A Bayesian model for growth curve analysis

An experiment involves K subjects where for subject i, ni, values yi1, yi2, ..., y(ini) of a random variable Y are observed at times ti1 ti2, ..., t(ini). Assume that yij = F(tij) +eij where eij are independently and identically distributed (i.i.d.) N(0, sigma2). We consider the estimation of the function F and the testing of the homogeneity hypothesis that, for [formula, see text] does not depend on t. The function F(i,t) is modelled as a Gaussian process which seeks to quantify the notions that for each i, F(i, t) is a slowly changing function of t and that for i is not equal to j, F(i, t), and F(j, T) are in some sense similar. We propose to estimate F (i, t) by its posterior mean given all of the data. This Bayes estimate is shown to be equivalent to a particular form of penalised likelihood estimation. We consider data-based methods for setting the parameters of the Gaussian process prior, develop a test of the homogeneity hypothesis, report the results of a Monte Carlo study illustrating the effectiveness of the proposed methodology, and apply the methods to a study of variations in temperature and blood pressure over the course of the menstrual cycle.