Boolean Algebras and Classification of Interactions in Sufficient-Component Cause Model

Abstract

A mathematical model of the sufficient-component cause framework is considered based on the theories of Boolean algebra. The model consists of the space of states of a binary experiment and a set of symmetries of the experiment. The space of states is a Boolean algebra of n Boolean variables where n is the number of the binary causes in the experiment. The set of symmetries of the experiment is a subgroup of the group of all automorphisms of Boolean algebra of the states of experiment. This subgroup is generated by transformations preserving a type of interaction. An experimenter should deduce these transformations from the peculiar properties of the experiment. Examples of such transformations are provided. Classification of interactions is obtained by the calculation of the orbits of action of the group of symmetries on the space of states of the experiment. It is shown that the classification of the interaction for the ordinary symmetries of sufficient causes is the same as reported in related works. Other symmetries of the binary experiment are considered as well. It is shown that the corresponding classification of the interaction types in a binary experiment depends substantially on the symmetries of the experiment. Statistical criteria of particular types of responses are proven and the problem

Joint Action of Binary Factors in the Sufficient Causes Theory and Its Classification

“…A more rigorous formal presentation of the binary sufficient causes theory is possible on the basis of finite Boolean algebras theory. That was shown in [22,23] for two variables and in [24] for general case of n variables. An adequate mathematical apparatus is presented therein for studying a classification of factors joint action types in the binary theory of sufficient causes.…”

“…A complete description of the formalism of binary sufficient causes theory based on Boolean algebras theory is presented in [22][23][24]. The paper [24] also describes the relationship between the theory of sufficient causes and the Neyman-Holland-Rubin causality model [25].…”

“…Briefly, the structure of the binary theory of sufficient causes can be described as follows [22][23][24]. The state space of a binary experiment forms a finite Boolean algebra consisting of the set of all binary vectors of length n. The response (outcome) is considered as a Boolean function on this space and the set of all responses forms the Boolean algebra of all Boolean functions defined on the state space.…”

“…Empirical symmetries [15,17,21] play an important role in the structure of a binary experiment. In the proposed formalization [22][23][24] these symmetries can be written as automorphisms on the algebra of Boolean functions.…”

“…In the usual epidemiological situation, these symmetries form a group of all symmetries of the n-dimensional cube [22][23][24]. As a result, it is impossible to distinguish the types of joint action which could be considered antagonistic (in a sense, weakening the total effect of agents) from the synergistic types (that are stronger than the sum of one-factor effects).…”