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# Growth curve (statistics)

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### Growth curve (statistics)

Table of height and weight for boys over time. The growth curve model (also known as GMANOVA) is used to analyze data such as this, where multiple observations are made on collections of individuals over time.

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.

## Contents

• Definition 1
• History 2
• Applications 3
• Other uses 4
• Footnotes 5
• References 6

## Definition

Growth curve model:[2] Let X be a p×n matrix, A a p×q matrix with q ≤ p, B a q×k matrix,C a k×n matrix with rank(C) + p ≤ n and let Σ be a positive-definite p×p matrix. Then

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

The growth curve model was invented by Potthoff and Roy in 1964;[3] they used it to analyze repeated measurements of animals or humans to obtain a biological growth curve.

## Applications

GMANOVA is frequently used for the analysis of surveys, clinical trials, and agricultural data.[4]

## 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.[5]

## Footnotes

1. ^ Kim, Kevin and 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, FL: Chapman & Hall/CRC.
2. ^ Kollo, Tõnu and 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.
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
4. ^ Pan, Jian-Xin and Fang, Kai-Tai (2002). Growth curve models and statistical diagnostics. Springer Series in Statistics. New York: Springer-Verlag.
5. ^ Seber, G. A. F. and 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.

## References

• Davidian, Marie; David M. Giltinan (1995). Nonlinear Models for Repeated Measurement Data. Chapman & Hall/CRC Monographs on Statistics & Applied Probability.
• Kshirsagar, Anant M. and Smith, William Boyce (1995). Growth curves. Statistics: Textbooks and Monographs 145. New York: Marcel Dekker, Inc.
• Pan, Jianxin; Fang, Kaitai (2007). Growth curve models and statistical diagnostics. Mathematical Monograph Series 8. Beijing: Science Press.
• Timm, Neil H. (2002). ""The general MANOVA model (GMANOVA)" (Chapter 3.6.d)". Applied multivariate analysis. Springer Texts in Statistics. New York: Springer-Verlag.
• Vonesh, Edward F. and Chinchilli, Vernon G. (1997).
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