Implementation of Rao's one-sample polynomial growth curve model using SAS

Longitudinal data are frequently treated with the classic analysis of variance and regression models. However, these models assume independence of observations. Hoel (1964) demonstrated that the use of least-squares methods on intercorrelated serial observations results in the rejection of the null hypothesis much too frequently. Although appropriate models for analyzing longitudinal data have been available for quite some time, they have remained inaccessible due to cumbersome matrix manipulations. We implement Rao's (1959) one-sample polynomial growth curve model using the programming capability and matrix language of SAS, which involves testing the goodness-of-fit and calculation of confidence bands for polynomial growth curves fit to data at equally spaced time points. Confidence intervals for the parameters themselves are also computed. The method and program (presented in the Appendix) are illustrated with examples involving mandibular ramus height in 12 young male rhesus monkeys. The data set, which spans a 4 year period (yearly observations), is fit adequately by a quadratic equation. The data spanning a 2 year period (half-year observations) are fit adequately by the linear equation. These examples illustrate the considerable widening of confidence bands that occurs when polynomial equations having more terms than are needed to meet the goodness-of-fit requirement are considered.