Growth curve (statistics)

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 N_p,n(0,I_p,_n).

This differs from standard MANOVA by the addition of C, a "postmatrix".^[3]

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.

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]

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.