Shifts of the psychometric function: distinguishing bias from perceptual effects.

Morgan, Dillenburger, Raphael, and Solomon have shown that observers can use different response strategies when unsure of their answer, and, thus, they can voluntarily shift the location of the psychometric function estimated with the method of single stimuli (MSS; sometimes also referred to as the single-interval, two-alternative method). They wondered whether MSS could distinguish response bias from a true perceptual effect that would also shift the location of the psychometric function. We demonstrate theoretically that the inability to distinguish response bias from perceptual effects is an inherent shortcoming of MSS, although a three-response format including also an "undecided" response option may solve the problem under restrictive assumptions whose validity cannot be tested with MSS data. We also show that a proper two-alternative forced-choice (2AFC) task with the three-response format is free of all these problems so that bias and perceptual effects can easily be separated out. The use of a three-response 2AFC format is essential to eliminate a confound (response bias) in studies of perceptual effects and, hence, to eliminate a threat to the internal validity of research in this area.

“Varopoulos paradigm”: Mackey property versus metrizability in topological groups.

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Außenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Díaz Nieto and Martín Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.