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The growth curve model in statistics is a specific multivariate linear model, also known as GMANOVA (Generalized Multivariate Analysis-Of-Variance).[1] It generalizes MANOVA by allowing post-matrices, as seen in the definition.
Definition
[edit]Growth curve model:[2] Let X be a p×n random matrix corresponding to the observations, A a p×q within design matrix with q ≤ p, B a q×k parameter matrix, C a k×n between individual design matrix with rank(C) + p ≤ n and let Σ be a positive-definite p×p matrix. Then
defines the growth curve model, where A and C are known, B and Σ are unknown, and E is a random matrix distributed as Np,n(0,Ip,n).
This differs from standard MANOVA by the addition of C, a "postmatrix".[3]
History
[edit]Many writers have considered the growth curve analysis, among them Wishart (1938),[4] Box (1950) [5] and Rao (1958).[6] Potthoff and Roy in 1964;[3] were the first in analyzing longitudinal data applying GMANOVA models.
Applications
[edit]GMANOVA is frequently used for the analysis of surveys, clinical trials, and agricultural data,[7] as well as more recently in the context of Radar adaptive detection.[8][9]
Other uses
[edit]In mathematical statistics, growth curves such as those used in biology are often modeled as being continuous stochastic processes, e.g. as being sample paths that almost surely solve stochastic differential equations.[10] Growth curves have been also applied in forecasting market development.[11] When variables are measured with error, a Latent growth modeling SEM can be used.
Footnotes
[edit]- ^ Kim, Kevin; Timm, Neil (2007). ""Restricted MGLM and growth curve model" (Chapter 7)". Univariate and multivariate general linear models: Theory and applications with SAS (with 1 CD-ROM for Windows and UNIX). Statistics: Textbooks and Monographs (Second ed.). Boca Raton, Florida: Chapman & Hall/CRC. ISBN 978-1-58488-634-1.
- ^ Kollo, Tõnu; von Rosen, Dietrich (2005). ""Multivariate linear models" (chapter 4), especially "The Growth curve model and extensions" (Chapter 4.1)". Advanced multivariate statistics with matrices. Mathematics and its applications. Vol. 579. Dordrecht: Springer. ISBN 978-1-4020-3418-3.
- ^ a b R.F. Potthoff and S.N. Roy, “A generalized multivariate analysis of variance model useful especially for growth curve problems,” Biometrika, vol. 51, pp. 313–326, 1964
- ^ Wishart, John (1938). "Growth rate determinations in nutrition studies with the bacon pig, and their analysis". Biometrika. 30 (1–2): 16–28. doi:10.1093/biomet/30.1-2.16.
- ^ Box, G.E.P. (1950). "Problems in the analysis of growth and wear curves". Biometrics. 6 (4): 362–89. doi:10.2307/3001781. JSTOR 3001781. PMID 14791573.
- ^ Radhakrishna, Rao (1958). "Some statistical methods for comparison of growth curves". Biometrics. 14 (1): 1–17. doi:10.2307/2527726. JSTOR 2527726.
- ^ Pan, Jian-Xin; Fang, Kai-Tai (2002). Growth curve models and statistical diagnostics. Springer Series in Statistics. New York: Springer-Verlag. ISBN 0-387-95053-2.
- ^ Ciuonzo, D.; De Maio, A.; Orlando, D. (2016). "A Unifying Framework for Adaptive Radar Detection in Homogeneous plus Structured Interference-Part I: On the Maximal Invariant Statistic". IEEE Transactions on Signal Processing. PP (99): 2894–2906. arXiv:1507.05263. Bibcode:2016ITSP...64.2894C. doi:10.1109/TSP.2016.2519003. S2CID 5473094.
- ^ Ciuonzo, D.; De Maio, A.; Orlando, D. (2016). "A Unifying Framework for Adaptive Radar Detection in Homogeneous plus Structured Interference-Part II: Detectors Design". IEEE Transactions on Signal Processing. PP (99): 2907–2919. arXiv:1507.05266. Bibcode:2016ITSP...64.2907C. doi:10.1109/TSP.2016.2519005. S2CID 12069007.
- ^ Seber, G. A. F.; Wild, C. J. (1989). ""Growth models (Chapter 7)"". Nonlinear regression. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: John Wiley & Sons, Inc. pp. 325–367. ISBN 0-471-61760-1.
- ^ Meade, Nigel (1984). "The use of growth curves in forecasting market development—a review and appraisal". Journal of Forecasting. 3 (4): 429–451. doi:10.1002/for.3980030406.
References
[edit]- Davidian, Marie; David M. Giltinan (1995). Nonlinear Models for Repeated Measurement Data. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. ISBN 978-0-412-98341-2.
- Kshirsagar, Anant M.; Smith, William Boyce (1995). Growth curves. Statistics: Textbooks and Monographs. Vol. 145. New York: Marcel Dekker, Inc. ISBN 0-8247-9341-2.
- Pan, Jianxin; Fang, Kaitai (2007). Growth curve models and statistical diagnostics. Mathematical Monograph Series. Vol. 8. Beijing: Science Press. ISBN 9780387950532.
- Timm, Neil H. (2002). ""The general MANOVA model (GMANOVA)" (Chapter 3.6.d)". Applied multivariate analysis. Springer Texts in Statistics. New York: Springer-Verlag. ISBN 0-387-95347-7.
- Vonesh, Edward F.; Chinchilli, Vernon G. (1997). Linear and Nonlinear Models for the Analysis of Repeated Measurements. London: Chapman and Hall.
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| Correlation and dependence | |||||||||
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| Regression as a statistical model |
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| Decomposition of variance | |||||||||
| Model exploration | |||||||||
| Background | |||||||||
| Design of experiments | |||||||||
| Numerical approximation | |||||||||
| Applications | |||||||||
