Axiomatic evaluation of k-multiplicativity in ratio scaling: Investigating numerical distortion

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Highlights

  • k-multiplicativity holds for separately used fractions and integers.

  • The function between mathematical and perceived numbers is of the form W(p)=kpω.

  • There is no single parametric instantiation for fractions and integers.

  • Directly estimated scale values can deviate from numbers they mathematically express.

Abstract

It is a well established empirical observation that most human participants do not process the numerical instructions used in production or estimation tasks veridically. Luce and collaborators (e.g., Luce, 2002; Steingrimsson and Luce, 2007) have analyzed the kind of “numerical distortion” that appears to be operating. They stated the relationship between perceived and mathematical numbers to be described by a power function, if the empirically testable axiom of k-multiplicativity holds. This study examined the validity of k-multiplicativity by testing whether the stimulus intensities resulting from successive adjustments ×p×q multiplied by a constant factor k are equal to the stimulus intensity resulting from single adjustments ×r. Therefore, the data of three different ratio production experiments with a total of N=35 participants were (re-)analyzed. In Experiment I, integers were used as ratio production factors (p1), while in Experiment II, only fractions (p<1) were applied. In Experiment III, both p1 and p<1 were intermixed. In Experiments I and II, k-multiplicativity held for all n=20 participants. Experiment III revealed axiom violations for 13 of n=15 participants. The failure of 1-multiplicativity confirms the common observation that the participants’ number representation is often not veridical. However, the validity of k-multiplicativity shows that the relationship between mathematical and perceived numbers follows a power function of the form W(p)=kpω with k1 and ω1. However, the numerical distortion differs for fractions compared to integers.

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 (x,p,t) represents a participant’s adjustment x, that is perceived to last p times as long as the standard t. First of all,

Conclusion

So in conclusion, the validity of 1-multiplicativity supports the assumption of a weighting function of the form W(p)=pω. 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 k-multiplicativity can be tested assuming W(p)p. 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 p1 were used, while in Experiment II, only fractions p<1 were employed. In Experiment III, both p1 and p<1 were intermixed.

Participants

Both Experiments I and II were completed by N=10 participants, respectively. N=15 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 k-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 k-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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