Abstract
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.
Highlights
A system’s behavior, be the system biological, social, or technological, can be thought of as a network of stochastically interdependent random entities
In this paper the joint distribution criterion is formulated in complete generality, for arbitrary sets of random entities and corresponding sets of external factors
Even for the finite case, the present definition is mathematically more rigorous, and it profits from the precision offered by the notation xα = (x, ‘α’) for factor points. It can be seen more immediately than the previous definitions to be reformulable into the joint distribution criterion for selective influence, as discussed
Summary
For a finite set of random variables the definition of selective influence was given in Dzhafarov (2003) and refined in Dzhafarov and Gluhovsky (2006) and Kujala and Dzhafarov (2008) Applying it to our example, (A, B) is selectively influenced by (α, β) if and only if one can find functions f and g and a random entity C whose distribution does not depend on α, β, such that (A,B)αβ ∼ ( f (α,C), g (β,C)),. The fact that for any given treatment A and B are stochastically related (i.e., paired, whether independent or interdependent) means in Kolmogorov’s probability theory that A and B are measurable functions of one and the same random entity.
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