Coevolution of Mathematics, Statistics, and Genetics

Genetics is the science of studying heredity. Heredity is the process of transmitting genetic materials from parents to offspring. In genetic studies, hypotheses derived from biological theories and mathematical models are tested with the data from experiments or observations of genetic phenomena using statistical methodologies. Throughout the history of genetics, mathematics and statistics have been extensively used for genetic studies, and genetics, in turn, has influenced many fields of mathematics and statistics. In this chapter, we describe some of the most important mathematical models and statistical methods in the history of genetics. We especially focus on three periods: (1) the early days, when the basic concepts in genetics were established, such as genes, evolution, and inheritance, and mathematical models of such genetic mechanisms were laid out; (2) the period of studying family data from twins or large pedigrees in the mid- to late twentieth century; and (3) the present period of exploring big genetic data by complex modeling and machine learning. We show that various probabilistic models, differential equations, and graph and network theories have been applied to the analysis of genetic data. We also illustrate how statistical issues involved with model fitting, estimation, and hypothesis testing have been raised and resolved in the context of genetic studies, contributing to the field of statistics as well as that of genetics. In the discussion, we suggest some promising mathematical and statistical methods to be applied in future genetic studies.