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.     

Double Skew-Dual Scaling: A Conjoint Scaling of Two Sets of Objects Related by a Dominance Matrix

Consider two sets of objects, {alpha(1), em leader, alpha(n)} and {beta(1), em leader, beta(m), such as n subjects solving m tasks, or n stimuli presented first and m stimuli presented second in a pairwise comparison experiment. Let any pair (alpha(i), beta(j)) be associated with a real number a(ij), interpreted as the degree of dominance of alpha(i) over beta(j) (e.g., the probability of alpha(i) relating in a certain way to beta(j)). Intuitively, the problem addressed in this paper is how to conjointly, in a "naturally" coordinated fashion, characterize the alpha-objects and beta-objects in terms of their overall tendency to dominate or be dominated. The gist of the solution is as follows. Let A denote the nxm matrix of a(ij) values, and let there be a class of monotonic transformations straight phi with nonnegative codomains. For a given straight phi, a complementary matrix B is defined so that straight phi(a(ij))+straight phi(b(ij))=const, and one computes vectors D(alpha) and D(beta) (the dominance values for alpha-objects and beta-objects) by solving the equations straight phi(A) straight phi(D(beta))/Sigma;straight phi(D(beta))=straight phi(D(alpha)) and straight phi(B(T)) straight phi(D(alpha))/Sigmastraight phi(D(alpha))=straight phi(D(beta)), where (T) is transposition, Sigma is the sum of elements, and straight phi applies elementwise. One also computes vectors S(alpha) and S(beta) (the subdominance values for alpha-objects and beta-objects) by solving the equations straight phi(B) straight phi(S(beta))/Sigmastraight phi(S(beta))=straight phi(S(alpha)) and straight phi(A(T)) straight phi(S(alpha))/Sigmastraight phi(S(alpha))=straight phi(S(beta)). The relationship between S-vectors and D-vectors is complex: intuitively, D(alpha) characterizes the tendency of an alpha-object to dominate beta-objects with large dominance values, whereas S(alpha) characterizes the tendency of an alpha-objects to fail to dominate beta-objects with large subdominance values. For classes containing more than one straight phi-transformation, one can choose an optimal straight phi as the one maximizing some measure of discrimination between individual elements of vectors straight phi(D(alpha)), straight phi(D(beta)), straight phi(S(alpha)), and straight phi(S(beta)), such as the product or minimum of these vectors' variances. The proposed analysis of dominance matrices has only superficial similarities with the classical dual scaling (Nishisato, 1980). Copyright 1999 Academic Press.     

Dissimilarity cumulation theory in arc-connected spaces

This paper continues the development of the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling [Dzhafarov, E.N and Colonius, H. (2007). Dissimilarity Cumulation theory and subjective metrics. Journal of Mathematical Psychology, 51, 290-304]. In arc-connected spaces the notion of a chain length (the sum of the dissimilarities between the chain’s successive elements) can be used to define the notion of a path length, as the limit inferior of the lengths of chains converging to the path in some well-defined sense. The class of converging chains is broader than that of converging inscribed chains. Most of the fundamental results of the metric-based path length theory (additivity, lower semicontinuity, etc.) turn out to hold in the general dissimilarity-based path length theory. This shows that the triangle inequality and symmetry are not essential for these results, provided one goes beyond the traditional scheme of approximating paths by inscribed chains. We introduce the notion of a space with intermediate points which generalizes (and specializes to when the dissimilarity is a metric) the notion of a convex space in the sense of Menger. A space is with intermediate points if for any distinct there is a different point such that (where D is dissimilarity). In such spaces the metric G induced by D is intrinsic: coincides with the infimum of lengths of all arcs connecting to In Universal Fechnerian Scaling D stands for either of the two canonical psychometric increments and (ψ denoting discrimination probability). The choice between the two makes no difference for the notions of arc-connectedness, convergence of chains and paths, intermediate points, and other notions of the Dissimilarity Cumulation theory.     

Multidimensional Fechnerian Scaling: Pairwise Comparisons, Regular Minimality, and Nonconstant Self-Similarity

Stimuli presented pairwise for same-different judgments belong to two distinct observation areas (different time intervals and/or locations). To reflect this fact the underlying assumptions of multidimensional Fechnerian scaling (MDFS) have to be modified, the most important modification being the inclusion of the requirement that the discrimination probability functions possess regular minima. This means that the probability with which a fixed stimulus in one observation area (a reference) is discriminated from stimuli belonging to another observation area reaches its minimum when the two stimuli are identical (following, if necessary, an appropriate transformation of the stimulus measurements in one of the two observation areas). The remaining modifications of the underlying assumptions are rather straightforward, their main outcome being that each of the two observation areas has its own Fechnerian metric induced by a metric function obtained in accordance with the regular variation version of MDFS. It turns out that the regular minimality requirement, when combined with the empirical fact of nonconstant self-similarity (which means that the minimum level of the discrimination probability function for a fixed reference stimulus is generally different for different reference stimuli), imposes rigid constraints on the interdependence between discrimination probabilities and metric functions within each of the observation areas and on the interdependence between Fechnerian metrics and metric functions belonging to different observation areas. In particular, it turns out that the psychometric order of the stimulus space cannot exceed 1.     

The equivalence of two ways of computing distances from dissimilarities for arbitrary sets of stimuli

Given a set endowed with pairwise dissimilarities, the Dissimilarity Cumulation procedure computes the (quasi)distance between any two elements of the set as the infimum of the sums of dissimilarities across all finite chains of elements connecting the two elements. For finite sets, this procedure is known to be equivalent to recursive corrections for violations of the triangle inequality in any sequence of ordered triads of points which contains every triad a sufficient number of times. This paper extends this equivalence to infinite set.     

Multidimensional Fechnerian Scaling: Probability-Distance Hypothesis

The probability-distance hypothesis states that the probability with which one stimulus is discriminated from another is a function of some subjective distance between these stimuli. The analysis of this hypothesis within the framework of multidimensional Fechnerian scaling yields the following results. If the hypothetical subjective metric is internal (which means, roughly, that the distance between two stimuli equals the infimum of the lengths of all paths connecting them), then the underlying assumptions of Fechnerian scaling are satisfied and the metric in question coincides with the Fechnerian metric. Under the probability-distance hypothesis, the Fechnerian metric exists (i.e., the underlying assumptions of Fechnerian scaling are satisfied) if and only if the hypothetical subjective metric is internalizable, which means, roughly, that by a certain transformation it can be made to coincide in the small with an internal metric; and then this internal metric is the Fechnerian metric. The specialization of these results to unidimensional stimulus continua is closely related to the so-called Fechner problem proposed in 1960's as a substitute for Fechner's original theory.     

Dissimilarity cumulation theory and subjective metrics

We present a new mathematical notion, dissimilarity function, and based on it, a radical extension of Fechnerian Scaling, a theory dealing with the computation of subjective distances from pairwise discrimination probabilities. The new theory is applicable to all possible stimulus spaces subject to the following two assumptions: (A) that discrimination probabilities satisfy the Regular Minimality law and (B) that the canonical psychometric increments of the first and second kind are dissimilarity functions. A dissimilarity function Dab for pairs of stimuli in a canonical representation is defined by the following properties: (1) a≠b?Dab>0; (2) Daa=0; (3) If and , then ; and (4) for any sequence {a_nX_nb_n}_n∈N, where X_n is a chain of stimuli, Da_nX_nb_n→0?Da_nb_n→0. The expression DaXb refers to the dissimilarity value cumulated along successive links of the chain aXb. The subjective (Fechnerian) distance between a and b is defined as the infimum of DaXb+DbYa across all possible chains X and Y inserted between a and b.