JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Ratio Comparisons of Color Differences

Y. SUGIYAMA* AND HILTON WRIGHT

Division of Applied Physics, National Research Council, Ottawa, Canada

(Received 15 May 1963)

A group of 14 observers made ratio comparisons among a number of color differences consisting of Munsellcolor samples. The same color differences had been judged previously in another experiment by the methodof paired comparisons. The results of the two experiments were found to be equivalent, indicating that bothmethods of judgment-the ratio method and the paired comparison method-can be used with good relia-bility for the scaling of color differences.

INTRODUCTION

TWO of the most common methods of judgment inpsychometrics are the method of paired compari-

sons and the method of ratio comparisons.' Both of thesemethods have been used in experiments dealing withthe scaling of color differences.2' 3 If the color differencesto be tested do not differ too widely from each other insize and the main object of the experiment is to studythe comparative sizes of these differences, then themethod of paired comparisons can be employed effec-tively. If, however, absolute differences between thepairs must be known, the ratio method is applied. Inthe method of paired comparisons, observers areusually required to judge whether a particular colordifference is greater than or less than some other colordifference, and in certain cases a rating scale may beused. The method of ratio comparisons makes con-siderably greater demands upon human observationalpowers since the use of this method requires the esti-mation of the ratio of one color difference to another.

Possibly, for a given experimental situation it wouldbe expected that the use of either method of judgmentwould provide the same scale values when the observa-tional data are analyzed. In general, however, this is notthe case since it has been found that ratio scales andinterval (or rating) scales each measure somethingdifferent. 4 If the attributes of a set of stimuli to bemeasured are qualitative rather than quantitative,'then the two methods may give results which arelinearly related. Galifret 6 has indicated that certaintypes of color differences, in particular saturation differ-ences, may lie on a metathetic sensory continuum. This

* Postdoctorate Fellow from Osaka Industrial Research Insti-tute, Ikeda Branch, Japan.

I J. P. Guilford, Psychomnetric Methods (McGraw-Hill, BookCompany, Inc., New York, 1954), 2nd ed.

2 y. Sugiyama and Hilton Wright, J. Opt. Soc. Am. 53, 1214(1963).

3 S. M. Newhall, Am. J. Psychol. 52, 394 (1939).4 S. S. Stevens, "Ratio Scales, Partition Scales and Confusion

Scales," in Psychological Scaling Theory and Applications, editedby H. Gulliksen and S. Messick (John Wiley & Sons, Inc., NewYork, 1960).

5 In psychology, qualitative attributes are sometimes referredto as "metathetic attributes," while quantitative attributes arereferred to as "prothetic attributes." A more complete descriptionof this terminology is given in Ref. 4.

6 Y. Galifret, Annie Psychol. 59, 35 (1959).

suggests, therefore, that for saturation differences alinear relationship would be expected to hold betweenthe scale values obtained by both the ratio and paired-comparison methods. The purpose of this paper is tocompare the results of two separate experiments; thefirst, in which pairs of colors (saturation differences)had been judged by the paired-comparison method,and the second in which the same color pairs werejudged by the ratio-comparison method. The firstexperiment is discussed in detail in a previous paper.2

The second experiment is discussed below.

RATIO COMPARISON EXPERIMENT

Fourteen observers, including 11 who participated inthe paired-comparison experiment, made ratio compari-sons among six color differences consisting of pairs ofMunsell colors. The experimental conditions wereidentical to those in the paired-comparison experiment,the details of which may be found in Ref. 2. The in-structions issued for judging the color differences by theratio method were as follows:

"The purpose of this experiment is to estimate thetotal color difference between pairs of colors. Determinewhich pair of colors gives the larger color difference,that is, the pair on the right or the pair on the left.State your answer. Estimate the ratio of the largercolor difference to the smaller color difference. In thecase where the color differences of both pairs are equalthe answer is one."

A few individuals used ratios ranging from 1.0 to6.0, but the majority of observers used ratios of 1.0,1.5, 2.0, 2.5, and 3.0.

Judgments were made on all possible pair combinationsfor all four modes of presentation (as shown in Fig. 1of Ref. 2). A complete trial consisted of 60 judgments.One practice trial was made first by each observer. Theresults of the first trial were recorded but were not em-ployed in the final analysis for the group. Half of thejudgments for a particular trial were made at one sittingand the usual time lapse between sittings was aboutl-j days. Each observer made three repeat runs of theexperiment, the average time between runs being aboutone week.

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JANUARY 1964VOLUME 54, NUMBER I

Y. SUGIYAMA AND HILTON WRIGHT

ANALYSIS AND RESULTS

The object of the analysis is to derive scale values,4i say,7 characterizing the absolute color differences ofthe color pairs. Assume that there are in pairs, andjudgments are made on all pair combinations includingthe reverse order of every combination. When the pairsare arranged in the order (i,j) such that Pair i is on theobserver's left and Pair j is on his right, the judgedratio, Jij, of the color difference of Pair i to the colordifference of Pair j is given by

RjŽ~:= tag/ij. (1)

One of the more important items of interest will be totest the relation stated by Eq. (1).

When the pairs are arranged in the reverse order(j,i) such that Pair j is on the observer's left and Pair iis on his right, the judged ratio, Rji(or 1/1ij), of thecolor difference of Pair j to the color difference of Pair iis given by

fP1 i= 1/~J~= 1li/. (2)

The geometric mean of the judgments, Rij, made onPairs i and j for the two orders of presentation (ij)and (j,i) is given by

(3)

GROUP I 1i

GROUP II i

G o

GROUP1 Ai

CASE C DATA0 A0O a

I0.0.5 1.0

a.I1.5

0 DA a A X

-1.5 -1.0 -0.5 0 0.5 1.0SCALE VALUES

13 i-6, *i5, Ai-4, A I 33 01 .- 2, X ill

-DIRECTION OF LARGER COLOR DIFFERENCE

FIG. 1. Scale values, Is, for Case C data for the group computedby the ratio method. Scale values, &, are computed by the Scheff 6method. The ii represent absolute color differences while the &jrepresent comparative color differences.

This condition means that the product of all the scalevalues 4i is unity. The scale of 4i depends upon the con-stant, zero, in Eq. (7), and changing the constantchanges the scale but not the origin. The solution toEq. (6) is found to be

1 mlogti=- _ logRij,

im i_=1or

1 mli=antilog - E logRij .

in j=l -

(8)

(9)

Normally it would be expected that the ratios Pi andl/RII would be equal, but in practice real differencesmay occur signifying an order effect. The order effectAi1 is given by

A\i j= M~sj~il'- (4)

Generally ,Xij will be nearly the same for all pair combi-nations, and therefore Aij can be represented by anaverage order effect A, which is given by

I' Aij (5)i<j

where

t hiji<j

represents the product of all Aij with i<j, and Al= 1m1 ( 1t- 1)/ 2.

According to Torgerson5 a least squares estimate ofthe scale values 4i may be obtained by minimizing

fll

Q= I_' (Iogli-logij- logf?,j)2,i<j

with the condition

E2 logi= 0.i=1

(6)

(7)

7 A circumflex over a symbol (e.g., it) refers to a sample esLi-mate, and a symbol without the circumflex (e.g., Is) refers to thepopulation value.

8 W. S. Torgerson, Theory and illetlhods of Scaling (John Wiley& Sons, Inc., New York, 1958).

The use of logarithms and antilogarithms in theleast squares fitting technique for finding the scalevalues 4i is not mathematically rigorous, since thequantity that is really desired to minimize is not Q but

mn ti

i<i 1tj

The justification for employing such a computationalprocedure, apart from convenience, apparently de-pends upon two conditions. First, that no pairs withzero color difference (i.e., two identical colors makingup a pair) are included amongst the set of stimuli and,second, that all color differences must be of the sameorder of magnitude. Failure of the first condition couldlead to infinitely large values of 1s, while failure of thesecond condition would result in a tendency to accordtoo much weight to the color pairs having the largercolor differences. The second difficulty could be over-come by introducing into Eq. (6) some function which

TABLE I. Comparison of scale values obtained in two experiments.

Paired comparison Ratio comparisonColor pair experiment experiment

i 0.347 (ai +2.98) ii

1 1.35 1.442 1.08 1.013 0.98 0.924 1.27 1.235 0.58 0.636 0.96 0.98

76 Vol. 54

Rij= E-Rul-RA".

J RATIO COMPARISONS OF

would modify the numerical contribution made byeach pair in the fitting procedure and give each colordifference an appropriate amount of weight in thesolution given by Eq. (9). Such a refinement, however,was not necessary in the case of the present experiment.

Figure 1 shows the scale values 4i for Case C data2

computed according to Eq. (9) for two groups ofobservers, Group I referring to the 14 observers andGroup II referring to the original 11 observers. The4i are almost the same for the two groups. The scalevalues e, for Case C data for Group II, derived by theScheff6 method in the paired-comparison experiment,are also shown in Fig. 1 (the signs of the as were changedin order to conform with the definitions in the presentexperiment). It can be noted that the two sets ofvalues, 4i and di, are plotted on different scales, thereason being that the ci represent comparative color

TABLE II. Comparison of the actual judged ratios Rli with thecorresponding calculated ratios li/tl (ratio comparisons), and(&i+,1)/(&j+fl) (paired comparisons) for Case C data for GroupII.

ti &!i +Pair combination Ria Ii &j+n

(1,2) 1.42 1.42 1.25(1,3) 1.50 1.56 1.38(1,4) 1.22 1.17 1.06(1,5) 2.26 2.28 2.33(1,6) 1.52 1.47 1.41

(2,3) 1.14 1.10 1.10(2,4) 0.82 0.82 0.85(2,5) 1.48 1.60 1.86(2,6) 1.10 1.03 1.13

(3,4) 0.75 0.75 0.77(3,5) 1.43 1.46 1.68(3,6) 0.98 0.94 1.02

(4,5) 1.92 1.95 2.19(4,6) 1.16 1.25 1.32

(5,6) 0.58 0.64 0.60

differences and the 4i represent absolute color differ-ences. Theoretically, if the ti and as are equivalent, thenthe ti should be related to the ai by a linear equation ofthe form

i = 4, (10)

where r is a constant of proportionality and 7 is anadditive constant. A direct comparison of the 4i withthe di was made by computing by a least squares tech-nique the constants v and 7 for Case C data for GroupII. The constants q and v for the present experimentwere 2.98 and 0.347, respectively. A comparison of thescale values t(adi+7) and 4i is given in Table I. Ninety-five percent confidence intervals for each P(ai+77)and each 4i were established by means of Scheff6's Stest.9 The intervals for P(di+-q) were z40.05 and for

H. Scheff6, The Analysis of Variance (John Wiley & Sons,inc., New York, 1958).

OBS. 3CASE ACASE B

ti

1i0.5

CASE A a, .CASE B d, * A

I I

-L.O -0.5OBS. 8CASE A lieCASEB j 0i

. I .

0.5

CASE A iCASE B A, e

-1.0

A A 00AA 00

1.0

A o 003 A 0

0 O.-

W

xX

1.5 2.0

l

t.0 1.5

AO

1.003 AOA

0_

x

I . 2 . . 0

15 2.O

1

I- 2?0 3.0

SCALE VALUES

FIG. 2. Scale values 1i and &c for Case A and Case B data of twotypical observers.

4 were 4 6% of 4i. The two sets of scale values listedabove are, therefore, not significantly different.

It is of interest to compare the actual judged ratiosRij with the corresponding estimated ratios /tiij forthe different pair combinations. The values Rij andiillj for Case C data for Group II are listed in thesecond and third columns of Table II, respectively.The fourth column shows the calculated ratios(di+,q)/1(%+-q) which were derived from the resultsof the first experiment. The 4/ij are in very good agree-ment with the Rij, which verifies the relation stated inEq. (1). The (dci+,)/(1%j+?) may also be regarded asgiving a good representation of the Rij.

The average order effect A calculated according toEq. (5) for Groups I and II is found to be 1.02 forCase A, and 1.12 for Case B. This means that, on theaverage, the absolute color difference of a pair placedon the observer's left appears to be 2%o higher for CaseA and 12% higher for Case B than when the same pairoccurs on the observer's right. It is difficult to make anexact comparison of the order effect a of the paired-comparison experiment with the order effect A of theratio-comparison experiment; however, an approximateformula relating a with A is

For the group data of the paired-comparison experi-ment S was found to be +0.10 for Case A and +0.41 forCase B. The corresponding values of A, calculatedaccording to Eq. (11) with 7= 2 .9 8 , are 1.03 for Case Aand 1.14 for Case B, which compare almost perfectlywith those values obtained in the ratio-comparisonexperiment.

GROUP I

CASE A

CASE B

CASE C

0

.

0.5A

SCALE VALUES, 2t

AOO A

ADO A

ADO A

1.0

xxx

1.5

FIG. 3. Scale values 1i for Cases A, B, and C data for Group I.

. .

77January 1964 COLOR DIFFERENCES

I

Y. SUGIYAMA AND HILTON WRIGHT

Figure 2 shows the values 4i and dc for Case A andCase B judgments (see Ref. 2) of two typical observers.It is evident that the scaling is not influenced by eithermode of orientation, A or B. A similar statement appliesto the Group I results shown in Fig. 3. The scale values4i for Case C data for Group I agree fairly closely withthose found for Case A and Case B data. These conclu-

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

sions check with the results obtained in the pairedcomparison experiment.

ACKNOWLEDGMENTS

The authors express their thanks to Dr. GilnterWyszecki and Dr. G. L. Howett for discussions whichresulted in clarifying many points of the manuscript.

VOLUME 54, NUMBER 1 JANUARY 1964

Phase Shift of Sinusoidally Alternating Colored Stimuli

P. L. WALRAVEN AND H. J. LEEBEEK

Institute for Perception RVO-TNO, Soesterberg, The Netherlands(Received 21 March 1963)

In order to avoid luminance flicker at equal luminance of two alternating colored stimuli de Lange foundthat a phase shift of the stimuli with respect to each other has to be introduced.

This compensation for the phase shift occurring in the retina-cortex system has been measured for a largenumber of wavelength combinations at a retinal illuminance of 2 trolands and a 20 field. The phase anglesare approximately additive.

An analysis is given in terms of more basic phase shifts between the red and green systems accordingto the Young-Helmholtz theory.

A fixed phase-shift-vs-frequency relationship between these systems explains the phase shifts measured,although second-order nonlinear effects cause small distortions of this simple model.

INTRODUCTION

ALTERNATING colored stimuli appear at low fre-quencies as a continuous change in color, e.g. from

red to green and the reverse. As frequency increases, theeye cannot follow these rapid changes completely, andbeyond a certain frequency these color alternationsvanish. The color flicker fusion frequency is then passed,although the field in the mixture color may still flickerin luminance. The luminance flicker can be minimizedor sometimes completely eliminated by a proper choiceof the beam fluxes, and it is assumed that the luminancesof the stimuliare then equal. The luminosity curve isusually determined in this way. We refer to minimizingthe luminance flicker, because de Lange",2 has found thatsometimes luminance flicker cannot be completelyeliminated by only adjustment of the beam fluxes.This residual flicker could, however, be eliminated bychanging the phase angle between the two stimuli.In this way one apparently compensates for a phaseshift between the responses of these stimuli somewherein the retina-cortex system.

De Lange' indicated that for sinusoidal stimuli sucha phase shift may be one of the sources which cause adifference between the determination of the luminositycurve by the flicker method and by the step-by-step

I H. de Lange Dzn, "Attenuation Characteristics and PhaseShift Characteristics of the Human Fovea-Cortex Systems inRelation to Flicker Fusion Phenomena," thesis, Technical Uni-versity of Delft, 1957.

'H. de Lange Dzn, J. Opt. Soc. Am. 48, 784-789 (1958).

method. Walraven, Leebeek, and Bouman' suggestedthat the differences found in the relation between color-fusion frequency and color difference along differentlines in the color diagram possibly are also due to thesephase shifts. Moreover, they may be related to phe-nomena like Benham's top.4 It seems worthwhile, there-fore, to extend the investigations by de Lange and tomeasure some further characteristics of this phase shift.

APPARATUS AND PROCEDURE

Experimental setup and procedure are about thesame as those used by de Lange.' 2 Both sides of thefilament of a band lamp (Fig. 1) are used as two lightsources, the one for the light path via Mirror Ml, theother for the light path via Mirror M2. InterferenceFilters IF, and IF2 are inserted in the light paths. Thefluxes of the two beams can be varied with the neutralwedges W, and W2 and by variation of the aperturesD, and D2 of the lenses L, and L2. The two light beamsare polarized by the Polaroids P, and P2. In this waythe light source is projected twofold on the frosted glassG via the rotating Polaroid P3. The combination of twoPolaroids in one light path, one fixed and the otherrotating, provides a sinusoidal flux variation of thelight beam'. Placing the polarization axes of the Polar-oids P, and P2 mutually perpendicularly, the two sinewaves have a phase difference of 1800. This means that

I P. L. Walraven, H. J. Leebeek, and M. A. Bouman, ActaElectronica 2, 50-55 (1958).

4 C. E. Benham, Nature 51, 113-114 (1894).

78 Vol. 54