Estimation of Time-Response Curves and Their Confidence Bands

The study of a biological process over a period of time is often of interest in physiological and medical investigations. Important examples of this are afforded by curves describing secretion and excretion processes, and by growth curves for various body measurements. Because of the dearth of suitable subjects for study of secretion and excretion processes, and because of the desirability of following contemporaries in establishing growth curves, a common practice is to take observations on each of a group of subjects over a period of time. Thus the basic data consist of responses at different time points for a number of individuals. (An example of this type of data is shown in Table 2.) The deviations of any one individual's responses from the expected responses for the whole population are often correlated, and, as far as estimating the population response curve is concerned, this type of data does not contribute as much information as the same number of observations on different individuals at each time point. Nevertheless confidence bands for the curve and its parameters can still be obtained by all appropriate modification of well known methods. Rao 1959 has presented a solution to the problem of estimating confidence bands for a response curve when the data are of the type just described. His method does not require that we assume independence among the observations on the same individual at different time points; he therefore assumes the errors have a multivariate distribution, and the solution depends on multivariate methods. In developing a solution, Rao assumes only that the deviations from the true response curve have a multivariate normal distribution; and so long as this assumption is not unrealistic, his method can be used. However, if we are considering only one type of measurement or