title: Three Boundary Conditions for Computing the Fixed-Point Property in Binary Mixture Data.
abstract: The notion of "mixtures" has become pervasive in behavioral and cognitive sciences, due to the success of dual-process theories of cognition. However, providing support for such dual-process theories is not trivial, as it crucially requires properties in the data that are specific to mixture of cognitive processes. In theory, one such property could be the fixed-point property of binary mixture data, applied-for instance- to response times. In that case, the fixed-point property entails that response time distributions obtained in an experiment in which the mixture proportion is manipulated would have a common density point. In the current article, we discuss the application of the fixed-point property and identify three boundary conditions under which the fixed-point property will not be interpretable. In Boundary condition 1, a finding in support of the fixed-point will be mute because of a lack of difference between conditions. Boundary condition 2 refers to the case in which the extreme conditions are so different that a mixture may display bimodality. In this case, a mixture hypothesis is clearly supported, yet the fixed-point may not be found. In Boundary condition 3 the fixed-point may also not be present, yet a mixture might still exist but is occluded due to additional changes in behavior. Finding the fixed-property provides strong support for a dual-process account, yet the boundary conditions that we identify should be considered before making inferences about underlying psychological processes.
authors.affiliation: Department of Psychology, University of Amsterdam, Amsterdam, The Netherlands.|Amsterdam Brain and Cognition (ABC), University of Amsterdam, Amsterdam, The Netherlands.
authors.forename: Leendert
authors.identifier: http://orcid.org/0000-0001-9120-1075
authors.initials: L
authors.lastname: van Maanen
authors.affiliation: Amsterdam Brain and Cognition (ABC), University of Amsterdam, Amsterdam, The Netherlands.|Amsterdam School of Economics (ASE), Faculty of Economics and Business (FEB), University of Amsterdam, Amsterdam, The Netherlands.
authors.forename: Joaquina
authors.initials: J
authors.lastname: Couto
authors.affiliation: Amsterdam Brain and Cognition (ABC), University of Amsterdam, Amsterdam, The Netherlands.|Amsterdam School of Economics (ASE), Faculty of Economics and Business (FEB), University of Amsterdam, Amsterdam, The Netherlands.
authors.forename: Mael
authors.initials: M
authors.lastname: Lebreton
mesh_terms: D003071:Cognition / Q000502:physiology*; D003198:Computer Simulation; D006801:Humans; D008960:Models, Psychological; D011930:Reaction Time / Q000502:physiology*