Reduced-rank growth curve models

The growth curve model proposed by Potthoff and Roy (Biometrika 51 (1964) 313) and studied further by Rao and others in several articles has useful applications in many fields. Typically, the model relates the mean response of a characteristic observed over several times, represented as a parametric function of time, to time-invariant covariates. In the context of multivariate regression of a set of response variables related to a set of predictors, previous work has demonstrated that more parsimonious modeling is possible through the assumption of reduced rank of the regression coefficient matrix. In this paper, we examine the use of reduced-rank structure for the coefficient matrix of the parameters in the growth curve model. The developments of the basic reduced-rank growth curve model are surveyed and some extensions are given, and related work is discussed. Details are provided on maximum likelihood estimation of the parameters of these reduced-rank growth curve models, as well as on likelihood ratio testing for the rank of the coefficient matrix. The connection between the reduced-rank models and the multivariate one-way ANOVA model and linear discriminant analysis is indicated. A numerical example is provided to illustrate the merits of the techniques.