PC program for obtaining orthogonal polynomial regression coefficients for use in longitudinal data analysis

Abstract

Much of longitudinal data analysis begins with dimensionality reduction, i.e., the replacement of the T observations x 1 , x 2 , …, x T on an individual taken at times t 1 , t 2 , …, t T (not necessarily equally spaced) by a smaller number, P, of parameters which are then used to describe and compare growth processes. We focus on the class of polynomial growth curve models for one-sample data matrices in which the P regression coefficients are estimated by an equation of the form documentclass{article}pagestyle{empty}begin{document}$ hat tau = ({rm W'W}) $end{document} 1 W′x and consider the choice of the design matrix W. The case in favor of using orthogonal polynomials to comprise the elements of W and provide a PC program, written in GAUSS, for obtaining them is presented. This program can be used instead of existing tables of orthogonal polynomials in the case of equally spaced time points, and to avoid laborious hand-computation to obtain them when the time points are not equally spaced. The program also computes the corresponding orthogonal polynomial regression coefficients documentclass{article}pagestyle{empty}begin{document}$ hat alpha = (Phi 'Phi)^{ - 1} Phi '{rm x} $end{document} , where Φ consists of orthogonal polynomials, which may then be input into other programs for subsequent analysis, e.g., to compare the growth profiles of several groups of individuals. Examples of the use of the program are given. Information on obtaining a copy of the program is provided in Appendix A. © 1992 Wiley-Liss, Inc.