Evaluating the Ratio Scalability of Perceived Duration An Axiomatic Study

• Main Author: Birkenbusch, Jana
• Format: Others
• Language: German; en
• Published: 2015
• Online Access: https://tuprints.ulb.tu-darmstadt.de/5025/1/Dissertation%20Birkenbusch%20Print%202.pdf; Birkenbusch, Jana <http://tuprints.ulb.tu-darmstadt.de/view/person/Birkenbusch=3AJana=3A=3A.html> (2015): Evaluating the Ratio Scalability of Perceived Duration An Axiomatic Study.Darmstadt, Technische Universität, [Ph.D. Thesis]

The relationship between the physical intensity t of a stimulus and its perceived magnitude φ(t) can be described by Stevens’ power law φ(t) = αt^β (Stevens, 1956). The exponent of the power function β, crucial for the shape of the psychophysical function, depends on the sensory modality studied and can be estimated via direct scaling experiments. As recent developments in axiomatic measurement theory have shown, the application of direct scaling is based on fundamental assumptions concerning the participants’ scaling behavior: The observers’ perception of the investigated modality needs to be ordered on a sensory continuum and has to be valid on a ratio scale. Furthermore, the numbers as presented in the experimental instructions have to be processed as exact mathematical values. Narens (1996) made these implicit assumptions empirically testable by expressing them in the behavioral axioms of monotonicity, commutativity and multiplicativity. However, rigorous axiomatic testing showed that most participants fail to veridically process the numerical instructions used in production or estimation tasks. Steingrimsson and Luce (2007) have thoroughly analyzed the kind of “numerical distortion” that appears to be operating and claimed that the relationship between perceived and mathematical numbers can be described by a power function. To make this assumption empirically testable, they formulated the axiom of k-multiplicativity. The present thesis aimed to empirically evaluate this axiomatic framework to the perception of short durations. This was accomplished by combining axiomatic testing strategies derived from different theoretical approaches (Augustin, 2008; Narens, 1996; Steingrimsson & Luce, 2007) in a single reinvestigation thereby affording a much more precise determination of the concept of ratio scalability than in the most earlier empirical studies. Furthermore, the application domain was human time perception, which had not been subjected to this kind of axiomatic approach before. The aim of Manuscript A (N = 25) was to find out whether the basic assumptions for the application of direct scaling methods are valid for the perception of short durations. Furthermore it was tested whether the estimated power law parameters are invariant under changes of the reference stimulus and thus psychologically relevant. In accordance with previous findings for other sensory continua, monotonicity held for the duration adjustments of most participants. Significant violations of the commutativity axiom were found in 12.5% of all pertinent tests, whereas multiplicativity was violated in 32% of such tests. The axioms of weak multiplicativity and invertibility were violated in over 50% of the tests, indicating a problem with psychological relevance. Manuscript B examined whether a relationship between mathematical and perceived numbers can be described by a power function with a constant exponent and whether there is a difference between the processing of integers and fractions. To that effect, the validity of k-multiplicativity was evaluated for N = 35 participants. The axiomatic tests showed a power function with a constant exponent to appropriately describe the relationship between mathematical and perceived numbers. However, different values of k were found for integers and fractions indicating that they are processed differently. Manuscript C investigated whether the functional relationship between standard duration and power law parameters can be determined. Furthermore, it tested whether the standard dependency of the power law parameters is an artifact of the ratio production procedure or whether this finding is stable even if other measures of sensitivity are used. The power law parameters were estimated for six different standard durations t (0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 s) and compared to the corresponding Weber fractions. The results of two experiments with the same N = 10 participants show a positive power relationship between the duration of the standard and the estimated exponent of Stevens’ power law, which can be described by the function β = 0.13t^0.3. A negative power relationship of the form W = 0.84t^−0.3 was found between the Weber fractions and the duration of the standard. In conclusion, the present doctoral thesis shows that if using ratio production of temporal intervals, the measurement is based on a sensory continuum and on a ratio scale. Therefore, the application of direct scaling methods in order to determine the power law parameters for perceived duration is legitimate. It was further found that a large proportion of the participants does not process the numerals that are presented in the experimental instructions at face value, i.e., an inherent numerical distortion impedes an unequivocal interpretation of the scale values. However, this numerical distortion does not reflect an entirely arbitrary or intractable interpretation of numbers, but a well-characterized mathematical relationship – a power function with a constant exponent. Because it was shown that fractions and integers are processed differently, they should not be intermixed within one ratio production experiment. Furthermore, the present thesis showed that modeling perceived time as a function of physical time, regardless of whether a power function or a linear relationship holds, is difficult: Even if both kinds of models seem to describe the relationship quite well, the estimated parameters depend on the magnitude of the reference stimulus used in the experiment and thus can hardly be interpreted in a psychologically relevant way. However, the influence of the standard on the size of the exponent seems to be systematic: Increasing standard durations go along with increasing exponents. Weber fractions measured under identical conditions were found to decrease with increasing standard durations and thus, combining both findings, it can be assumed that differential sensitivity for duration perception increases between 100 and 400 ms and remains at a constant level between 400 and 600 ms. A bias due to the ratio production procedure is thus ruled out.