ELSEVIER International Journal of Bio-Medical Computing 37 (1994) 105-I 12 Extension of the Carter-Yang polynomial growth curve model to allow unique times of measurement for subjects Elizabeth A. Maugera, Charles J. Kowalskib, Emet D. Schneiderman*“, Stephen M. Willis’ aDepartment of Biostatistics, bDepartment of Biologic and Materials Sciences, The University of Michigan. Ann Arbor, MI 48109, USA CDepartment of Oral and Maxillofacial Surgery and Pharmacology, Baylor College of Dentistry, PO Box 660677, Dallas, TX 752664677, USA Received 16 November 1993;accepted 20 December 1993 Abstract A PC program extending the procedure due to Carter and Yang (Commun Stat: Theory Methods, 8 (1986) 2507-2526) to allow unique times of measurement for subjects is described, illustrated and made available. Given lon- gitudinal observations on each of N subjects comprising a single group, this program determines the lowest degree polynomial in time adequate to tit the average growth curve (AGC); estimates this curve and provides confidence bands for the AGC, and confidence intervals for the corresponding polynomial regression coeffkients; and so-called predic- tion intervals which, with a given level of confidence, will contain the growth curve of a ‘new’ subject from the same population of which the N subjects constitute a random sample. Two kinds of missing data are accommodated. First, in the context of studies planned so that subjects will be measured at identical times and, second, in unstructured stud- ies where subjects may present with their own, unique times of measurement. Keywords: Longitudinal data; Missing values; Unbalanced designs; PC program 1. Introduction A longitudinal study which is planned so that subjects will be measured at the same set of time points is said to be balanced (or to have a balanced design). If there are no missing data, the resulting data set is said to be complete. In an earlier paper [ 11,we described, illustrated, and made available a menu-driven PC program implementing Rao’s [2] * Corresponding author. so-called two-stage, or random coefficients, polynomial growth curve model. Given a data set which is both balanced and complete, this pro- gram can be used to (a) determine the degree of the polynomial adequate to fit the average growth curve (AGC) of the sample, and (b) estimate and compute confidence bands for the AGC. More recently [3], we implemented the Carter-Yang [4] extension of this procedure which accommodates missing data, but still requires a balanced design. This program produces output similar to that in 0020-7101/94/$07.00 0 1994 Elsevier Science Ireland Ltd. All rights reserved SSDI 0020-7101(94)01002-I