The mathematical sciences have made significant contributions to many areas of science of special importance to mankind, and they, in turn, have been enriched by these contributions. One obvious example is the interactions between statistics and problems in medicine, epidemiology, and public health. These contributions have led to an entire subfield of the mathematical sciences, biostatistics, to address applications in these fields.
One of the most important individuals in the development of statistics was John Graunt. In 1662, Graunt turned his attention to the Bills of Mortality, weekly reports by London parish clerks giving the number and the causes of death. These had been instituted to help authorities detect the onsets of epidemics. Graunt published an analysis of these data and developed what is now known as a life table, which allows for the calculation of life expectancy. His work was especially ingenious in that he did not have basic population data from censuses to work with. The ideas and methods Graunt established more than 300 years ago have been highly refined and now form the basis for the life insurance industry and modern survival analysis.
The refinements of census methodology, especially in health and vital statistics, are one of the most important aspects of epidemiology, a body of methods designed to determine which group is likely to become ill, the reasons for illness, and what can be done to control an illness. An important method in medical science for determining the efficacy of treatments is the clinical trial. The initial formalization of the clinical trial method used the work of the famous statistician R.A. Fisher on randomization of treatments to comparable groups of experimental units. Fisher had developed these methods to obtain proper statistical tests of significance for agricultural variables such as soil type and fertilizer. With the inception of rigorous drug approval processes, the importance of clinical trials has intensified, as has the need for innovations to make the trials as informative and as efficient as possible. A variety of innovations have been introduced, for example sequential methods (see the case study on martingale theory in Appendix A). The methods developed to solve medical problems, such as determining the causes of certain diseases and evaluating the efficacy of various therapies, have become core elements of the discipline of statistics and have been applied to many other substantive areas.
Another important aspect of biostatistics involves the mathematical and statistical modeling of biological and biomedical phenomena. Formal models have contributed greatly to an understanding of the time course of epidemics, the carcinogenesis process at the cellular level, the dose-response behavior of animals or humans to drugs or toxic substances, the point processes of neuron firings, and the pharmacokinetics and pharmacodynamics of drugs in the bloodstream. Much of the early mathematical modeling involved relatively simple systems of differential equations or Markov process models. As the behavior of biological processes becomes known in much greater detail and more sophisticated scientific technologies are developed to measure biological processes, more complex mathematical models are required. For example, a new dimension of models is now being developed to gain an understanding of scientific phenomena such as protein folding, cognitive neuroscience, and genomics. These biological problems will ultimately create new branches of statistics and mathematics, and the mathematical and statistical sciences will ultimately help to improve our understanding of these areas.
SOURCE: Based on Colton and Armitage (1998); Johnson and Kotz (1982); and Stigler (1986).