Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T10:59:48.421Z Has data issue: false hasContentIssue false

Analyzing growth trajectories

Published online by Cambridge University Press:  12 October 2011

I. W. McKeague*
Affiliation:
Department of Biostatistics, Columbia University, New York, NY, USA
S. López-Pintado
Affiliation:
Department of Biostatistics, Columbia University, New York, NY, USA
M. Hallin
Affiliation:
ECARES, Université libre de Bruxelles, Bruxelles, Belgium ORFE, Princeton University, Princeton, USA CentER, Tilburg University, The Netherlands ECORE, Bruxelles, Belgium Académie Royale de Belgique, Brussels, Belgium
M. Šiman
Affiliation:
Institute of Information Theory and Automation of the ASCR, Pod Vod′arenskou věží 4, Prague 8, Czech Republic
*
*Address for correspondence: Prof. I. W. McKeague, Department of Biostatistics, Columbia University, 722 West 168th Street, 6th Floor, New York, NY 10032, USA. (Email im2131@columbia.edu)

Abstract

Growth trajectories play a central role in life course epidemiology, often providing fundamental indicators of prenatal or childhood development, as well as an array of potential determinants of adult health outcomes. Statistical methods for the analysis of growth trajectories have been widely studied, but many challenging problems remain. Repeated measurements of length, weight and head circumference, for example, may be available on most subjects in a study, but usually only sparse temporal sampling of such variables is feasible. It can thus be challenging to gain a detailed understanding of growth patterns, and smoothing techniques are inevitably needed. Moreover, the problem is exacerbated by the presence of large fluctuations in growth velocity during early infancy, and high variability between subjects. Existing approaches, however, can be inflexible because of a reliance on parametric models, require computationally intensive methods that are unsuitable for exploratory analyses, or are only capable of examining each variable separately. This article proposes some new nonparametric approaches to analyzing sparse data on growth trajectories, with flexibility and ease of implementation being key features. The methods are illustrated using data on participants in the Collaborative Perinatal Project.

Type
Original Articles
Copyright
Copyright © Cambridge University Press and the International Society for Developmental Origins of Health and Disease 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Dasgupta, P, Hauspie, R. Perspectives in Human Growth, Development and Maturation, 2001. Kluwer Academic Publishers, Springer-Verlag, New York.Google Scholar
2. Sumiya, T, Tashima, T, Nakahara, H, Shohoji, T. Relationships between biological parameters of Japanese growth of height. Environmetrics. 2001; 12, 367382.CrossRefGoogle Scholar
3. Li, N, Das, K, Wu, R. Functional mapping of human growth trajectories. J Theor Biol. 2009; 261, 3342.Google Scholar
4. López-Pintado, S, McKeague, IW. Recovering gradients from sparsely observed functional data. Biometrics. 2011, under revision; http://www.columbia.edu/~im2131/ps/growthrate-package-reference.pdf Google Scholar
5. López-Pintado, S, Romo, J. On the concept of depth for functional data. J Amer Stat Assoc. 2009; 104, 718734.CrossRefGoogle Scholar
6. Hallin, M, Lu, Z, Paindaveine, D, Šiman, M. Local bilinear multiple-output quantile regression and regression depth. Preprint, 2011.Google Scholar
7. Ferraty, F, Vieu, P. Nonparametric Functional Data Analysis, 2006. Springer, New York.Google Scholar
8. Ramsay, JO, Silverman, BW. Functional Data Analysis, 2005. Springer, New York.Google Scholar
9. Hall, P, Müller, HG, Wang, JL. Properties of principal components methods for functional and longitudinal data analysis. Ann Stat. 2006; 34, 14931517.CrossRefGoogle Scholar
10. James, G, Hastie, TJ, Sugar, CA. Principal component models for sparse functional data. Biometrika. 2000; 87, 587602.Google Scholar
11. Rice, J, Wu, C. Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics. 2000; 57, 253259.Google Scholar
12. Yao, F, Müller, HG, Wang, JL. Functional data analysis for sparse longitudinal data. J Amer Stat Assoc. 2005; 100, 577590.CrossRefGoogle Scholar
13. Yao, F, Müller, HG, Wang, JL. Functional linear regression analysis for longitudinal data. Ann Stat. 2005; 33, 28732903.Google Scholar
14. Kirsch, A. An Introduction to the Mathematical Theory of Inverse Problems, volume 120 of Applied Mathematical Sciences, 1996. Springer-Verlag, New York.Google Scholar
15. Nashed, MZ, Wahba, G. Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind. Math Comput. 1974; 28, 6980.Google Scholar
16. Wahba, G. Practical approximate solutions to linear operator equations when the data are noisy. SIAM J Numer Anal. 1977; 14, 651667.CrossRefGoogle Scholar
17. Liu, B, Müller, HG. Estimating derivatives for samples of sparsely observed functions, with application to online auction dynamics. J Amer Stat Assoc. 2009; 104, 704717.Google Scholar
18. López-Pintado, S, McKeague, IW. Growthrate: Bayesian reconstruction of growth velocity. R package version 1.0, 2011. http://CRAN.R-project.org/package=growthrate Google Scholar
19. Shohoji, T, Kanefuji, K, Sumiya, T, Qin, T. A prediction of individual growth of height according to an empirical Bayesian approach. Ann Inst Stat Math. 1991; 43, 607619.CrossRefGoogle Scholar
20. Barry, D. A Bayesian model for growth curve analysis. Biometrics. 1995; 51, 639655.Google Scholar
21. Arjas, E, Liu, L, Maglaperidze, N. Prediction of growth: a hierarchical Bayesian approach. Biom J. 1997; 39, 741759.Google Scholar
22. Cai, T, Liu, W, Luo, X. A constrained 1 minimization approach to sparse precision matrix estimation. J Amer Stat Assoc. 2011; 106, 594607.Google Scholar
23. R Development Core Team. R: A Language and Environment for Statistical Computing, 2011. R Foundation for Statistical Computing, Vienna, Austria, http://www.R-project.org/ Google Scholar
24. Mahalanobis, PC. On the generalized distance in statistics. Proc Nat Acad Sci India. 1936; 13, 13051320.Google Scholar
25. Tukey, JW. Mathematics and the picturing of data. Proceedings of the International Congress of Mathematicians, Vancouver, B.C., 1974, 1975, vol. 2, pp. 523531. Canad. Math. Congress, Montreal, Que.Google Scholar
26. Oja, H. Descriptive statistics for multivariate distributions. Stat Probab Lett. 1983; 1, 327332.Google Scholar
27. Liu, R. On a notion of data depth based on random simplices. Ann Stat. 1990; 18, 405414.Google Scholar
28. Fraiman, R, Meloche, J. Multivariate L-estimation. Test. 1999; 8, 255317.CrossRefGoogle Scholar
29. Zuo, Y, Serfling, RJ. General notions of statistical depth function. Ann Stat. 2000; 28, 461482.Google Scholar
30. Liu, R, Singh, K. A quality index based on data depth and multivariate rank test. J Amer Stat Assoc. 1993; 88, 257260.Google Scholar
31. Liu, R. Control charts for multivariate processes. J Amer Stat Assoc. 1995; 90, 13801388.CrossRefGoogle Scholar
32. Yeh, A, Singh, K. Balanced confidence sets based on Tukey depth. J R Stat Soc Ser B. 1997; 3, 639652.CrossRefGoogle Scholar
33. Rousseeuw, P, Leroy, AM. Robust Regression and Outlier Detection, 1987. Wiley, New York.CrossRefGoogle Scholar
34. Liu, R, Parelius, JM, Singh, K. Multivariate analysis by data depth: descriptive statistics, graphics and inference. Ann Stat. 1999; 27, 783858.Google Scholar
35. Zuo, Y. Multidimensional trimming based on projection depth. Ann Stat. 2006; 34, 22112251.Google Scholar
36. Cuesta-Albertos, JA, Nieto-Reyes, A. The random Tukey depth. Comput Stat Data Anal. 2008; 52, 49794988.CrossRefGoogle Scholar
37. Kong, L, Mizera, I. Quantile tomography: using quantiles with multivariate data. Preprint, arXiv:0805.0056v1, 2010.Google Scholar
38. Hallin, M, Paindaveine, D, Šiman, M. Multivariate quantiles and multiple-output regression quantiles: from L 1 optimization to halfspace depth. Ann Stat. 2010; 38, 635669.Google Scholar
39. Paindaveine, D, Šiman, M. Computing multiple-output regression quantile regions. Comput Stat Data Anal, to appear, 2011.Google Scholar
40. Wei, Y. An approach to multivariate covariate-dependent quantile contours with application to bivariate conditional growth charts. J Amer Stat Assoc. 2008; 103, 397409.Google Scholar
41. Sun, Y, Genton, MG. Functional boxplots. J Comput Graphical Stat. 2011; 20, 316334.Google Scholar