Conditionally Selective Dependence of Random Variables on External Factors

Abstract

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.

Unconditionally Selective Dependence of Random Variables on External Factors

“…This meaning, also discussed in Dzhafarov (1999), is derived from Dzhafarov (1992Dzhafarov ( , 1997 and Dzhafarov and Schweickert (1995). According to the definition proposed in these papers, [X 1 , ..., X n ] are unconditionally selectively influenced by, respectively, [1 1 , ..., 1 n ] if these random variables can be presented as X 1 =X 1 (P 1 , ..., P n ; 1 1 ) } } } _ X i =X i (P 1 , ..., P n ;…”

“…What then becomes of the hypothesis that [X 1 , X 2 ] are selectively influenced by # 1 and # 2 , respectively ? Generalizing, what is the meaning in which one can say that stochastically interdependent random variables [X 1 , ..., X n ] (such as hypothetical response time components or observable aptitude scores) are selectively influenced by, respectively, subsets [1 1 , ..., 1 n ] of a factor set 8 upon which the joint distribution of [X 1 , ..., X n ] is known to depend? Dzhafarov (1997Dzhafarov ( , 1999) distinguishes two different meanings in which such selectiveness of influence can be understood.…”

“…Random variables [X 1 , ..., X n ] are conditionally selectively influenced by, respectively, [1 1 , ..., 1 n ] if, for any i, the conditional distribution of X i , given any fixed values for the rest of the variables, depends only on the factors comprising 1 i . Dzhafarov (1999) provides a complete characterization (the necessary and sufficient structure of the joint density function) for this form of selective influence. When applied to the problem posed earlier, this characterization yields the following result.…”

“…One meaning, termed in Dzhafarov (1999) conditionally selective influence, is derived from the idea of indirect nonselectivity proposed in Townsend (1984) and studied in Townsend and Thomas (1994). Random variables [X 1 , ..., X n ] are conditionally selectively influenced by, respectively, [1 1 , ..., 1 n ] if, for any i, the conditional distribution of X i , given any fixed values for the rest of the variables, depends only on the factors comprising 1 i .…”

Selective attention modulates the effect of target location probability on redundant signal processing

“…All analyses were conducted separately for each individual. We first tested the selective-influence assumption, a critical assumption of SFT (Dzhafarov, 1999;Kujala & Dzhafarov, 2008;Townsend & Thomas, 1994). In the current experimental setting, the selective-influence assumption holds when the brightness manipulation on a location only influences the information processing speed of that location.…”