| Research Methods in Nutritional Anthropology (1989) |
|6. Elementary mathematical models and statistical methods for nutritional anthropology|
The objective of this paper is to provide a non-rigorous overview of a few elementary mathematical models and analytic techniques that appear to have utility for furthering the goals of nutritional anthropology. Among these goals are recognizing and specifying the conditions under which regularities occur in how people conceptualize, procure, process, distribute, and use food, and the attendant consequences. As in any science, the aim of nutritional anthropology is to provide reliable and valid information in a form suitable for practical use. Striving to achieve these goals entails several operations (not always in this order): observation, concept formation, measurement, enumeration, comparison, classification, proposition formulation, verification, and extrapolation. And these in turn require, among other things, a precise language, logic, and calculus for: (a) defining relations between concepts and variables, and relations between these and empirical phenomena and substantive theory; (b) formulating and verifying propositions; and (c) deriving implications and extrapolations beyond the facts observed. We conjecture that mathematics can meet these needs, for it is simultaneously an abstract, yet precise, language, logic, and calculus for defining the relations among a set of elements. If correspondences between empirical phenomena or substantive theory and mathematical systems can be successfully established, then the full conceptual and analytic power of mathematics can be harnessed and put to use.
The process of mapping empirical data and substantive theory onto abstract, mathematical systems (or the converse) is known as "mathematical modelling." Mathematical models (like other kinds of models, such as toys, games, and maps) are abstract expressions or representations of part of the real world. They consist of statements about relationships among a set of variables. These statements normally take the form of a set of equations or rules for establishing identities among quantities. Like other models, they are designed to be observed, contemplated, manipulated, tested, and revised. As Hoffmann comments, "A calculus is an abstract mathematical structure, a model its interpretation within an empirical context" (1971, pp. 189-190).
The value of building a mathematical model is considerable. It provides a medium for precise description and a logic for reasoning through complex arguments and for examining the logical validity of statements. Also, a model greatly augments analytic power by enabling the decomposition and reduction of data to intelligible proportions and by allowing for complex, subtle, and extended derivations. A model serves to define problems concisely and to generate verifiable predictions and extrapolations in a way that makes practical implications more obvious. A model may also indicate significant gaps in data and measurements requisite to resolving certain problems, and provide direction for the collection of new kinds of data and the development of new kinds of measurement procedures. Moreover, a model facilitates the transfer of data, concepts, and patterns from one field to another by formalizing representations and descriptions of empirical phenomena and substantive theory at a level abstract enough to be integrated with models of similar phenomena in other disciplines, e.g. economics, physics, and biology.
At the outset we acknowledge that the scope and treatment of mathematical models in this chapter will be quite limited, for several reasons. First, we have tried to focus on models with rather obvious connections to both pure and applied research in nutritional anthropology, as is often indicated by their prior use. Second, as far as possible we want to avoid conjuring up artificial examples and to demonstrate the application of mathematical models to real data from actual research in the field. While the problems and data to which we have applied certain models are attenuated, we hope they will show how analysis can proceed and how analogous research concerns can be tackled. Third, our presentation is limited to modelling, for the most part, with pre-calculus mathematics. Because we have assumed little more than an elementary background in algebra and statistics, we have allocated considerable space for illustrating basic computational procedures. We forgo enunciating theorems, discussing proofs, and making extensive derivations. Finally, we should stress that we see our purpose as offering a smorgasbord of appetizers, prepared to convey some of the flavour of mathematical analysis, but not to be mistakenly consumed as a substitute for a more substantially nourishing main course. If we motivate a desire to explore the benefits of mathematics a little more than before, then we will have succeeded.
The models we have selected to describe are, in order: (a) prediction models, using explicit functions to infer the values of unknowns from the values of knowns; (b) preference relations for the analysis of the properties of relations in a set of elements to construct a preference order; (c) decision-making models for optimally allocating scarce means to alternative ends, with surrounding constraints; (d) input-output models for specifying the structure of relations of interdependence among a set of components conceived as an endogenous system and the relations of this system to exogenous environmental variables; and (e) stochastic process models for representing phenomena as a sequence of random outcomes that are governed probabilistically.