on-function regression to analyze EE data (38). These approaches allowed the impacts of diurnal PA patterns on obesity-related outcomes to be assessed. Augustin et al. (12) also considered semiparametric approaches to describe PA patterns and used a histogram of the PA distribution as a predictor in their regression model. Using data from the Avon Longitudinal Study of Parents and Children, Augustin et al. established that their approach provided better fits than summary statistics–based methods.

Our current work was motivated by the stand-biased desk study in which a school-based PA intervention study was assessed (3). The cluster-randomized study was conducted from 2011 to 2013 in three elementary schools within the College Station Independent School District (CSISD) (3). The study is described in detail elsewhere (3); briefly, at the beginning of the 2011–2012 academic year, 24 teachers from three elementary schools were recruited and randomly assigned to the use of either stand-biased desks with stools [Stand2learn LLC College Station, TX, USA, stand-biased desk (model S2LK04) and stool (model S2LS04)] or traditional desks (model 2200 FBBK Series by Scholar Craft Products, Birmingham, AL) with chairs (9000 Classic Series, by Virco Inc., Torrance, CA, USA) for in-class activities (3). A total of 374 students in second through fourth grades were included in the study at baseline. To calculate BMI, each student’s height and weight were measured at the start of each semester by trained research assistants. Study participants were required to wear calibrated BodyMedia SenseWear® armband devices (BodyMedia, Pittsburgh, PA) during school hours for 1 school week in each semester from Fall 2011 to Spring 2013. The devices recorded subject-specific step counts and caloric EE per minute while worn. All study participants consented or assented to participate in the study, and consent to participate was obtained from the parents or legal guardians of all participants. The study protocol was approved by the Institutional Review Board, Human Subjects Program at Texas A&M and the CSISD Review Board.

Using a hierarchical linear mixed effects model, Benden et al. (3) showed that children in stand-biased desk classrooms had significantly higher EE than children in traditional desk classrooms (estimate [Est.] = 0.16, standard error [SE] = 0.04, p < 0.001) in the Fall semester, after adjusting for grade, race and gender. However, using a summary value in the hierarchical linear mixed model does not take advantage of the high dimensionality of EE. Figure 1 illustrates such high dimensionality by showing EE data gathered every minute over 5 school days for a randomly selected student participant in our motivating stand-biased desk study. Nonparametric smoothing was used to approximate the average EE recorded over the five school days for a randomly selected student. By smoothing the mean, we uncover underlying patterns in the data while retaining important features (20, 39). The hierarchical linear mixed model does not provide the ability to assess the impact of these underlying patterns on childhood obesity, thus limiting interpretability.

In this manuscript, we use different modeling approaches to examine the relationship between PA and body weight status, as indicated by measures of BMI. We describe the use of a linear mixed-effects model (LMEM), a quantile mixed-effects model (QMEM), a functional mixed-effects model (FMEM), and a functional quantile mixed-effects model (FQMEM) to study the relationship between school-day EE (SDEE) and BMI. We include random effects in all models to account for clustering and treat SDEE as both a scalar-valued summary and a function-valued predictor variable. We discuss the advantages and disadvantages of each modeling approach. The manuscript is organized as follows. In Section 2, we describe the statistical models employed in our applications. The results from our analyses are provided in Section 3, and we offer some concluding remarks in Sections 4 and 5.

In this manuscript, we analyzed data collected at baseline (Fall 2011). Our analytic sample of 256 participants excluded those with large proportions of missing or incomplete accelerometer data. Mean hourly SDEE values were obtained by calculating hourly averages of minute-level device-measured observations across the 5 days of a school week during which the devices were worn. In our analytic sample, all students had exactly 30 mean hourly SDEE values. For all models, we used log (BMI), the natural logarithm of BMI as the response variable and we adjusted for the following covariates: age, sex (boys vs. girls), and intervention (stand-biased desks vs. traditional desks). Given the cluster-randomized design, we attempted to account for the clustering effects of teachers within schools. However, due to computational and convergence issues, we only included a random intercept for schools in all models. We implemented linear regression and quantile regression models with the R software packages lme4 (40) and lqmm (41, 42), respectively. Below, we provide descriptions of the different models considered.

In the remaining of this section, we provide descriptions of the models considered.

We demonstrated and compared mean regression and quantile regression methods for examining the effects of device-based EE measures on BMI. Our analyses indicate that the association between EE and BMI varies depending on how high-dimensional. Device-based EE measures are summarized. When EE measures were summarized into an overall averaged scalar value, we obtained a statistically significant effect of overall mean SDEE on log (BMI). However, when SDEE was considered to be a functional variable, a statistically significant effect of SDEE on log (BMI) was observed for only specific intervals during the device wear time. Thus, spline-based approaches may uncover patterns and allow additional interpretations that are not possible when using approaches that require device-based EE data to be summarized into scalar values. Our results illustrate the complexity of analyzing data collected by devices intended to monitor PA in school-based studies of obesity and body weight status. Despite the existence of much literature describing the obesity-reducing impacts of increasing EE or PA during the school day, we observed that the choice of statistical model is important for accurately assessing the extent of any such relationship.

Across all our models, boys tended to have lower log (BMI) values than girls, and students assigned to stand-biased desks had lower log (BMI) values than those assigned to traditional desks. Age did not have a statistically significant effect on log (BMI), and white students had significantly lower log (BMI) values than non-white students in quantile models at the 10th and 50th quantile levels of log (BMI) only. Based on the 95% pointwise confidence intervals obtained for the FMEM and FQMEM, we observe that the pointwise confidence intervals excluded zero within short time intervals only across the entirety of device wear times, suggesting a significant effect of EE on BMI during these time intervals only and indicated that EE patterns were not independent of time in this study sample. Thus, treating device-measured EE as a function-valued predictor rather than a scalar-valued one may allow for more thorough interpretations of intervention data.

Based on our analyses, no statistical associations between EE and BMI were detected in any of the fitted models. Ad hoc comparisons of AICs tended to favor models using a function-valued predictor. For example, the random intercept LMEM produced an AIC of −358.8, whereas the FMEM produced an AIC of −365.7. The small difference in AICs confirmed a slight advantage for models that treat EE as a function-valued predictor when analyzing our data. Similar comparisons were observed when comparing the same quantiles among quantile-based regression models. When comparing across quantiles, AIC values ranged from approximately −400 to 40, illustrating that EE may have differential effects on different BMI categories. Overall, and based on AIC comparisons, regressions at the 25th quantile appeared to provide the best model fit for our analyses.

We used different regression-based methods to investigate the impacts of EE on BMI among elementary school–aged children recruited from a Texas school district. Using the LMEM, we assessed the effect of overall mean EE on BMI; however, this approach does not account for potential diurnal EE patterns, and the analysis is focused on assessing the conditional BMI mean. Using B-splines in the FMEM uncovered EE patterns and provided more interpretability by modeling objective EE measures as curves. Compared with the spline methodology, methods using the overall mean to represent EE resulted in the loss of information. While both the LMEM and the FMEM enable assessment of covariate effects on the conditional BMI mean, the QMEM and the FQMEM enables assessments of covariate effects across the entire BMI distribution.

One limitation of using quantile regression is related to sample size. Smaller samples tend to limit the implementation of quantile regression models, especially when estimating quantiles of the outcome variable distribution tails or when a large number of covariates are included in the model. Thus, although functional quantile regression models are advantageous for assessing how EE patterns over time affect BMI and enable the effects of these patterns to be assessed across the entire BMI distribution, we recommend the use of functional quantile models for large sample sizes only. For small to moderate sample sizes, estimations around BMI distribution tails may be problematic. Overall, when interest lies in assessing how a function-valued predictor affects children of all body types, quantile regression–based methods are recommended. However, the analytic sample must be sufficiently large to ensure adequate statistical power to assess the effects of PA across the entire BMI distribution.

In our analyses, we accounted for clustering by including random effects for schools using a mixed-effects framework. Failure to account for the cluster-randomized study design may lead to invalid statistical inference, especially for highly clustered data (51, 52). In our analytic sample, the intraclass correlation coefficient was approximately 10%, which highlights the importance of accounting for the clustered effect in our analyses.

The data analyzed in this study is subject to the following licenses/restrictions: The data set and the R codes for the analyses performed in the present manuscript are available from the corresponding author on request. Requests to access these datasets should be directed to GH, gilson.honvoh@cchmc.org.

The studies involving humans were approved by Texas A&M University and College Station Independent School District Institutional Review Boards. The studies were conducted in accordance with the local legislation and institutional requirements. Written informed consent for participation in this study was provided by the participants’ legal guardians/next of kin.

GH: Writing – original draft, Writing – review & editing. RZ: Writing – review & editing. AG: Writing – review & editing. MB: Data curation, Writing – review & editing. CT: Writing – original draft, Writing – review & editing.