# Growth curve (statistics)

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

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

${\displaystyle X=ABC+\Sigma ^{1/2}E}$

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

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

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

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]

## Footnotes

1. 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.
2. 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. 579. Dordrecht: Springer. ISBN 978-1-4020-3418-3.
3. 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
4. Wishart, John (1938). "Growth rate determinations in nutrition studies with the bacon pig, and their analysis". Biometrika. 30: 16–28. doi:10.1093/biomet/30.1-2.16.
5. Box, G.E.P. (1950). "Problems in the analysis of growth and wear curves". Biometrics. 6: 362–89. doi:10.2307/3001781.
6. Radhakrishna, Rao (1958). "Some statistical methods for comparison of growth curves". Biometrics. 14: 1–17. doi:10.2307/2527726.
7. 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.
8. 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): 1–1. arXiv:1507.05263. Bibcode:2016ITSP...64.2894C. doi:10.1109/TSP.2016.2519003.
9. 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): 1–1. arXiv:1507.05266. Bibcode:2016ITSP...64.2907C. doi:10.1109/TSP.2016.2519005.
10. 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.
11. Meade, Nigel (1984). "The use of growth curves in forecasting market development—a review and appraisal". Journal of Forecasting. 3: 429–451. doi:10.1002/for.3980030406.

## References

• 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. 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. 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.