The statistical modelling of growth

Abstract

In many fields of application, such as biology, psychology, agriculture, geology, botany, engineering and medicine, experiments are conducted in which a number of responses are repeatedly measured on each of a number of experimental units under differing experimental conditions. Longitudinal data which consists of observations that are ordered by time or position in space, for example the height of a child measured annually or monthly over a period of time, is considered. The study of growth in height is an excellent model for the investigation of other forms of growth and is considered in the practical applications of this thesis. Any progress made in measuring and modelling physical growth will serve as a good basis when attempts are made in future on the more difficult task of describing cognitive, affective or social development. The Richards growth function, a generalisation of nonlinear functions with a flexible point of inflection, often used to describe and compare growth curves, is considered. The generalised least squares, the maximum likelihood and the asymptotic distribution free frequentist estimation procedures for linear and nonlinear random parameter models are discussed. Two algorithms namely the Fisher scoring algorithm and the Expected Maximization (EM) algorithm are discussed. The Gauss quadrature numerical integration technique, which usually provide reliable approximations when closed form solutions for integrals are not available, is considered. The Bayes and Maximum Aposteriori (MAP) estimators are discussed, for linear and nonlinear models. An empirical Bayes method for the estimation of unknown model parameters is applied to an incomparable collection of longitudinal human growth records begun at the Fels institute in 1929, as well as to the Berkeley human growth records ( see Tuddenham and Snyder, 1954). The nonlinear fixed and random parameter Richards models with time series deviations (ARMA(l,l)), for non-consecutive data, are considered and applied to different datasets. Most of the theory discussed has been implemented in computer programs