Conditionally Selective Dependence of Random Variables on External Factors

Selective influence of experimental factors upon observable or hypothetical random variables is a key concept in the analysis of processing architectures and response time decompositions. This paper deals with the notion of conditionally selective influence, defined as follows. Let {X1, em leader, Xn} be stochastically interdependent random variables (e.g., hypothetical components of response time), and let Phi be a set of external factors affecting the joint distribution of {X1, em leader, Xn}. A subset of factors &Lambdai conditionally selectively influences Xi if at any fixed values of the remaining random variables the conditional distribution of Xi only depends on factors inside &Lambdai. The notion of conditional selectivity generalizes the relationship between factors and random variables described in Townsend (1984) as "indirect nonselectivity." This paper establishes the structure of the joint distribution of {X1, em leader, Xn} that is necessary and sufficient for {X1, em leader, Xn} to be conditionally selectively influenced by (not necessarily disjoint) factor subsets {&Lambda1, em leader, Gamman}, respectively. The notion of conditional selectivity is compared to that of unconditional selectivity, defined as follows. A subset of factors &Gammai unconditionally selectively influences Xi if the latter can be presented as a deterministic function of &Gammai and of some random variables (the same for all Xi, i=1, em leader, n) whose joint distribution does not depend on any factors from Phi. The two forms of selective influence are generally incompatible. Copyright 1999 Academic Press.