Unconditionally Selective Dependence of Random Variables on External Factors.

What is the meaning of saying that random variables {X(1), em leader, X(n)} (such as aptitude scores or hypothetical response time components), not necessarily stochastically independent, are selectively influenced respectively by subsets {Gamma(1), em leader, Gamma(n)} of a factor set Phi upon which the joint distribution of {X(1), em leader, X(n)} is known to depend? One possible meaning of this statement, termed conditionally selective influence, is completely characterized in Dzhafarov (1999, Journal of Mathematical Psychology, 43, 123-157). The present paper focuses on another meaning, termed unconditionally selective influence. It occurs when two requirements are met. First, for i=1, em leader, n, the factor subset Gamma(i) is the set of all factors that effectively change the marginal distribution of X(i). Second, if {X(1), em leader, X(n)} are transformed so that all marginal distributions become the same (e.g., standard uniform or standard normal), the transformed variables are representable as well-behaved functions of the corresponding factor subsets {Gamma(1), em leader, Gamma(n)} and of some common set of sources of randomness whose distribution does not depend on any factors. Under the constraint that the factor subsets {Gamma(1), em leader, Gamma(n)} are disjoint, this paper establishes the necessary and sufficient structure of the joint distribution of {X(1), em leader, X(n)} under which they are unconditionally selectively influenced by {Gamma(1), em leader, Gamma(n)}. The unconditionally selective influence has two desirable properties, uniqueness and nestedness: {X(1), em leader, X(n)} cannot be influenced selectively by more than one partition {Gamma(1), em leader, Gamma(n)} of the factor set Phi, and the components of any subvector of {X(1), em leader, X(n)} are selectively influenced by the components of the corresponding subpartition of {Gamma(1), em leader, Gamma(n)}. Copyright 2001 Academic Press.