The Joint Distribution Criterion and the Distance Tests for Selective Probabilistic Causality

A general definition and a criterion (a necessary and sufficient condition) are formulated for an arbitrary set of external factors to selectively influence a corresponding set of random entities (generalized random variables, with values in arbitrary observation spaces), jointly distributed at every treatment (a set of factor values containing precisely one value of each factor). The random entities are selectively influenced by the corresponding factors if and only if the following condition, called the joint distribution criterion, is satisfied: there is a jointly distributed set of random entities, one entity for every value of every factor, such that every subset of this set that corresponds to a treatment is distributed as the original variables at this treatment. The distance tests (necessary conditions) for selective influence previously formulated for two random variables in a two-by-two factorial design (Kujala and Dzhafarov, 2008, J. Math. Psychol. 52, 128–144) are extended to arbitrary sets of factors and random variables. The generalization turns out to be the simplest possible one: the distance tests should be applied to all two-by-two designs extractable from a given set of factors.