THE ROLE OF STATISTICAL-MODELS

The formal theory of statistical analysis depends on the abstractions that the data are observed values of random variables, that a family of probability distributions (a model) has, at least tentatively, been formulated and that the objective of the analysis is connected in some way with the underlying 'true' distribution. These ideas, while very powerful, are idealizations and in particular the choice of the model is of critical importance. In addition, many challenging aspects of applied statistics are, of course, concerned with such issues as the meaning and quality of data, and the interpretation of carefully chosen tables and graphs, matters where the probabilistic aspect is barely visible. It is argued that, while the recognition of this is important, the moral is not that descriptive statistics should be separated from probabilistic statistics, but rather that the common ground should be explored; what is the formal theory underlying descriptive methods and how can the results of analyses based on elaborate theory be presented in a striking and easily understood way? There are various aspects to the choice of models. A first distinction is between primary aspects, which in effect serve to define the aspects of substantive interest, and secondary aspects which complete the specification and which thereby indicate appropriate efficient methods of analysis and how to calculate measures of uncertainty. Often, but not necessarily, the latter concern assumptions of distributional shape and of independence; they are secondary not in the sense of being unimportant but rather in not being the primary focus of substantive interest. Models, especially in their primary aspects, are of three broad types, substantive (i.e. based on special subject-matter knowledge or theory), empirical representing usually smooth dependencies without any strong subject-matter input and indirect models which are useful, for example, for calibration of descriptive methods. All three types can be subdivided in various ways; see Cox (1990). These ideas were illustrated in some detail by discussing various kinds of model for precipitation, for example hourly rainfalls, and by the issues involved in predicting the AIDS epidemic. The importance of interplay between substantive and empirical ideas was emphasized.