Despite the mounting evidence supporting the value of regular physical activity (PA), anywhere from 56% to 96% of adults do not meet the 2018 Physical Activity Guidelines for Americans of 150 minutes of moderate PA per week [1–4]. In some populations, such as those with clinical conditions or older adults, even fewer are routinely achieving 150 minutes of moderate-to-vigorous PA [5–7]. Independent of meeting physical activity guidelines, sedentary time is also a risk factor for poorer health outcomes [8]. The health benefits of regular light PA and/or reducing sedentary time across a wide range of populations have been postulated and supported by strong evidence across a range of populations [5, 9, 10]. Thus, effective means to differentiate sedentary, light, moderate, and vigorous PA are needed.

Objective measures of PA have gained popularity and acceptance in commercial and more recently, healthcare domains [11–13], providing some advantages over the traditional self-report methodologies. However, while activity measurement has been validated in numerous studies, largely in young adults [2, 14–21], there remain considerable variations in the algorithms used to estimate energy expenditure from triaxial acceleration signals [22, 23] with relatively few parsimonious wrist-based algorithms to choose from.

Older approaches to estimating PA from hip-worn accelerometry found the relationship between activity “counts” and activity intensity often depended on the type and intensity of the activity itself. In fact, the use of “Crouter equations” involved the use of two linear equations to fit what is essentially a nonlinear relationship [16]. In response, multiple complex machine learning and random forest approaches have been developed [21], yet without clear or consistent improvements in energy expenditure estimates over more simple algorithms [23]. Advances in PA measurement have moved away from the use of “counts”, which are often ill-defined and can vary from device to device, towards the use of raw accelerations to extract mean and variance of accelerations per unit time, such as that proposed by Hildebrand et al. [19]. However, the linear nature of the Hildebrand approach makes it unable to differentiate sedentary from light activity, as the algorithm intercept produces a minimum or floor energy expenditure estimate of approximately 2 metabolic equivalents (METs). To address this limitation, a follow-up study identified 45 milli g/minute (mg/min) as a cut-off for classifying sedentary behavior that can be applied separately [24]. While this is an improvement, it is not fully able to address the inherent nonlinearities observed. For example, using their linear algorithm, the 45 mg/min cut-off is equivalent to 2.49 METs. Thus, when this correction is applied, the linear algorithm is incapable of identifying any light activity in the range from 1.6 METs to 2.48 METs. This level of PA translates to energy expenditure levels expected for many activities of daily living (ADLs) and slow walking [25, 26], common types of activities for many patient cohorts [27].

Traditionally, hip-worn accelerometers were commonplace with numerous algorithms proposed and validated in the literature. However, wrist-worn devices are increasingly popular due to improved compliance and ease of use [20]. Indeed, there has been a substantial increase in consumer-based activity monitoring devices worn on the wrist [28]. However, studies have demonstrated that algorithms to estimate PA levels typically differ between hip and wrist-worn measurements [20], with wrist accelerations often higher than hip for the same activities [19].

While the linear Hildebrand approach produced the most consistent estimates of moderate to vigorous PA (MVPA) relative to a validated hip measure of activity out of several different algorithms, its inability to detect a region of light PA is a substantial barrier to its use in both healthy and clinical populations [29]. Nonlinearities in the relationships between acceleration outcomes and energy expenditure are often reported (despite the use of linear regression [16, 19]). We hypothesized that a modified Hildebrand approach, using a nonlinear algorithm, could resolve this problem and eliminate the need to apply a separate cut-off to differentiate sedentary from light behavior, while maintaining the accuracy in estimating MVPA. Thus, the primary purpose of this study was to test this hypothesis using mean raw acceleration signals measured at the wrist and the hip in healthy adults across the lifespan performing a range of different controlled laboratory tasks from sedentary to vigorous intensity. These findings would then provide the needed validation as a basis for its use in patient cohorts, who may be expected to have generally lower levels of PA. Secondarily we aimed to compare hip and wrist accelerations across activities to investigate if a simple transformation may be applicable between wear locations and to explore whether this may be applicable for improving comparisons across methodologies.

A combination of previously reported [19] and newly collected accelerometry data, with their corresponding measured or estimated oxygen consumption (VO₂), were best fit to linear and nonlinear equations. To supplement the previously reported laboratory tasks, additional data involving a wider range of task intensities, particularly higher intensity tasks, were collected. The process of collecting and extracting the data, fitting the parameter values for each model, and the statistical comparisons to assess these models are outlined below.

While PA is not without risks, it is widely recognized as providing many physical and mental health benefits, including older adults and those with clinical conditions [5, 6, 9, 10, 33]. The use of wearable sensors has great potential to aid clinicians in caring for their patients as well as provide objective assessments for patients to self-monitor their health and provide a degree of locus of control over their self-management. The use of objective wearables may be of value in clinical rehabilitation and/or hospital settings to better track activity levels throughout a patient’s day and not only when nursing or rehabilitation therapies can assist with movement [11]. However, there remains a critical need for researchers to have validated algorithms to accurately interpret the acceleration signals, particularly at the lower PA intensities when targeting patient populations.

A wide range of processing models have been proposed to estimate PA metrics from raw acceleration signals, including simple, linear regression [19] to more complex decision tree models [21]. Following the law of parsimony, which suggests that the simplest model to adequately represent a phenomenon should be chosen, finding the most accurate yet straightforward model is optimal. The current study supports the use of a single nonlinear equation to define activity intensity based simply on the mean accelerometer ENMO output. Indeed, the estimated energy expenditure for light, moderate, and vigorous activities in the current study was nearly identical to the previously validated Hildebrand linear approach, supporting that this basic approach is generalizable. However, the nonlinear algorithm has the added advantage of being able to discriminate sedentary from light activity which is not possible using the original linear Hildebrand approach without implementing a 2nd step, which adds potentially unnecessary complexity. Certainly, identifying specific behaviors as sedentary using techniques such as machine learning is an alternate strategy to assigning low energy expenditures as sedentary, but is beyond the scope of this study.

While linear models provide a simple and traditional approach, their drawback is the intercept value defining the lowest VO₂ estimate when accelerations are zero. Hildebrand reported wrist and hip model intercept values of 7.28 and 6.67 mLO₂/kg/min, respectively, using a similar methodology as this current study [19]. This translates to 2.1 and 1.9 METs, even when no activity is measured. Thus, sedentary behavior (based on typical cut-offs of < 1.5 METs or < 1.6 METs) would be identified as light activity, regardless of the true energy expenditure required. In recognition of this limitation, they proposed acceleration cut-offs to separately identify sedentary activity before applying their linear model [24]. This correction factor, 45 mg or below for the wrist (47 mg for the hip) in adults was reported to have excellent sensitivity (> 95%) but only moderate specificity (74–78%) [24]. However, the sedentary cut-off values identified: 44.8 mg/min and 47.4 mg/min for wrist and hip, respectively, correspond to 2.49 METs and 2.66 METs using their linear equation [19]. This correction results in erroneously identifying light activity occurring between 1.5 METs and 2.5–2.7 METs as sedentary. Accordingly, this correction factor approach cannot identify any light activity until it exceeds 2.5 METs when this correction is applied. Whereas the nonlinear model provides similar estimates as the linear-alternate model without the limitations associated with this two-step approach.

Crouter and colleagues [16] observed a similar nonlinear phenomenon, addressing the nonlinearities between activity counts measured using hip-worn devices and energy expenditure with the use of two, discontinuous linear equations. While the commonly referred to “Crouter equation” approach relies on the variance of the acceleration signal as the equation determinant, this variance is somewhat collinear with intensity. That is, greater variation was observed at the lower intensities (i.e., lifestyle activities), and less variation was observed with walking, jogging, and running, which were the higher-intensity activities evaluated. Further, this two-regression model approach has not been adapted to process raw accelerometry data, rather it uses only “count” data for the hip and is unavailable for wrist-worn measurement [15, 16]. Corrections available in software platforms, such as ActiLife to adjust wrist-based measurement for use with hip-based algorithms induce greater error than observed differences between hip and wrist wear during walking [34]. Thus, single nonlinear equations that are specific to hip and wrist wear provide easily defined and user-friendly approaches to estimating energy expenditure across the intensity spectrum from acceleration data that can be transparently processed by the end user.

Two studies comparing four different processing algorithms further support the use of this nonlinear modification. The first is similar to the current study, in that it involved lab-based activities but in young adults [23]; the second assessed daily free-living PA in women with fibromyalgia [35]. Both compared the linear and nonlinear approaches included in the current study, relying only on mean accelerations per minute, as well as two more complex approaches that relied on acceleration variability metrics (SD of the signal) and/or random forest methods [21]. Only this nonlinear algorithm resulted in estimates of low, moderate, and high PA that correlated consistently with self-reports in the patient cohort; the linear Hildebrand showed significant correlations only with estimated time spent in MVPA, and the random forest methodology only significantly correlated with sedentary time [35]. In the laboratory-based activities, the nonlinear model was the best algorithm to match measured VO₂ for sedentary and light tasks, was excellent at representing moderate walking, but tended to underestimate VO₂ for the moderate activities involving disproportionate arm use (i.e., walking while texting on a phone) [23]. Overall, these additional investigations support the current findings suggesting the utility of a simple nonlinear algorithm but also show that no algorithm is perfectly able to reflect all forms of PA.

Studies have demonstrated that wrist- and hip-based measurements are not interchangeable, typically requiring separate algorithms [18–20]. Our findings support that hip and wrist assessments can be significantly different yet show that these differences vary with activity. Overall, we observed wrist ENMO accelerations were approximately 75% greater than hip accelerations, particularly for the sweeping and running activities, with the hip-wrist relationship displaying a strong linear component, particularly through the intensity range of daily activities. Our results suggest that if desired, any wrist correction should be moderate at most and that either a linear or a weak nonlinear curve explains roughly 87–90% of the variance observed between hip and wrist ENMO assessments, making their comparison across studies not as problematic as some have suggested.

The increasing use of wrist-worn devices has clearly outpaced the development of wrist-worn algorithms. While the accuracy of wrist-based measurement has been called into question [20], the current study, along with the previous linear Hildebrand model, collectively suggest that similar if not better estimates can be obtained from wrist-based algorithms. We observed R² for the wrist that slightly surpassed the hip algorithms when using the nonlinear model. Previous reports using the linear model reported only slightly worse R² values at the wrist (i.e., 0.75 to 0.76) compared to the hip (i.e., 0.79–0.81) [19]. Accordingly, we can confidently conclude that the nonlinear wrist algorithm is at least as good as the nonlinear hip algorithm across a wide range of activities and intensities.

In addition to the global limitations of accelerometry to assess PA, specific limitations of the current study should be noted. First, we assessed the non-dominant wrist without holding any purses, bags, or assistive devices during ambulation, which may limit generalizability when assessing lifestyle PA estimates under “real world” conditions and in patient cohorts. These results are limited to the wear sites assessed and may not generalize to other sites used such as the thigh, ankle, or low back. Further, we did not assess VO₂ at the individual level but used observation and previous literature to generate mean (SD) estimates of energy expenditure. However, we combined our results with previously reported data using indirect calorimetry [19] to minimize this limitation and reduce potential bias. Additionally, in another study using indirect calorimetry to measure VO₂, supporting results were reported [23]. Order effects of the controlled tasks cannot be ruled out, but the repeatability of the 3 tasks tested twice—with the second test being at the end of the protocol, suggests order had minimal impact on accelerometry measurements. Lastly, the fact that our linear model closely matched the original Hildebrand linear model, but with extra data at a wider range of task intensities, supports that this limitation minimally affected our outcomes. However, despite the inclusion of young and older adults, these findings may not fully translate to clinical populations. Future studies involving patient cohorts would be beneficial to further assess these models in the populations being targeted.

In conclusion, these findings support the use of a simple nonlinear algorithm for wrist- or hip-worn accelerometry for estimating energy expenditure for most tasks, particularly if sedentary behavior and/or light PA is of interest. However, if only identification of MVPA is needed, the linear algorithms produce nearly identical results and thus are appropriate for this purpose. Recognizing that the acceleration magnitudes can differ between locations, distinct wear location algorithms are optimal to estimate energy expenditure most accurately, and caution should be used when attempting to transform between hip and wrist accelerations due to the nonlinear and inconsistent nature of this relationship, particularly for low intensity or functional tasks.