Dichotomization and Estimation of Interaction through a Boolean Framework

The paper develops an approximate approach to data analysis based on measured values of arbitrary nature factors and a response. It makes possible to detect joint action of these factors as well as to estimate the strength (degree) of their joint action in the given response. This approach is founded on the Boolean model of the binary theory of sufficient causes developed by the authors. In particular, the concept of degree of joint action of a different number of factors (Boolean variables) in a given binary response (Boolean function) is introduced, as well as its properties are studied in this framework. The application of this theory becomes possible after the dichotomization of raw data and construction a probability distribution of a Boolean function-valued random variable based on the dichotomized data. In turn, the latter probability distribution allows one to construct a probability distribution of the degree of joint action of Boolean variables as a random variable. In addition, some basic concepts of fuzzy logic, probabilistic and statistical methods are used for theoretical justifications and computer data simulation. The results of computer simulations discussed in the paper confirm the developed approach. The presented theoretical results and examples can be used to study joint action of factors in the analysis of real data in epidemiology, toxicology, evidence-based medicine, etc.