A case study in the modelling of growth

Abstract

This paper discusses a variety of problems which arise in the statistical modelling of a developmental process in biology. The data under study come from studies of ovarian follicular growth in sheep. Variations within and/or between individuals can be given a probabilistic interpretation, which due to characteristic features of biological data is seldom straightforward. Commonly, the system to be described is complex, being determined by hormonal or other influences which may not be fully understood or capable of being adequately observed. Mathematical and statistical models of such systems are necessarily exploratory and approximate; the nature of population and growth processes is such that they are also typically nonlinear in the parameters. In the work reported here, sampling was destructive and replicate measurements on the same individuals through time were therefore not available. A comprehensive but exploratory model for the development of ovarian follicles in mammals and small rodents has been developed, and in this paper we address its application to the data and review the difficulties and limitations attending its use. The model represents the probability, that a given ovarian follicle is in healthy growth and with size between specified limits, as a function of 9 parameters. The ultimate purpose of the analysis of each species is to make inferences on a single set of population parameters, about which the parameters specific to individual animals are assumed to be randomly distributed. A distinctive feature of growth modelling within individuals across different stages of development is the fact that the information matrices of the model estimates will, as functions of the ruling parameters, vary over time. Asymptotic Normal theory for the combination of unequally informative data for inference on a common parameter set has been given by Read and Berry [6], along with an iterative numerical estimation procedure.