# Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data and Preprocessing

#### UK Biobank: Accelerometry Sub-Study

#### 2.2. Methodology and Analysis

#### Physical Activity Acceleration Profiles

#### Curve Registration

#### Functional Regression

#### Computation, Software, and Reproducibility

`mgcv::bam()`function in the

`mgcv`[34] package, and registration is performed using the

`registr`package [35].

`R`package

`rnhanesdata`[7,37]. Our analysis is reproduced using the NHANES data via a supplementary markdown file to be uploaded to Github upon publication.

#### 2.3. Populations of Comparison

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BLSA | Baltimore Longitudinal Study of Aging |

ENMO | Euclidean Norm Minus One |

FDA | Functional Data Analysis |

FoSR | Function on Scalar Regression |

FPCA | Functional Principal Component Analysis |

NHANES | National Health and Nutrition Examination Survey |

PA | Physical Activity |

## References

- Shephard, R.J. Aging, Physical Activity, and Health; Human Kinetics Publishers: Champagne, IL, USA, 1997. [Google Scholar]
- Office of the Surgeon General (US). The Surgeon General’s Vision for a Healthy and Fit Nation; Office of the Surgeon General (US): Rockville, MD, USA, 2010.
- Vogel, T.; Brechat, P.H.; Leprêtre, P.M.; Kaltenbach, G.; Berthel, M.; Lonsdorfer, J. Health benefits of physical activity in older patients: A review. Int. J. Clin. Pract.
**2009**, 63, 303–320. [Google Scholar] [CrossRef] [PubMed] - Pedersen, B.K.; Saltin, B. Evidence for prescribing exercise as therapy in chronic disease. Scand. J. Med. Sci. Sport
**2006**, 16, 3–63. [Google Scholar] [CrossRef] [PubMed] - Koster, A.; Caserotti, P.; Patel, K.V.; Matthews, C.E.; Berrigan, D.; Van Domelen, D.R.; Brychta, R.J.; Chen, K.Y.; Harris, T.B. Association of Sedentary Time with Mortality Independent of Moderate to Vigorous Physical Activity. PLoS ONE
**2012**, 7, e37696. [Google Scholar] [CrossRef] - Matthews, C.E.; Keadle, S.K.; Troiano, R.P.; Kahle, L.; Koster, A.; Brychta, R.; Van Domelen, D.; Caserotti, P.; Chen, K.Y.; Harris, T.B.; et al. Accelerometer-measured dose-response for physical activity, sedentary time, and mortality in US adults. Am. J. Clin. Nutr.
**2016**, 104, 1424–1432. [Google Scholar] [CrossRef][Green Version] - Leroux, A.; Di, J.; Smirnova, E.; Mcguffey, E.J.; Cao, Q.; Bayatmokhtari, E.; Tabacu, L.; Zipunnikov, V.; Urbanek, J.K.; Crainiceanu, C. Organizing and Analyzing the Activity Data in NHANES. Stat. Biosci.
**2019**, 11, 262–287. [Google Scholar] [CrossRef] - Smirnova, E.; Leroux, A.; Cao, Q.; Tabacu, L.; Zipunnikov, V.; Crainiceanu, C.; Urbanek, J.K. The Predictive Performance of Objective Measures of Physical Activity Derived From Accelerometry Data for 5-Year All-Cause Mortality in Older Adults: National Health and Nutritional Examination Survey 2003–2006. J. Gerontol. Ser. A
**2019**, 75, 1779–1785. [Google Scholar] [CrossRef] - Leroux, A.; Xu, S.; Kundu, P.; Muschelli, J.; Smirnova, E.; Chatterjee, N.; Crainiceanu, C. Quantifying the Predictive Performance of Objectively Measured Physical Activity on Mortality in the UK Biobank. J. Gerontol. Ser. A
**2020**. [Google Scholar] [CrossRef] - Sallis, J.F.; Saelens, B.E. Assessment of Physical Activity by Self-Report: Status, Limitations, and Future Directions. Res. Q. Exerc. Sport
**2000**, 71, 1–14. [Google Scholar] [CrossRef] - Schrack, J.A.; Zipunnikov, V.; Goldsmith, J.; Bai, J.; Simonshick, E.M.; Crainiceanu, C.M.; Ferrucci, L. Assessing the “Physical Cliff”: Detailed Quantification of Aging and Physical Activity. J. Gerontol. Med. Sci.
**2014**, 69, 973–979. [Google Scholar] [CrossRef] [PubMed][Green Version] - Troiano, R.P.; Berrigan, D.; Dodd, K.W.; Masse, L.C.; Tilert, T.; McDowell, M. Physical activity in the United States measured by accelerometer. Med. Sci. Sport Exerc.
**2008**, 40, 181–188. [Google Scholar] [CrossRef] - Sudlow, C.; Gallacher, J.; Allen, N.; Beral, V.; Burton, P.; Danesh, J.; Downey, P.; Elliott, P.; Green, J.; Landray, M.; et al. UK Biobank: An open access resource for identifying the causes of a wide range of complex diseases of middle and old age. PLoS Med.
**2015**, 12, e1001779. [Google Scholar] [CrossRef] [PubMed][Green Version] - Di, J.; Spira, A.; Bai, J.; Urbanek, J.; Leroux, A.; Wu, M.; Resnick, S.; Simonsick, E.; Ferrucci, L.; Schrack, J.; et al. Joint and individual representation of domains of physical activity, sleep, and circadian rhythmicity. Stat. Biosci.
**2019**, 11, 371–402. [Google Scholar] [CrossRef] [PubMed] - Lock, E.F.; Hoadley, K.A.; Marron, J.S.; Nobel, A.B. Joint and individual variation explained (JIVE) for integrated analysis of multiple data types. Ann. Appl. Stat.
**2013**, 7, 523–542. [Google Scholar] [CrossRef] - Cui, E.; Crainiceanu, C.M.; Leroux, A. Additive Functional Cox Model. J. Comput. Graph. Stat.
**2020**, 1–31. [Google Scholar] [CrossRef] - Ramsay, J.O.; Silverman, B.W. Functional Data Analysis; Springer: New York, NY, USA, 2005. [Google Scholar]
- Goldsmith, J.; Zipunnikov, V.; Schrack, J. Generalized multilevel function-on-scalar regression and principal component analysis. Biometrics
**2015**, 71, 344–353. [Google Scholar] [CrossRef][Green Version] - Backenroth, D.; Shinohara, R.T.; Schrack, J.A.; Goldsmith, J. Nonnegative decomposition of functional count data. Biometrics
**2020**, 76, 1273–1284. [Google Scholar] [CrossRef] - Reiss, P.T.; Goldsmith, J.; Shang, H.L.; Ogden, R.T. Methods for Scalar-on-Function Regression. Int. Stat. Rev.
**2017**, 85, 228–249. [Google Scholar] [CrossRef] - Wrobel, J.; Zipunnikov, V.; Schrack, J.; Goldsmith, J. Registration for exponential family functional data. Biometrics
**2019**, 75, 48–57. [Google Scholar] [CrossRef][Green Version] - Xiao, L.; Huang, L.; Schrack, J.A.; Ferrucci, L.; Zipunnikov, V.; Crainiceanu, C.M. Quantifying the lifetime circadian rhythm of physical activity: A covariate-dependent functional approach. Biostatistics
**2014**, 16, 352–367. [Google Scholar] [CrossRef][Green Version] - McDonnell, E.I.; Zipunnikov, V.; Schrack, J.A.; Goldsmith, J.; Wrobel, J. Scale-invariant time registration of 24-hour accelerometric rest-activity profiles and its application to human chronotypes. bioRxiv
**2020**. [Google Scholar] [CrossRef] - Doherty, A.; Jackson, D.; Hammerla, N.; Plötz, T.; Olivier, P.; Granat, M.H.; White, T.; van Hees, V.T.; Trenell, M.I.; Owen, C.G.; et al. Large Scale Population Assessment of Physical Activity Using Wrist Worn Accelerometers: The UK Biobank Study. PLoS ONE
**2017**, 12, e0169649. [Google Scholar] [CrossRef] - Bycroft, C.; Freeman, C.; Petkova, D.; Band, G.; Elliott, L.T.; Sharp, K.; Motyer, A.; Vukcevic, D.; Delaneau, O.; O’Connell, J.; et al. The UK Biobank resource with deep phenotyping and genomic data. Nature
**2018**, 562, 203–209. [Google Scholar] [CrossRef][Green Version] - Elliott, L.T.; Sharp, K.; Alfaro-Almagro, F.; Shi, S.; Miller, K.L.; Douaud, G.; Marchini, J.; Smith, S.M. Genome-wide association studies of brain imaging phenotypes in UK Biobank. Nature
**2018**, 562, 210–216. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fry, A.; Littlejohns, T.J.; Sudlow, C.; Doherty, N.; Adamska, L.; Sprosen, T.; Collins, R.; Allen, N.E. Comparison of Sociodemographic and Health-Related Characteristics of UK Biobank Participants With Those of the General Population. Am. J. Epidemiol.
**2017**, 186, 1026–1034. [Google Scholar] [CrossRef] [PubMed][Green Version] - Reiss, P.T.; Huang, L.; Mennes, M. Fast Function-on-Scalar Regression with Penalized Basis Expansions. Int. J. Biostat.
**2010**, 6. [Google Scholar] [CrossRef] [PubMed] - Bauer, A.; Scheipl, F.; Küchenhoff, H.; Gabriel, A.A. An introduction to semiparametric function-on-scalar regression. Stat. Model.
**2018**, 18, 346–364. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R. Varying-Coefficient Models. J. R. Stat. Soc. Ser. B
**1993**, 55, 757–779. [Google Scholar] [CrossRef] - Park, S.Y.; Staicu, A.M.; Xiao, L.; Crainiceanu, C.M. Simple fixed-effects inference for complex functional models. Biostatistics
**2018**, 19, 137–152. [Google Scholar] [CrossRef][Green Version] - Wood, S.N.; Li, Z.; Shaddick, G.; Augustin, N.H. Generalized Additive Models for Gigadata: Modeling the U.K. Black Smoke Network Daily Data. J. Am. Stat. Assoc.
**2017**, 112, 1199–1210. [Google Scholar] [CrossRef][Green Version] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2020. [Google Scholar]
- Wood, S.N. Generalized Additive Models: An Introduction with R; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Wrobel, J. registr: Registration for Exponential Family Functional Data. J. Open Source Softw.
**2018**, 3, 557. [Google Scholar] [CrossRef] - Centers for Disease Control and Prevention (CDC); National Center for Health Statistics (NCHS). National Health and Nutrition Examination Survey Data; U.S. Department of Health and Human Services, Centers for Disease Control and Prevention: Hyattsville, MD, USA, 2003–2006.
- Leroux, A. Rnhanesdata: NHANES Accelerometry Data Pipeline. R Package Version 1.0. 2018. Available online: https://github.com/andrew-leroux/rnhanesdata (accessed on 1 January 2021).
- Varma, V.R.; Dey, D.; Leroux, A.; Di, J.; Urbanek, J.; Xiao, L.; Zipunnikov, V. Re-evaluating the effect of age on physical activity over the lifespan. Prev. Med.
**2017**, 101, 102–108. [Google Scholar] [CrossRef] [PubMed] - Strączkiewicz, M.; Urbanek, J.; Fadel, W.; Crainiceanu, C.; Harezlak, J. Automatic car driving detection using raw accelerometry data. Physiol. Meas.
**2016**, 37, 1757. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Toy Registration Example. Black and gray colors denote two separate subjects, and panels represent (

**A**) binary activity profiles, (

**B**) unregistered probability curves, (

**C**) warping functions, and (

**D**) binary activity profiles and probability curves that have been aligned using the warping functions in panel (

**C**).

**Figure 2.**Estimated population average acceleration (ENMO) by age and time of day for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). Color intensity in panels A/B indicates lower (purple) versus higher (green) levels of activity. Color intensity in panel C indicates Males are less (purple), the same (white), or more (green) active as compared to Females, with a solid black line demarcating the transitions between regions where males are more or less active than females (i.e., the difference is 0). Estimates of these surfaces in panels A/B are obtained by fitting a function-on-scalar regression (FoSR) model separately by sex where the population average ENMO is allowed to vary smoothly in both time of day and age. The FoSR model is fit by modelling the average ENMO as a tensor product smooth of marginal spline bases, with cyclic cubic regression splines used in the time domain to respect the cyclic nature of time. Panel C is obtained by taking the difference of panels A and B.

**Figure 3.**Estimated population cumulative average acceleration (ENMO) by time of day for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). separately for ages 50 (black), 60 (red), and 70 (blue) years old. Solid lines denote point estimates and dashed lines represent point-wise confidence intervals. Estimates of the curves presented in panels A/B are obtained by fitting a function-on-scalar regression (FoSR) model separately by sex where the population average activity count is allowed to vary smoothly in both time of day and age, then numerically integrating the estimated population average activity count over the 12 a.m.–12 a.m. period. The FoSR model is fit by modelling the average ENMO as a tensor product smooth of marginal spline bases, with cyclic cubic regression splines used in the time domain to respect the cyclic nature of time. Panel C is obtained by taking the difference of panels A and B.

**Figure 4.**Estimated population average probability of being active (or, more specifically, generating ENMO greater than or equal to 30 milli-gravity units) by age and time of day for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). Color intensity in panels A/B indicates lower (purple) versus higher (green) probability of being active. Color intensity in panel C indicates Males are less (purple), the same (white), or more (green) likely to be active as compared to Females, with a solid black line demarcating the transitions between regions where males are more or less likely to be active than females (i.e., the difference is 0). Estimates of these surfaces in panels A/B are obtained by fitting a generalized function-on-scalar regression (FoSR) model separately by sex where the log odds of being active is allowed to vary smoothly in both time of day and age. The FoSR model is fit by modelling the log odds of being active as a tensor product smooth of marginal spline bases, with cyclic cubic regression splines used in the time domain to respect the cyclic nature of time using binary active/inactive profiles. Panel C is obtained by taking the difference of panels A and B.

**Figure 5.**Estimated population average warping functions for Males (

**panel A**), Females (

**panel B**), and the estimated difference between Males and Females (

**panel C**). separately for ages 50 (black), 60 (red), and 70 (blue) years old. Estimates of the lines presented in panels A/B are obtained by fitting a function-on-scalar regression (FoSR) model separately by sex where the population average warping function is allowed to vary smoothly in both time of day and age. The FoSR model is fit by modelling the average warping function as a tensor product smooth of marginal spline bases with cubic regression splines used in both the time and age domains. Panel C is obtained by taking the difference of panels A and B.

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**MDPI and ACS Style**

Wrobel, J.; Muschelli, J.; Leroux, A.
Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study. *Sensors* **2021**, *21*, 1545.
https://doi.org/10.3390/s21041545

**AMA Style**

Wrobel J, Muschelli J, Leroux A.
Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study. *Sensors*. 2021; 21(4):1545.
https://doi.org/10.3390/s21041545

**Chicago/Turabian Style**

Wrobel, Julia, John Muschelli, and Andrew Leroux.
2021. "Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study" *Sensors* 21, no. 4: 1545.
https://doi.org/10.3390/s21041545