Multilevel models in human growth and development research.

The analysis of change is an important issue in human growth and development. In longitudinal studies, growth patterns are often summarized by growth 'models' so that a small number of parameters, or the functions of them can be used to make group comparisons or to be related to other measurements. To analyse complete and balanced data, growth curves can be modelled using multivariate analysis of variance with an unstructured variance-covariance matrix; for incomplete and unbalanced data, models such as the two-stage model of Laird and Ware (1982) or the multilevel models of Goldstein (1987) are necessary. The use of multilevel models for describing growth is recognized as an important technique. It is an efficient procedure for incorporating growth models, either linear or nonlinear, into a population study. Up to now there is little literature concerning growth models over wide age ranges using multilevel models. The purpose of this study is to explore suitable multilevel models of growth over a wide age range. Extended splines are proposed, which extend conventional splines using the '+' function and by including logarithmic or negative power terms. The work has been focused on modelling human growth in length, particularly, height and head circumference as they are interesting and important measures of growth. The investigation of polynomials, conventional splines and extended splines on data from the Edinburgh Longitudinal Study shows that the extended splines are better than polynomials and conventional splines for this purpose. It also shows that extended splines are, in fact, piecewise fractional polynomials and describe data better than a single segment of a fractional polynomial. The extended splines are useful, flexible, and easily incorporated in multilevel models for studying populations and for the estimation and comparison of parameters.