Axiomatic evaluation of k-multiplicativity in ratio scaling: Investigating numerical distortion
Section snippets
Theoretical background
The relationship between perceived and mathematical numbers has been studied in the framework of what is called a “separable” representation (Luce, 2002; Steingrimsson, this volume) postulating a transformation of the physical stimulus level to a perceived intensity, that is independent of the way the numerical response is generated in a scaling experiment. In a magnitude estimation experiment, this numerical response is the numeral the participant assigns to a given stimulus magnitude, whereas
Empirical background
In a previous study, Birkenbusch, Ellermeier, and Kattner (2015) investigated, whether the implicit assumptions fundamental to Stevens’ direct scaling methods hold for the perception of short durations, i.e., whether individual duration perception is based on a ratio scale and whether the numbers used in the experiment and by the participants can be interpreted at face value. The three fundamental axioms formulated by Narens (1996) were tested: Monotonicity, commutativity and multiplicativity.
Mathematical background
The focus of this study is to empirically investigate the relationship between perceived and mathematical numbers and therefore, in the following, the axiomatic approach of Steingrimsson and Luce (2007) on which the present experiment is based is recapitulated and explained in detail. In this study, the notation introduced by Narens (1996) is used according to which represents a participant’s adjustment , that is perceived to last times as long as the standard . First of all,
Conclusion
So in conclusion, the validity of 1-multiplicativity supports the assumption of a weighting function of the form . Because testing 1-multiplicativity empirically yielded the axiom to be violated in nearly all tests performed, its fundamental assumption has to be refuted. Thus, the axiom of -multiplicativity can be tested assuming . If this weaker condition holds, the relationship between perceived and mathematical numbers follows an power function with a positive and constant
Methods
Three ratio production experiments were carried out differing in the ratio production factors and the length of the standard durations. In Experiment I, integer ratio production factors were used, while in Experiment II, only fractions were employed. In Experiment III, both and were intermixed.
Participants
Both Experiments I and II were completed by participants, respectively. participants took part in Experiment III. The total sample consisted of 13 male and 22 female participants
Results
The data of the three experiments were analyzed separately and thus, they are described in three separate sections. Furthermore, the validity of the axioms was tested individually for each participant and each trial type. In the following, the results for the axiomatic analyses of monotonicity, commutativity and 1-multiplicativity are recapped for each experiment. Afterwards, the axioms of -multiplicativity is tested, respectively. Finally, the results are briefly summarized. A summary on the
Axiomatic testing
The aim of this experiment was to investigate the relationship between perceived and mathematical numbers by testing the axioms of 1-multiplicativity and -multiplicativity using data from three different duration production experiments.
In Experiments I and II, the axiomatic testing procedure as proposed by Narens (1996) yielded the fundamental axioms of monotonicity and commutativity to hold for most participants.
Thus, it may be assumed that participants operate on a sensory continuum and make
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