Tolerance Bands for Growth Curves

Abstract

Growth curve models and analyses have received considerable attention from a number of authors. Recent papers which reference earlier work are Potthoff and Roy [1964], Elston [1964], Rao [1965], and Krishniaiah [1967]. The main concern of these and earlier papers was to establish a model for growth curves and then to perform tests of hypotheses on the population mean growth curve or to construct confidence intervals or bands for the population mean growth curve. In this paper the model of Rao [1965] is used to construct tolerance bands for a population of individual growth curves. In the model employed each individual in the population has a growth curve whose expectation can be expressed as a polynomial in time. The tolerance bands desired are bands such that a given proportion, 7r, of the individuals in the population have their expected growth curves entirely within the bands. In the case where the parameters of the expected growth curves are unknown, a solution is given which contains a proportion ir of the expected growth curves with confidence level approximately (1 a). These bands would be useful, for example, in determining when to move fish from one hatchery pond to another for larger fish. At any time with any set confidence, limits can be found for the size of 90%, say, of the population. Given a set of tolerance bands based on a large number of fish growth curves, the manager can decide at what time any given run of fish can be expected to have a proportion ir of their numbers in a particular size range. In section 5 a numerical example is presented for a sample of fish with approximate linear growth for the time-span of interest. Consider the model for polynomial growth curves