An empirical Bayes approach to growth curve analysis

Consider an experiment designed to compare G treatments in terms of the pattern of change over time in a variable I: Suppose that treatment group i consists of n(i) subjects and that, for subject j in treatment group i, values y(ij1), y(ij2),..., y(ijK) of the variable Y are observed at times t(1), t(2),..., t(K) Assume that y(ijk) = F(i, t(k)) + b(ij) + e(ijk) where {b(ij)} are independent and identically distributed (IID) N(0, sigma(1)(2)) and {e(ijk)} are IID N(0, sigma(2)). We consider the estimation of the function F and the testing of the homogeneity hypothesis that, for i not equal j, F(i, t) - F(j, t) 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 not equal j, F(i, t) and F(j, t) are in some sense similar. We estimate F(i, t) by its posterior mean given all the data. This Bayes estimate is shown to be equivalent to a particular form of penalized likelihood estimation. We propose a data-based method 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 comparing the responses over time of rabbits to four insulin treatments.