Comparing responses when each response is a curve.

Abstract

We describe, generalize, and demonstrate the application of a method (W. H. Lawton, A. Sylvestre, and M. S. Maggio, Technometrics 14: 513-532, 1972) that can be used to partially analyze population data. The data from each subject consist of a series of responses observed at distinct values of a predictor variable. The method assumes that all subjects' data originate from a common process but differ because the "units" of predictor and response variables differ among subjects. For example, if the predictor variable is time, time can be "faster" or "slower" from subject to subject. We deal with two different problems. In the first one the response at x for the ith subject is of the form beta 1i + beta 2iG[(x - beta 3i)/beta 4i] + epsilon, where G(x) is a mathematical "shape" function (of the predictor variable x) representing the process and epsilon is observation error. The units of observed and predictor variable are then defined by the values of beta 1i, beta 2i and beta 3i, beta 4i, respectively. In particular beta 1i and beta 3i express shifts and beta 2i, beta 4i express scales of the observed and predictor variables, respectively. In the second problem the response at x for the ith subject is of the form beta 1i + beta 2i integral of x0 G[(s - beta 3i)/beta 4i].Hi(x - s) ds + epsilon, where Hi(x) is a known function. In both problems, the method estimates the common "shape" function G(x) nonparametrically and the parameters beta 1i, beta 2i, beta 3i, and beta 4i for each subject.(ABSTRACT TRUNCATED AT 250 WORDS)