A ratio scale for brightness perception derived from difference and ratio judgments

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A ratio scale for brightness perception derived from difference

and ratio judgments

Citation for published version (APA):

de Ridder, H. (1992). A ratio scale for brightness perception derived from difference and ratio judgments. (IPO rapport; Vol. 869). Instituut voor Perceptie Onderzoek (IPO).

Document status and date: Published: 26/08/1992

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lnstitute for Perception Research

PO Box 513, 5600MB Eindhoven

Rapport no. 869

HdR/hdr 92/14

26.08.1992

A ratio scale for brightness perception

derived from difference and ratio judgments

H. de Ridder

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A ratio scale for brightness perception derived

from difference and ratio judgments

1

H uib de Ridder

Institute for Perception Research (IPO)

P.O. Box 513, 5600 MB Eindhoven

The Netherlands

Abstract

The often observed nonlinearity between magnitude estimation and category sealing has raised the question whether subjects can judge differences in sensory magnitude as well as ratios of sensory magnitude whenever instructed to do so. Experiments are described showing that subjectscan distinguish between brightness differences and brightness ratios; the rank order of difference judgments differs from the rank order of ratio judgments. Nonmetric analysis of these rankings yields a ratio scale for brightness perception that is (1) a power function of luminanee (power: 0.13), (2) linearly related to direct category ratings, (3) nonlinearly rela~ed to direct magnitude estimations. The latter is probably due to nonlinear number handling by subjects.

1. lntrod netion

1.1 Magnitude estimation vs category sealing

The most direct method for measuring the strength of unidimensional sensations like brightness or loudness is sealing. Several methods are available. Magnitude estimation and numerical category sealing are old rivals. For brightness perception, Stevens' psychophysical relation based on magnitude estimation suggests a power function between perceived brightness B and luminanee L, or

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L ~ Lo.

For a 5°-field, dark background, brightness exponent

f3

equals 0.33 (Stevens, 1975). A background which prevents bias due to adaptation raises exponent

f3

(Stevens and Stevens, 1963; de Ridder and Roufs, 1987). An example of magnitude estimation is shown in Figure 1. The parameters are: 1°, 256 ms, 70 Td background, 4 subjects,

f3

= 0.56.

1_{This report is an extended version of a poster presented at the XXVth International Congress} of Psychology, Brussels (Belgium), August 19-24, 1992 (de Ridder, 1992)

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2 . - - - .

Magnitude estimation slope=0.56

1.5-Q)

E

m

1-0>

0

0.5-0-r---.,---.,---~,~

1·

₂

₃

₄

ret. illum. (log Td)

/ Figure 1/

If category sealing based on a limited amount of numbers is used, the following equation fits satisfactory:

B = a

* (

L - L0 )~ + b.

(2)

Now, exponent (3 equals 0.13 (see Figure 2). This agrees with somerecent literature (Rule and Curtis, 1982).

1.2 Ratio vs difference judgments

Magnitude estimation is considered a ratio estimation technique, implying that judg-ment ratios represent ratios of sensory magnitude. Category sealing, on the other hand, is considered an interval estimation technique, implying that judgment dif-ferences represent di:fdif-ferences in sensory magnitude. If so, then category ratings are linearly related to magnitude estimations. In general, however, a negatively acceler-ated nonlinear relation is observed (Krueger, 1989). This also holds for the present study as can be seen in Figure 3 where the category ratings of Figure 2 have been plotted as a function of the magnitude estimations of Figure 1. ·

The above-mentioned nonlinearity raises the question whether subjectscan judge di:fferences in sensory magnitude as well as ratios of sensory magnitude whenever instructed to do so. Torgerson (1961) argued that subjects perceive one relation only, but that this relation is reported di:fferently depending on instructions. Experimental

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(/) ()

al

0>

0

1.3

- r - - - ,

Category sealing

1.2 _

slope=0.13

"o-'.c

1. 1-

0.9-0.8

1 r

2

=0.992

oO"

/ /Ç>//

"a-'

0-'ó

o-'_.

.,,'""'

/Ó

I I I

2

3

4 ret. illum. (log Td)

I

Figure

2j

support for this notion comes predominantly from e:x:periments in which differences had to be assessed by means of interval sealing techniques and ratios by means of ratio sealing techniques. (Birnbaum, 1982; Poulton, 1989). The results suggest that subjects perceive differences, not ratios (Birnbaum's 'subtractive model').

If subjects perceive only one relation between sensory magnitudes, then the rank orders of difference judgments and ratio judgments should be the same. This is confirmed by nonmetric analysis of experimental results supporting Birnbaum's sub-tractive model. Occasionally, however, clear differences in the rank orders of differ-ence judgments and ratio judgments have been observed: Parker et al. (1975) and Orth (1982) for perceived line length, Ruleet al. (1981) for heaviness and Schneider et al. (1982) for pitch. These experiments have in common that the same sealing technique was used to assess ratios and differences.

Krantzet al. (1971) have shown theoretically that a ratio scale of sensory magni-tude can be derived from the rank orders of difference and ratio judgments if these rank orders are different and if both ratio and difference judgmynts are based on a common sensory scale. Such ratio scale implies that sensory magnitudes are known but for a multiplication factor.

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9

0 Q)

co

(.)

7 en

>

...

0

5 C>

Q) +o-1

co

₃

(.)

0

20

40

60 magnitude estimate

I

Figure

31 1.3 Aim of the present study

• To show that subjects can judge brightness ditierences as well as brightness ratios whenever instructed to do so,

• To establish a ratio scale for brightness perception based on ratio and difference judgments,

• To compare this ratio scale with brightness scales obtained by direct magnitude estimation and direct category sealing.

2. Experiments

2.1 Difference judgments

Under similar conditions as described insection 1.1, the same 4 subjects determined the rank order of 28 pairs of light flashes. Criterion: perceived brightness difference. The method of tetrads ('pairs of pairs') was used. For the brightnesses of the 8 stimuli from which the 28 pairs were formed, nonmetric analysis of the rank order yields projection values PD on a unidimensional scale that is unique up to multiplication and addition, i.e. an interval scale. Consequently,

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Figure 4 shows projection values PD as a function of retina! illuminance expressed in log units. The nonlinear relation indicates that the brightness-luminance relation is not a logarithmic function but a power function.

E

c

aJ

2~---~

1 -

0-

-1-Nonmetric sealing

ranking of _differences

.sg

s·fi

r:,"~

_"_,.(}

9(

/Ó

6---:Ç''

_§'/0

---·

-2~~----~,~---~,---~,~

1

2

3

4 ret. illum. (log Td)

I

Figure

41 2.2 Ratio judgments

0

GS

0

HR

~

LT

(>TH

The same subjects repeated the previous experiment, but now the criterion was perceived brightness ratio. If subjects do perceive brightness ratios, then a non-metric analysis of the resulting rank order should yield projection values PR for the brightnesses of the 8 stimuli on a log-interval scale, or

PR = c*log(B)

+

d.

(4)

Figure 5 shows that these projection values are linearly related to retina! illuminance expressed in log units. Knowing that the brightness-luminance relation is a power function, this implies that subjects are indeed able to judge sensation ratios.

3. Ratio scale for brightness perception

Another way to prove that subjects can judge sensation differences as well as sensa-tion ratiosis by plotting projecsensa-tion values PD as a function of projection values PR.

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E

c

CJ

0)

0

2~---~

1-

0-

-1-Nonmetric sealing ranking of ratios

/.Ar

Ai,-/~"

__ "_ .. g:'

0 §8'

,'~

".B-"

-2-r----~~,---~,---~,~

1

2

3

4 ret. illum. (log Td)

I

Figure

si

0 GS

o

HR

6.

LT

(>TH

If subjects are not able to distinguish between ratios and differences, then the pro-jection values PD should lie on a straight line. Figure 6 shows that this does not

hold for brightness perception.

From eqs (3) and ( 4) it fellows that

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k1

=

1/c and k2

=

-djc

+log(

a).

In Figures 4, 5 and 6, the dashed lines denote the least-squares fit of this equation to the experimental data (r2 ₌_0.995).

A ratio scale for brightness perception can be derived from this equation, because both PD - b and 10('"l'"Pn) are proportional to brightness B (Krantz et al., 1971).

This ratio scale has been plotted as a function of retinal illuminance (Figure 7). The brightness-luminance relation turns out to be a power function with exponent

f3

equal to 0.13.

4. N onmetric sealing vs direct sealing

If category sealing is an unbiased interval estimation technique, then category rat-ings

Be.

should be linearly related to brightness

Bnm

derived from projection

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val-E

c

aJ

Cl

0 en

Q)

::J

co

>

I

Cl

a_

1-

0-

-1-Nonmetrie sealing

,P_"",

, ' ,,'' ,,'0

",o'

,"""','

,"o'

ç/

,6

ç/

, , , ,Ö -2~--~-~,~--~,--~~,--~

-2

-1

0

1 PR-values

I

Figure

61

0.8~---~

Nonmetrie sealing

slope=0.13

0.7-

2

r =0.997

0.6-0.5-

""".6

_""9'/

""o·

/ / d

/6

""a'

Ç) ,0 0.4~---~~---~~~--~~~~

1

2

3

4 ret. illum. (log Td)

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ues PD and PR. Similarly, if magnitude estimation is an unbiased ratio estima-tion technique, then magnitude estimaestima-tions Bme should be proportional to bright-ness Bnm· Figure 8 shows that category ratings are indeed linearly related to Bnm·

Category sealing

2

r

=0.995

ÇJ/' / I

3 9'

()/ , / / ,,,"Ó ' I

4 ,o'

'

,6

Bnm

I

Figure

8j

' /

,90/

I

5 9/

,J....J ' I

6

Magnitude estimations, on the other hand, are not proportional to Bnm since the slope of the straight line that fits the relation between log Bme and log Bnm (Fig-ure 9, dashed line) deviates significantly from one (Fig(Fig-ure 9; solid line ). The ob-served deviation is probably due to nonlinear number handling by subjects (Rule and Curtis, 1982; Wagenaar, 1982; Krueger, 1989).

5. Conclusions

• Subjects can judge brightness ratios as wellas brightness differences whenever instructed to do so

• A ratio scale for brightness perception can be derived from a combination of difference and ratio judgments ·

• Brightness-luminance relation is a power function

• The value of the brightness exponent is substantially less than the value of 0.33 proposed by Stevens and others

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Q)

E

aJ

C>

0

2~---~

1.5

1

0.5 Magnitude estimation

slope=4.49

r

2

=0.97

0-+---r----,---r---r---.,.-l

0.3

0.5

0.7 log

Bnm

j Figure 9/

• Direct category sealing is linearly related to nonmetric sealing

• Direct magnitude estimation is nonlinearly related to nonmetric sealing.

6. References

Birnbaum, M.H. (1982) Controversies in psychological measurement. In: B. We-gener (Ed.), Social attitudes and psychophysical measurement (pp. 401-485). Hillsdale, NJ Lawrence Erlbaum Associates Inc.

Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A. (1971) Foundations of

measure-ment. Volume 1: Additive and polynomial representations. Academie Press,

New York.

Krueger, L.E. (1989) Reconciling Fechner and Stevens: Toward a unified psychophys-icallaw. Behaviaral and Brain Sciences, 12: 251-320.

Orth, B. (1982) A theoretica! and empirica! study of scale properties of magnitude-estimation and category-rating scales. In: B. Wegener (EÇ-. ), Social attitudes

and psychophysical measurement (pp. 351-377). Hillsdale, NJ Lawrence

Erl-baum Associates Inc.

Parker, S., Schneider, B., Kanow, G. (1975) Ratio scale measurement of the per-ceived lengthof lines. Journalof Experimental Psychology: Human Perception

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Poulton, E.C. (1989) Bias in quantifying judgments. Hillsdale, NJ Lawrence Erl-baum Associates Inc.

Ridder, H. de (1992) A ratio scale for brightness perception derived from difference and ratio judgments. International Journalof Psychology, 27 (3/4): 33. Ridder, H. de, Roufs, J.A.J. (1987) Brightness of time-dependent signals: A model

to test the mutual consistency of sealing and matching. Proceedings of 21st

CIE session, Venice, Italy, June 17-25, 1987. CIE publ. no. 71: 46-49. Rule, S.J., Curtis, D.W. (1982) Levelsofsensoryandjudgmentalprocessing:

Strate-gies for the evaluation of a model. In: B. Wegener (Ed.), Social attitudes and

psychophysical measurement (pp. 107-122). Hillsdale, NJ Lawrence Erlbaum Associates Inc.

Rule, S.J., Curtis, D.W., Mullin, C. (1981) Subjective ratios and differences in per-ceived heaviness. Journalof Experimental Psychology: Human Perception and

Performance, 7, 459-466.

Schneider, B., Parker, S., Upenieks, E.G. (1982) The perceptual basis of judgments of pitch differences and pitch ratios. Canadian Journalof Psychology, 36: 4-23. Stevens, J.C., Stevens, S.S. (1963) Brightness function: Effects of adaptation.

Jour-na[ of the Optical Society of America, 53: 375-385.

Stevens, S.S. (1975) Psychophysics: Introduetion to its perceptual, neural, and

so-cial prospects. John Wiley & Sons, New York.

Torgerson, W.S. (1961) Distances and ratiosin psychological sealing. Acta

Psycho-logica, 19: 201-205.

Wagenaar, W.A. (1982) Misperception of exponential growth and the psychological magnitude ofnumbers. In: B. Wegener (Ed.), Social attitudes and

psychophys-ical measurement (pp. 283-301). Hillsdale, NJ Lawrence Erlbaum Associates Inc.