1 Stevens Direct Scaling Methods and the Uniqueness Problem: Empirical Evaluation of an Axiom fundamental to Interval Scale Level

2 Overview Hypothesis – The Interval Scale Axiom Method Participants Stimuli Procedure Results Testing the Interval Scale Axiom Violations of the Interval Scale Axiom Discussion Commutativity Mathematical knowledge

3 Simple visual task: Ratio Production Experiment: Subjects were required to adjust the area of variable squares to a prescribed ratio p. Is a single subject able to produce adjustments on an interval scale level?

4 The Interval Scale Axiom The hypothesis that Φ is a subscale of an interval scale is equivalent to the assumption that the following interval scale axiom is satisfied: Let us assume real numbers p, q, r, r´ and stimuli s, t, x, y such that:

5 ___________(t, q, x) E, x q t __________________ (s, p, t) E, t p s ______________________________(s, r, x) E, x r s ______________________________________(y, r´, x) E, x r´ y Then and ___________________________(y, p´, t) E t p´ y

6 The Interval Scale Axiom can be put to an empirical test

7 Participants - 10 students and 7 graduates - 9 female and 8 male - Age: between 21 and 35 years old ( 26.9 years) - None had prior knowledge of hypothesis - Normal vision or corrected to normal vision Method

8 Stimuli White squares on a black background were projected on a white screen by means of a video projector

9 Apparatus White screen Video projector (Sony VPL-X1000E) Keyboard (Cherry G83), PC Software Orange TA (Michael Kickmeier)

10 Procedure Welcome... Thank you very much for participating... Collection of personal data name, age, normal sight? (y/n), student (y/n) Instruction Practice session (4 trials) test trial – collecting data darkened room (Heizhaus, SR 12.31) participants distance from the screen was 3 m

11 Instruction... The task is to adjust the right square in such a way, that its area appers to be n times as large as the area of the standard stimulus on the left. Use the and the keys to adjust the right square and indicate a satisfactory match by pressing the Return key. If the projector screen appears to be to small to adjust the area to the prescribed ratio n, then you have the opportunity to press the Esc key and the next task will appear...

12

13 2 standard squares (x) - 80 Pixel (13.3 cm on the screen) Pixel side length (20 cm) 11 ratio production factors - (q = 2,3,4,....,12) for each (x, q) combination: 10 adjustments = 220 adjustments Session 1

14 Evaluation of Session 1 Calculating the mean areas of 10 adjustments adjustments adjustments adjustments (will be the standard squares in Session 2)

15 Session 2 4 standard squares (different for each participant) ratio production factors (p = 2,3,4,5,6) for each (x q, p) combination: 10 adjustments = 200 adjustments

16 Notation: mean area for the 10 adjustments : 80 2,3 In both Sessions: Standard square and ratio production factors were randomly intermixed The 2 Sessions were conducted on 2 different days

17 Results A few descriptives - One participant had to be excluded after Session 1 -Esc key was not used no ceiling effects - Session 1 took an average time of 46 min -Session 2 took an average time of 30 min

18 Mean adjustments in Session 1. Standard = 80 Pixel side length Mean adjustments in Session 1. Standard = 120 Pixel side length

19 Testing the interval scale axiom Analyzable Quadruple: There exists a quadruple (x, q, p,r) such that the q p adjustment x q,p is statistically indistinguishable from the r adjustment x r. Potentially 4 analyzable quadruples for any participant (1) 80 2 p 1 = 80 r 1 (2) 80 3 p 2 = 80 r 2 (3) p 3 = 80 r 3 (4) p 4 = 80 r 4

20 Data Selection Rank Order The interval scale axiom is based on the assumption, that the adjustments preserve the mathematical order of the ratio production factors. n adjustments < (n+1) adjustments 2 < 3 3 < 4

21 For any participant all such tuples were analyzed: We evaluated only those participants with less than 2 rank order violations 4 particpants were excluded p (rank order violation) 1/36 <.03

22 After Data Selection: 33 opportunities to check for the Interval Scale Axiom

23 One single case (Participant M.A) Standard square: 80 Pixel side length q p = r 3 2 = 5 z(U)=.23, p =.82, n.s q = 3 p = 2 r = 5 Analyzable Quadruple: (80, 3, 2, 5) A natural number r´ > r was fixed in such a way, that the ratio production factor p´ is an integer.

24 Adjustments generated by a single participant (M.A.) starting from the standard square having 80 Pixel side length. Tripling the standard and doubling the outcome is statistically indistinguishable from the 5 adjustment. Therefore, tripling the standard and then tripling the outcome should be statistically indistinguishable from the 7 adjustment. This is not the case. The axiom is violated.

25... r´=6... r´=7

26 If adjustments are on an interval scale level, then: If (80 3) 2 = 80 5 then there also must hold (80 3) 3 = 80 7 Formally: 80 3,2 = ,3 = 80 7

27 The area 80 3,3 is different from the area 80 7 (z(U) = 3.41, p <.01) The axiom is violated.

28 Violations of the interval scale axiom in the whole sample 33 Tests: 18 violations vs. 15 non-violations Only 3 of the (remaining) 12 participants showed no violation of the axiom

29 Transform the pattern of outcomes into an overall statistical statement: binomial distribution Probability for a single violation by chance is.05 Probability for observing 18 or more violations in 33 tests is: p = Assuming that the interval scale axiom holds for the whole sample, then the above pattern is highly unlikely.

30 Conclusion For area production of squares: Single subject is not able to produce adjustments on an interval scale level

31 Discussion The study generalizes earlier findings by Ellermeier and Faulhammer (2000) indicating that multiplicativity fails to hold. Loudness Production: Data are not on a Ratio Scale Level

32 Commutativity Doubling a standard and tripling the outcome converges on the same final outcome as first tripling the (same) standard and then doubling the outcome 2 3 = 3 2 (?) Ellermeier & Faulhammer (2000), Zimmer (in press) found Commutativity to hold. 32 tests (16 participants, 2 standard squares) Commutativity was violated in 15 cases. Commutativity is not a general law that holds across all sensory modalities.

33 Mathematical knowledge 4 and 9... play a special role Phyiscal area of a square can be quadrupled by doubling the side length Coefficient of Variation separately for the 11 ratio production factors

34 Mean coefficients of variation calculated separately for each ratio production factor p. The 4 and 9 adjustments show less variability than their respective neighbors. Mathematical knowledge:

35 Participants applied mathematical knowledge Teghtsoonian (1965)...ability to make accurate area-judgements depends on (1) his ability accurately to estimate length, and (2) his knowledge that the area of a two-dimensional figure is proportional to the square of a linear dimension

36 Stevens psychophysical functions are based on mean values over the whole sample The aim of the present study was to investigate, whether a single subject is able to produce adjustments on an interval scale level. area production of squares: no (too many violations of the interval scale axiom) Consequences: Parametric statistics have to be replaced by nonparametric methods.

37 Further studies: - different sensory modalities (loudness, brightness,...) - other direct scaling methods (ratio estimation) - are different scale-types more appropriate

38 Thank you very much

39 Anderson, N. H. (1970). Functional Measurement and Psychophysical Judgment. Psychological Review, 77, Anderson, N. H. (1976). Integration Theory, Functional Measurement and the Psychophysical Law. In H. G. Geissler, & Y. M. Zabrodin (Eds.), Advances in Psychophysics. Berlin: VEB Deutscher Verlag der Wissenschaften. Ellermeier, W., & Faulhammer, G. (2000). Empirical Evaluation of Axioms Fundamental to Stevens's Ratio-Scaling Approach: I. Loudness Production. Perception & Psychophysics, 62, Graham, C. H. (1958). Sensation and Perception in an Objective Psychology. Psychological Review, 65, Luce, R. D. (2002). A Psychophysical Theory of Intensity Proportions, Joint Presentations, and Matches. Psychological Review, 109, Luce, R. D. (2004). Symmetric and Asymmetric Matching of Joint Presentations. Psychological Review, 111, McKenna, F. P. (1985). Another Look at the \New Psychophysics". British Journal of Psychology, 76, Narens, L. (1996) A Theory of Ratio Magnitude Estimation. Journal of Mathematical Psychology, 40, Narens, L. (2002) The Irony of Measurement by Subjective Estimations. Journal of Mathematical Psychology, 46, Peiner, M. (1999) Experimente zur direkten Skalierbarkeit von gesehenen Helligkeiten [Experiments on the direct scalability of perceived brightness]. Unpublished master's thesis, Universitat Regensburg. Shepard, R. N. (1978). On the Status of \Direct" Psychological Measurement. In C. W. Savage (Ed.), Minnesota Studies in the Philosophy of 17 Science (Vol. 9, pp ). Minneapolis: University of Minnesota Press. Shepard, R. N. (1981). Psychological Relations and Psychophysical Scales: On the Status of \Direct" Psychological Measurement. Journal of Math- ematical Psychology, 24, Stevens, S. S. (1936). A Scale for the Measurement of a Psychological Magnitude: Loudness. Psychological Review, 43, Stevens, S. S., & Guirao, M. (1963). Subjective Scaling of Length and Area and the Matching of Length to Loudness and Brightness. Journal of Experimental Psychology, 66, Teghtsoonian, M. (1965). The Judgment of Size. American Journal of Psycholoy, 78, Zimmer, K. (in press). Examining the Validity of Numerical Ratios in Loudness Fractionation. Perception and Psychophysics. Anderson, N. H. (1970). Functional Measurement and Psychophysical Judgment. Psychological Review, 77, Anderson, N. H. (1976). Integration Theory, Functional Measurement and the Psychophysical Law. In H. G. Geissler, & Y. M. Zabrodin (Eds.), Advances in Psychophysics. Berlin: VEB Deutscher Verlag der Wissenschaften. Ellermeier, W., & Faulhammer, G. (2000). Empirical Evaluation of Axioms Fundamental to Stevens's Ratio-Scaling Approach: I. Loudness Production. Perception & Psychophysics, 62, Graham, C. H. (1958). Sensation and Perception in an Objective Psychology. Psychological Review, 65, Krider, R. E, Raghubir, P., & Krishna, A. (2001) "Pizzas: or Square? Psychophysical Biases in Area Comparisons," Marketing Science, 20(4), Fall, Luce, R. D. (2002). A Psychophysical Theory of Intensity Proportions, Joint Presentations, and Matches. Psychological Review, 109, Luce, R. D. (2004). Symmetric and Asymmetric Matching of Joint Presentations. Psychological Review, 111, McKenna, F. P. (1985). Another Look at the New Psychophysics. British Journal of Psychology, 76, References

40 Narens, L. (1996) A Theory of Ratio Magnitude Estimation. Journal of Mathematical Psychology, 40, Narens, L. (2002) The Irony of Measurement by Subjective Estimations. Journal of Mathematical Psychology, 46, Orth, B. (1982). Zur Bestimmung der Skalenqualität bei,direkten´ Skalierungsverfahren. Zeitschrift für experimentelle und angewandte Psychologie, Band XXIX, Heft 1, S Peißner, M. (1999) Experimente zur direkten Skalierbarkeit von gesehenen Helligkeiten [Experiments on the direct scalability of perceived brightness]. Unpublished master's thesis, Universität Regensburg. Shepard, R. N. (1978). On the Status of Direct Psychological Measurement. In C. W. Savage (Ed.), Minnesota Studies in the Philosophy of Science (Vol. 9, pp ). Minneapolis: University of Minnesota Press. Shepard, R. N. (1981). Psychological Relations and Psychophysical Scales: On the Status of Direct" Psychological Measurement. Journal of Math- ematical Psychology, 24, Stevens, S. S. (1936). A Scale for the Measurement of a Psychological Magnitude: Loudness. Psychological Review, 43,

41 Stevens, S. S., & Guirao, M. (1963). Subjective Scaling of Length and Area and the Matching of Length to Loudness and Brightness. Journal of Experimental Psychology, 66, Teghtsoonian, M. (1965). The Judgment of Size. American Journal of Psycholoy, 78, Zimmer, K. (in press). Examining the Validity of Numerical Ratios in Loudness Fractionation. Perception & Psychophysics. Allgemein zur Methodik Huber, O. (2000) Das psychologische Experiment: Eine Einführung. Hans Huber Verlag, Bern. Bortz, Lienert, Boehnke: (2000) Verteilungsfreie Methoden in der Biostatistik. Springer Verlag, Berlin. Vorberg, D. & Blankenberger, S. (1999). Die Auswahl statistischer Tests und Maße. Psychologische Rundschau, 50,

42 Ergebnistabelle. Ein |z(U)| Wert > 1.96 bedeutet, daß die beiden Flächen die lt. Axiom gleich sein sollten, verschieden groß sind. In diesen Fällen wurde das Axiom verletzt (fettgedruckt). Das Vorzeichen gibt die Richtung der Verletzung an. Ein negatives Vorzeichen bedeutet, daß die q p´ Schätzung kleiner ist als die r´ Schätzung.

43 Es geht in dieser Untersuchung um das Schätzen von Flächen. Du wirst an der Wand eine Fläche sehen. Darüber steht eine Aufforderung, zb. 2. Das bedeutet, daß es Deine Aufgabe ist, eine 2 mal so große Fläche herzustellen. Es erscheint eine weitere Fläche, dessen Größe Du mit der Pfeil aufwärts () - Taste oder mit der Pfeil abwärts () - Taste entsprechend verändern kannst. Stelle also einfach eine Fläche her, welche Dir entsprechend der Aufforderung zb. 2 mal, 3 mal,... so groß erscheint. Wenn Du denkst, die von Dir hergestellte Fläche ist entsprechend der Aufforderung 2 mal, 3 mal,... so groß, dann drücke bitte die Return-Taste. Wenn Du denkst, die Leinwand ist zu klein für Deine Schätzung, dann drücke einfach die Esc-Taste und die nächste Aufgabe erscheint. Es gibt keinen Zeitdruck. Trotzdem solltest du nicht allzu lange an einer Aufgabe herumgrübeln. Die heutige Sitzung besteht aus ca. 200 Schätzungen. Nach 20 min gibt´s eine kurze Pause. Du kannst auch jederzeit selber mal durchatmen, die Augen fest schließen,.... Bitte versuche, dich so gut wie möglich zu konzentrieren. Instruktion