Generalized multilevel function-on-scalar regression and principal component analysis
This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scient...
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| Published in: | Biometrics Vol. 71; no. 2; pp. 344 - 353 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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United States
Blackwell Publishing Ltd
01.06.2015
International Biometric Society |
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| ISSN: | 0006-341X, 1541-0420, 1541-0420 |
| Online Access: | Get full text |
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| Abstract | This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject-specific and subject-day-specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within-function and within-subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross-sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time-specific change in probability of being active over a 24-hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects. |
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| AbstractList | Summary
This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject‐specific and subject‐day‐specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within‐function and within‐subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross‐sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time‐specific change in probability of being active over a 24‐hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects. Summary This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject-specific and subject-day-specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within-function and within-subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross-sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time-specific change in probability of being active over a 24-hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects. This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject-specific and subject-day-specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within-function and within-subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross-sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time-specific change in probability of being active over a 24-hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects.This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject-specific and subject-day-specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within-function and within-subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross-sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time-specific change in probability of being active over a 24-hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects. This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly 600 subjects over 5 days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject-specific and subject-day-specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within-function and within-subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross-sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time-specific change in probability of being active over a 24-hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects. This manuscript considers regression models for generalized, multilevel functional responses: functions are generalized in that they follow an exponential family distribution and multilevel in that they are clustered within groups or subjects. This data structure is increasingly common across scientific domains and is exemplified by our motivating example, in which binary curves indicating physical activity or inactivity are observed for nearly six hundred subjects over five days. We use a generalized linear model to incorporate scalar covariates into the mean structure, and decompose subject-specific and subject-day-specific deviations using multilevel functional principal components analysis. Thus, functional fixed effects are estimated while accounting for within-function and within-subject correlations, and major directions of variability within and between subjects are identified. Fixed effect coefficient functions and principal component basis functions are estimated using penalized splines; model parameters are estimated in a Bayesian framework using Stan, a programming language that implements a Hamiltonian Monte Carlo sampler. Simulations designed to mimic the application have good estimation and inferential properties with reasonable computation times for moderate datasets, in both cross-sectional and multilevel scenarios; code is publicly available. In the application we identify effects of age and BMI on the time-specific change in probability of being active over a twenty-four hour period; in addition, the principal components analysis identifies the patterns of activity that distinguish subjects and days within subjects. |
| Author | Goldsmith, Jeff Zipunnikov, Vadim Schrack, Jennifer |
| AuthorAffiliation | 4 Longitudinal Studies Section, Translational Gerontology Branch, National Institute on Aging, National Institutes of Health 1 Department of Biostatistics, Mailman School of Public Health, Columbia University 3 Department of Epidemiology, Bloomberg School of Public Health, Johns Hopkins University 2 Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University |
| AuthorAffiliation_xml | – name: 2 Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University – name: 3 Department of Epidemiology, Bloomberg School of Public Health, Johns Hopkins University – name: 1 Department of Biostatistics, Mailman School of Public Health, Columbia University – name: 4 Longitudinal Studies Section, Translational Gerontology Branch, National Institute on Aging, National Institutes of Health |
| Author_xml | – sequence: 1 givenname: Jeff surname: Goldsmith fullname: Goldsmith, Jeff email: jeff.goldsmith@columbia.edu organization: Department of Biostatistics, Mailman School of Public Health, Columbia University, New York, New York, U.S.A – sequence: 2 givenname: Vadim surname: Zipunnikov fullname: Zipunnikov, Vadim organization: Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, Maryland, Baltimore, U.S.A – sequence: 3 givenname: Jennifer surname: Schrack fullname: Schrack, Jennifer organization: Department of Epidemiology, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, Maryland, U.S.A |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/25620473$$D View this record in MEDLINE/PubMed |
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| References_xml | – reference: Stan Development Team. (2013). Stan Modeling Language User's Guide and Reference Manual, Version 1.3. – reference: Bishop, C. M. (1999). Bayesian PCA. Advances in Neural Information Processing Systems 382-388. – reference: Ruppert, D., Wand, M. P., and Carroll, R. J. (2003). Semiparametric Regression. Cambridge: Cambridge University Press. – reference: Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. New York: Springer. – reference: Serban, N., Staicu, A.-M., and Carrol, R. J. (2013). Multilevel cross-dependent binary longitudinal data. Biometrics 69, 903-913. – reference: Lunn, D., Spiegelhalter, D., Thomas, A., and Best, N. (2009). The BUGS project: Evolution, critique and future directions (with discussion). Statistics in Medicine 28, 3049-3082. – reference: Bai, J., He, B., Shou, H., Zipunnikov, V., Glass, T. A., and Crainiceanu, C. M. (2014). Normalization and extraction of interpretable metrics from raw accelerometry data. Biostatistics 15, 102-116. – reference: Tipping, M. E. and Bishop, C. (1999). Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B 61, 611-622. – reference: Hall, P., Müller, H.-G., and Yao, F. (2008). Modelling sparse generalized longitudinal observations with latent Gaussian processes. Journal of the Royal Statistical Society, Series B 70, 703-723. – reference: Morris, J. S. and Carroll, R. J. (2006). Wavelet-based functional mixed models. Journal of the Royal Statistical Society, Series B 68, 179-199. – reference: van der Linde, A. (2009). A Bayesian latent variable approach to functional principal components analysis with binary and count. Advances in Statistical Analysis 93, 307-333. – reference: Brumback, B. and Rice, J. (1998). Smoothing spline models for the analysis of nested and crossed samples of curves. Journal of the American Statistical Association 93, 961-976. – reference: Yao, F., Müller, H., and Wang, J. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association 100, 577-590. – reference: Shiroma, E. J., Freedson, P. S., Trost, S. G., and Lee, I. M. (2013). Patterns of accelerometer-assessed sedentary behavior in older women. Journal of the American Medical Association 310, 2562-2563. – reference: Reiss, P. T., Huang, L., and Mennes, M. (2010). Fast function-on-scalar regression with penalized basis expansions. International Journal of Biostatistics 6, Article 28. – reference: Goldsmith, J., Greven, S., and Crainiceanu, C. M. (2013). Corrected confidence bands for functional data using principal components. Biometrics 69, 41-51. – reference: Troiano, R. P., Berrigan, D., Dodd, K. W., Masse, L. C., Tilert, T., and McDowell, M. (2008). Physical activity in the united states measured by accelerometer. Medicine & Science in Sports & Exercise 40, 181-188. – reference: Atkinson, H. H., Rosano, C., Simonsick, E. M., Williamson, J. 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| SubjectTerms | Accelerometry Accelerometry - statistics & numerical data age Aged Aged, 80 and over Aging - physiology Bayes Theorem Bayesian inference Biometry body mass index Computer Simulation data collection DISCUSSION PAPER Female Generalized functional data Generalized linear models Hamiltonian Monte Carlo Humans Linear Models Male Middle Aged Models, Statistical Monte Carlo Method Motor Activity Penalized splines physical activity Principal Component Analysis Principal components analysis probability Regression Analysis |
| Title | Generalized multilevel function-on-scalar regression and principal component analysis |
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