Vol. 550. J. SOVERS AND L. J. BODI

can be matched by IG. For IG/IB= 1, and a green

primary with a fairly narrow emission band, Fig. 12shows that -qG/nBn0.8, which makes flG slightly smallerthan the present green efficiency. On the other hand,

IG/IR= 1 gives qn/-qR= 1.5 which also implies a smallerefficiency than obtained at present.

SUMMARY

Spectral energy distributions have been representedby functions which are Gaussian in frequency and whichdepend upon the peak wavelength X0, and the band-width at half-peak v. Calculations have related (xy),(uv), and mean luminosities to the parameters (Xo,o).Analysis of these results permits evaluation of different

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

(Xo,o-) combinations as primaries for color reproduction

in cathode-ray tubes. For a primary characterized by aparticular (Xo,o) pair, we may estimate: (a) the changein color area on the CIE 1960 diagram, (b) the change inluminosity afforded by the new primary, and (c) theradiance ratios required to reproduce an arbitrary colorpoint. The latter have been calculated for 9300'Kwhite and establish the requisite relations between elec-tron-gun currents and the efficiencies of conversion ofelectrical to radiant energy.

ACKNOWLEDGMENT

We would like to thank Dr. Alberthelpful comments on the manuscript.

VOLUME 55, NUMBER 12

K. Levine for

DECEMBER 1965

Precision of Color Differences Derived from a MultidimensionalScaling Experiment

HILTON WRIGHT

Division of AI pplied Physics, National Research Council, Ottawa, Canada

(Received 4 February 1965)

A multidimensional ratio-scaling method was used to analyze one observer's color difference judgments

made on two sets of colored tiles of equal luminous reflectance. The precision of the observed color differ-

ences was found to be approximately i30%. Taking this precision into account, the analysis indicated

that all colors could be represented by points in a two-dimensional Euclidean space in which distances

between two points were proportional to observed color differences independent of the location of the points.

A method involving relatively simple computations is used to derive the perceptual space for a large group

of colors by dividing the group into several subgroups and then overlapping the scaling solutions obtained

for each subgroup.

INTRODUCTION

THE statistical analysis of color-difference judg-ments by multidimensional scaling methods pro-

vide a useful means for investigating certain propertiesof the space of color perception. Several authors'- 8 have

applied these methods to develop a uniform color space

in which distances between pairs of points representobserved color differences.

The study of the precision of observed color differ-

ences, characteristic roots, and color coordinates forrepeated runs of a color-difference experiment has not

been explored. This paper describes an experiment per-

formed by one observer (H.W.) who judged ratios ofcolor differences provided by two sets of colored tiles.

1 W. S. Torgerson, Theory and 1etlhods of Scaling (John Wiley& Sons, Inc., New York, 1958).

2 Carl E. Helm, J. Opt. Soc. Am. 54, 256 (1964).R. Shepard, Psychometrika 27, 125 (1962).

4R. Shepard, Psychometrika 27, 219 (1962).T. Indow and T. Uchizono, J. Exp. Psychol. 59, 321 (1960).

6 T. Indow and K. Kanazawa, J. Exp. Psychol. 59, 330 (1960).T. Indow and T. Shioso, Japan Psychol. Res. 3, 45 (1956).

8 T. Indow, Acta Chromatica 1, 60 (1963).

The results are used to illustrate the observer's pre-cision of judgment in terms of the multidimensionalscaling variables.

EXPERIMENT

The observer judged ratios of color differences byarranging the colored tiles in triads as shown in Fig. 1.There were no separations between the members (ij,k)of a triad. In making a judgment on a particular triad,the observer first decided which color difference-dikdenoting the color difference formed by pair (i,k), ordJk denoting the color difference formed by the pair(j,k)-was the larger. Then the observer judged theratio of the larger color difference to the smaller. Alltriads of colors were presented in random sequence.

Two sets of colored tiles9 were used in the experiment.

The first set A consisted of seven tiles. The second set B

consisted of nineteen tiles. The tiles in each set varied

9 The samples were taken from special sets of colored tilesand colored papers previously used in experiments by the OSACommittee on Uniform Color Scales.

1650

December1965 MULTIDIMENSIONAL SCALING OF COLOR DIFFERENCES 1651

TABLE I. Specification of colored tiles used in the experiment.

CIE (x,y,Y) coordinatesSet Tile number i X y Y(%)

1 0.321 0.334 30.22 0.356 0.307 32.73 0.384 0.382 32.8

(A) 4 0.355 0.418 33.05 0.288 0.370 33.26 0.257 0.302 32.57 0.287 0.274 32.9

8 0.352 0.401 32.99 0.383 0.425 32.1

10 0.414 0.430 30.811 0.356 0.375 31.312 0.388 0.387 31.513 0.419 0.403 31.414 0.452 0.420 30.315 0.334 0.331 30.516 0.360 0.347 31.7

(B) 17 0.389 0.359 31.318 0.424 0.377 32.119 0.460 0.389 31.720 0.336 0.307 31.221 0.365 0.324 31.122 0.395 0.334 30.623 0.431 0.349 31.224 0.314 0.270 29.925 0.340 0.284 29.726 0.370 0.297 30.5

in hue and saturation but all had about the sameluminous reflectance. The CIE (x,y, Y) coordinates ofthe two sets of tiles computed with respect to the actualsource used to illuminate the tiles are given in Table I.

All of the tiles were hexagonal in shape. Each of thetiles had an outside diameter of 8 in. and subtended anangle of approximately 30 at the eye position. Thesurround was black.

The tiles were viewed perpendicularly in a booth withdiffuse artificial daylight providing an illuminance ofabout 650 lm/m2. The chromaticity coordinates of thesource were x=0.318 and y=0.333.

The experiment was run ten times on the set A tiles.A single experimental run required making 105 judg-ments, thus totalling 1050 judgments in all. Ratiosranged from one to four. The judgments were per-formed over a period of two weeks.

The procedure of presentation used on the set B tileswas slightly different. This set was divided into sevengroups. Each group consisted of seven tiles, some ofwhich occurred also in other groups (see Table II). Asingle experimental run required 105 judgments for eachof the seven groups, that is, a total of 735 judgments for

k

FIG. 1. Presentation of coloredtiles in triads (ij,k).

TABLE II. Grouping of set-B tiles.

Group Tile

1 12 13 16 17 18 21 222 9 10 12 13 14 17 183 13 14 17 18 19 22 234 17 18 21 22 23 25 265 16 17 20 21 22 24 256 11 12 15 16 17 20 217 8 9 11 12 13 16 17

the entire set B. The experiment was run four times.Ratios ranged from one to three. The judgments wereperformed over a period of six weeks.

ANALYSIS

The most recent and advanced methods of analyzingdata by multidimensional scaling methods are thosedeveloped by Shepard3' 4 and Kruskal.10"1' The analysisof observational data by these so-called nonmetric tech-niques requires knowing the similarities or proximitymeasures between pairs of colors. The earlier, metricmethods such as the complete method of triads developedby Torgerson,l require knowing the actual interpointdistances between all pairs of colors. The mathematicalassumptions of the metric methods may be more difficultto satisfy experimentally; however, the determinationof accurate similarity measures required for the non-metric methods is usually as involved as calculatinginterpoint distances between pairs of colors. Further-more, there is evidence which shows that analysis of thesame observational data by either scheme yields essen-tially the same results.8' 2 In the present experiment,all of the judgments were analyzed by Torgerson'scomplete method of triads.

The main objective of the analysis is to assign co-ordinates to each color so that the judged ratios of colordifferences between pairs of colors are equal to corre-sponding calculated ratios of distances between pairsof points in a Euclidean space of a minimum number ofdimensions.

The first step of the analysis is to compute from thejudged ratios so-called "interpoint distances" (equiva-lent to color differences) dij, between all pairs of points(colors) i, j. From the interpoint distances a matrix isconstructed whose elements are scalar products of theinterpoint distances referred to an origin at the centroidof all the points. This matrix is subjected to character-istic-vector analysis to obtain the projections of thepoints on a set of orthogonal axes which define thespace."3 The total number of characteristic roots andcharacteristic vectors computed is equal to the number

10 J. B. Kruskal, Psychometrika 29, 1 (1964).1 J. B. Kruskal, Psychometrika 29, 115 (1964).12 L. R. Mutalipassi and R. M. Hanson, Psychol. Rept. 15,

899 (1964).13 H. H.rHarman,7Modern Factor Analysis (University of

Chicago Pr'ess, Chicago, Illinois, 1960).

Vol.55HILTON WRIGHT1652

of points, but not all vectors are necessarily significant.In a color experiment not more than three vectors wouldbe expected to be significant, since color evidently is

perceptually three dimensional and may, for example,be expressed in terms of hue, saturation, and lightness.In the special case where the colors involved in the

experiment exhibit constant lightness, only two signifi-cant characteristic vectors are to be expected. Thepresent experiment closely relates to this special casesince the colored samples have about the same luminous

reflectance Y which correlates approximately tolightness.

RESULTS

(i) Set A Tiles

The interpoint distances dij computed from thejudged ratios of pairs of color differences between all

pairs of colors (ij) of the set A tiles for each experi-

mental run and for the (geometric) average of the ten

runs are listed in Table III. This table gives an indica-

tion of how precisely color differences for the same pair

can be judged by an observer. Discrepancies of 30%are quite common.

Proceeding with the analysis of the data, we com-

puted characteristic roots and characteristic vectors forthe various scalar product matrices constructed fromthe dij's. Table IV gives the characteristic roots.

X;, (a= 1, 2, ... 7) corresponding to the ten experimentalruns and the average. Each of the roots has been ordered

according to size, the largest root is put at the top

(a= 1), the smallest at the bottom (a= 7). The relative

.8

0.

-1.2 -. 8 -.4

fX2

03

.4 .8 [2

-. 4

-. 8

FiG. 2. Two-dimensional space with points representing thecolors of set-A tiles for the average data. XI and X2 are the prinic-pal axes of this space corresponding to the two largest charac-teristic roots Xi and X2 given in Table IV.

sizes of the roots in each run are of importance. A large

positive root means a significant characteristic vectoror principal dimension, while a small positive root, zeroroot, or negative root means an insignificant character-istic vector and thus negligible dimension.

In all the cases given in Table IV the first two rootsare considered significant. The value of the third rootdrops sharply and its importance will be discussed later.The fact that two large positive characteristic rootshave been obtained is in agreement with the expectationthat the colors used in the experiment vary in two attri-butes (hue and saturation) only and can thus be repre-sented as points in a two dimensional space. This space

TABLE III. Interpoint distances (color differences) dij of set-A tiles obtained for each experimental run and for the average.

Pair Experimental Run

(Yi) 1 2 3 4 5 6 7 8 9 10 Average

0.92 0.89 -(J9 -d5 U. "I0.70 0.64 0.780.51 0.60 0.540.51 0.63 0.810.61 0.71 0.680.94 0.73 0.830.42 0.52 0.49

0.951.262.002.290.81

1.061.411.812.480.73

0.92 0.89 ().'360.68 0.68 0.740.75 0.69 0.760.97 0.84 0.771.13 0.98 1.030.63 0.65 0.67

0.801.151.611.880.67

0.881.171.531.990.72

0.77 0.81 0.66 0.741.27 1.10 1.04 1.151.69 1.58 1.65 1.721.13 1.11 1.07 1.27

0.73 0.78 0.68 0.711.48 1.65 1.26 1.271.13 1.11 1.13 1.05

0.881.091.511.690.83

0.780.760.871.020.70

0.911.151.481.830.80

U.82 u.7 /0.82 0.780.83 0.760.97 0.871.02 1.100.77 0.76

0.841.101.541.540.79

0.72 0.80 0.801.21 1.21 1.181.60 1.51 1.481.26 1.13 1.05

0.741.331.06

0.911.111.641.820.79

0.721.211.450.95

0.79 0.78 0.771.24 1.28 1.250.99 1.08 0.98

V.ou

0.670.710.850.960.60

0.911.221.701.950.75

0.751.181.651.14

0.731.341.07

1.01 1.03 1.16 1.09 1.04 0.95 0.89 0.93 0.90 1.14 0.99

1.29 1.38 1.43 1.28 1.38 1.34 1.33 1.12 1.10 1.18 1.29

0.79 0.79 0.85 0.88 0.88 0.90 0.72 0.85 0.82 0.80 0.82

0.850.660.821.011.000.46

1.011.432.001.870.56

1.071.471.972.280.70

1, 21, 31,41, 51, 61, 7

2, 32,42, 52, 62, 7

3, 43, 53, 63, 7

4, 54, 64,7

5, 65,7

6,7

0.67 0.891.04 1.371.76 2.091.02 1.37

0.63 0.671.32 1.351.09 1.14

3

December1965 MULTIDIMENSIONAL SCALING OF COLOR DIFFERENCES

TABLE IV. Characteristic roots Xa for ten experimental runs and for the average of set A tiles.

Experimental Run

a 1 2 3 4 5 6 7 8 9 10 Average

1 2.66 3.53 3.22 3.38 2.25 1.75 2.01 2.04 2.45 2.03 2.432 1.65 1.76 1.53 1.39 1.38 1.05 1.28 0.930 1.29 1.06 1.303 0.157 0.149 0.278 0.420 0.174 0.251 0.257 0.374 0.247 0.349 0.2194 0.021 0.050 0.108 0.317 0.097 0.213 0.177 0.231 0.182 0.225 0.1195 0.000 0.000 0.000 0.096 0.030 0.084 0.067 0.128 0.000 0.088 0.0006 -0.262 -0.236 -0.109 -0.048 0.000 0.000 0.000 -0.068 -0.053 0.000 -0.0157 -0.341 -0.757 -0.832 -0.933 -0.300 -0.008 -0.273 -0.220 -0.414 -0.197 -0.335

is defined by the two characteristic vectors correspond-ing to the two largest positive characteristic roots andeach color can be represented by a point in it.

Figure 2 shows a plot of points representing the colorsof the set A tiles for the average observational data.X1 and X2 are the principal axes of the space corre-sponding to the two largest characteristic roots A1 andX2 given in Table IV. Distances between any two points(ij) are expected to be proportional to the correspond-ing color differences dij given in Table III ("average"column). The interpoint distances in Fig. 2 were con-structed on the basis of two characteristic vectors onlyand consequently the plotted, distances do not agreeperfectly with the average observed distances given inTable III.

A plot of the points representing the colors on thethird largest principal axis, corresponding to the thirdlargest characteristic root X3 (average column in TableIV), was also made but is not shown here. The three-dimensional coordinates of the color points did not seemto be systematically related to the two-dimensionalcoordinates, which suggests that a two-dimensional con-figuration of points is appropriate. A detailed compari-son of the reconstituted two-dimensional interpointdistances (shown in Fig. 2) and the corresponding ob-served interpoint distances for the average data is givenin columns two and three of Table V. In general theagreement is good. Only four reconstituted interpointdistances (3, 4; 4, 5; 2, 7; and 6, 7) lie outside the ob-served range, and of these only (3,4) is perhaps seriouslydifferent from its corresponding reconstituted value.The average observed dij for pair (3,4) is 0.75 and itsreconstituted interpoint distance based on the averagedata is 0.48. In experimental run No. 1, however, theobserved dij for pair (3,4) was 0.67 (see column two ofTable III) and its corresponding reconstituted inter-point distance was 0.59, which is not a significant dis-crepancy.1 4 Hence, in view of the precision of the actualobservations, the digs and reconstituted interpointdistances may be said to be linearly related.

The analysis was extended to include the third largestcharacteristic root, thus expanding the space from twoto three dimensions. In this case the reconstituted inter-

4 G. Wyszecki and H. Wright, J. Opt. Soc. Am. 55, 1166 (1965).

point distances are in better agreement with the corre-sponding average observed distances, as may be seen bycomparing the results given in columns two and four ofTable V. However, the over-all improvement is not con-sidered significantly better so as to warrant the use ofthree dimensions. It appears that the third dimensionprimarly accounts for the discrepancies between theaverage observed and reconstituted-two-dimensionalinterpoint distances involving colors Nos. 4 and 7.

To compare the results of the individual runs withthe average, it is generally necessary to translate, rotate,and sometimes reflect the coordinate axes until theseven points of each individual run coincide as closelyas possible with the corresponding average points. Thisis because the analysis does not provide a coordinatesystem which has a unique origin and orientation. Aleast-squares method was used to determine translation

TABLE V. Comparison of observed and reconstituted inter-point distances on the basis of two and three dimensions forthe average data.

Pair(ij)1, 21, 31,41, 51,61, 7

Averageobserved dij(taken fromTable III)

0.860.670.710.850.960.60

2, 32,42, 52, 62, 7

3,43, 53,63, 7

4, 54,64, 7

5, 65, 7

6, 7

0.911.221.701.950.75

0.751.181.651.14

0.731.341.07

0.991.29

0.82

Reconstituted interpointdistances computed for:

Two dimensions Three(as in Fig. 2) dimensions

0.95 0.990.65 0.670.54 0.680.84 0.861.06 1.060.57 0.61

0.831.221.741.930.98

0.481.081.661.12

0.601.321.11

1.001.30

1.03

0.931.231.741.940.98

0.751.141.681.19

0.631.371.13

1.011.30

1.05

1653

1 ILTON \WRIGHT V 55

03

,X-a +_

0

0

0

-12 -. 8 -4 8 12

s co%o no -41

0

0 6 0

FIG. 3. Two-dimensional space with points representing thecolors of set A tiles for ten experimental runs and the averagedata. Points obtained for individual runs are illustrated by opencircles; average points are illustrated by solid circles. X1 and X2are the principal axes of this space, corresponding to the twolargest characteristic roots X1 and X2 given in Table IV.

and rotation of the points in each individual run so asto produce the best mutual fit with the correspondingaverage points. Figure 3 shows a common plot of thepoints obtained for the ten individual runs and for theaverage. X1 and X2 are the principal axes of the spacecorresponding to the two largest characteristic roots X,and N2 given in Table IV. The open circles represent thepoints corresponding to the individual runs, the closedcircles are those for the average. The scatter of thepoints is a direct indication of the observer's precisionof judgment.

A quantitative measure of the agreement between thepoints of the individual runs with the correspondingaverage points may be determined by calculating theproportion of coordinate variance which each individualset of points has in common with the average.1 Theresults for the set A tiles are given below.

Proportion of coordinate variance common to points in eachindividual run with the average points

Run1 2 3 4 5 6 7 8 9 10

0.99 0.99 0.99 0.96 0.99 0.98 0.99 0.99 0.99 0.98

The above values indicate high correlation between thecoordinates of each individual run with the coordinatesof the average, since perfect agreement would yield avalue of unity.

(ii) Set B Tiles

The nineteen colors comprising the set B tiles werenot judged as a single group, but rather in seven groupsof seven colors as indicated in Table II. The tiles were

15 R. Shepard, J. Exp. Psychol. 55, 509 (1958).

divided into the different groups so as to avoid theproblem of judging large ratios of color differences, inwhich case ratios of perhaps 7:1 or even 10:1 wouldhave been encountered. Such large ratios cannot bejudged with satisfactory precision; with the tilesarranged in the particular groups indicated the actualratios did not exceed 3:1.

The use of the tiles in smaller groups also reduces theamount of labor involved in handling the tiles and inrecording the observational data. For example, the totalnumber of triad combinations for nineteen tiles whenordered in all possible ways is 2907. By experimentingwith the tiles in the seven groups, the total number oftriad combinations was reduced to 735.

For analysis of the judgments made on the set B tiles,the characteristic roots were calculated for the averageof the four runs made on each group. The roots areshown in Table VI. In each case two large positive roots

TABLE VI. Characteristic roots NS. obtained for average of fourexperimental runs made on seven groups of set B tiles.

GroupIt 1 2 3 4 5 6 7

1 2.08 2.37 2.44 2.85 2.78 1.97 2.462 1.43 1.51 1.42 1.06 1.09 1.87 1.313 0.191 0.286 0.199 0.220 0.319 0.257 0.2294 0.056 0.075 0.027 0.054 0.053 0.000 0.0305 -0.072 0.000 0.000 0.000 0.000 -0.008 0.0006 -0.174 -0.045 -0.056 -0.063 -0.086 -0.117 -0.0917 -0.882 -0.405 -0.271 -0.207 -0.293 -0.160 -0.166

were found; this is similar to the results of the experi-ment with the set A tiles. The characteristic vectorsassociated with the two large positive roots in each groupwere used to construct a two-dimensional space intowhich the corresponding seven colors could be em-

8+

11%

I 6-+ 09 -

1.2 I

12 /

.4

16 1

-[.6 -,2 -.B -.A

-. 824 025

010

I314

1 8

.4 i'2 X 9.8 t19

-1.2 T 26

SMOL IOu213i4u5IItIlSY M O L | o|01A||+|

FIG. 4. Two-dimensional space with points representing thecolors of set B tiles. Colors that belong to the same group (seeTable II) are represented by the same symbol. Contours enclosethose points representing each color location. Y, and Y2 are theprincipal axes of this space, corresponding to the two largestcharacteristic roots X, and X2 given in Table VI.

1654 Vol. 55

December1965 MULTIDIMENSIONAL SCALING OF COLOR DIFFERENCES

bedded. This leads to seven different spaces which canbe superimposed after properly translating, rotating andreflecting the individual coordinate systems such that allpoints which are common to one or other systems co-incide as closely as possible. As an example of the agree-ment between the coordinates of different groups, theproportion of coordinate variance common to the pointsof group 1 and the points of the other groups is shownbelow. The correlation of points indicated is slightly

Proportion of coordinate variance common to pointsin group 1 with points in other groups

Group2 3 4 5 6 7

0.96 0.99 0.98 0.92 0.93 0.99

poorer than that found for the set A tiles but it is stillconsidered good.

The common space in which the nineteen colors wereembedded is illustrated in Fig. 4. Y, and Y2 are theprincipal axes of the space corresponding to the twolargest characteristic roots Xi and X2 given in Table VI.Some of the colors are represented by more than one

point. This is because these colors were observed in oneor more groups in accordance with Table II. Colors thatbelong to the same group are given by the same symbol.For example, the tile No. 21 was observed in groups 1,4, 5, and 6 and, consequently, for this color four differentpoints (enclosed by a contour) are shown. The pointsdo not coincide exactly, but the discrepancies apparentlyfall within the spread of repeated observations as maybe deduced from Fig. 3.

This result is of particular interest as it shows that amultidimensional-scaling experiment can be carried outwith a large number of colored samples without havingto treat them as a single group. The samples can bedivided into overlapping subgroups and the resultsobtained for each subgroup can be combined to arriveat a space that is representative for all samples. Thenumber of required observations is greatly reduced andthe analysis, which then involves smaller matrices, isless elaborate.

ACKNOWLEDGMENT

The author thanks Dr. G. Wyszecki for his manyvaluable comments on all phases of the experiment.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 55, NUMBER 12 DECEMBER 1965

Category Judgments as Functions of Flash Luminance and Duration*MARK F. LEwist

Columbia University, New York, New York 10027

(Received 18 May 1965)

A method for obtaining suprathreshold, constant-response functions from category judgments was de-veloped and illustrated by obtaining judgments for flashes of varying luminance and duration in the fovea.The contours show agreement with Bloch's law; no Broca-Sulzer effect was obtained. A second experimentindicated that the method is sensitive to context effects. No reliable variation in the critical duration wasfound with changes in luminance level.

THIS research represents a demonstration of aTpsychophysical approach to category judgments.The method may also be used with magnitude estima-tion studies. The technique assumes, as do Graham andRatoosh,l only that the responses made by subjects inestimation or category judgment experiments are sepa-rate operants and should not be treated as numbers.

The problem was to develop a method for analyzingthe data from category judgments or estimation experi-ments, a method which would yield constant response

* This investigation was supported (in part) by a Public HealthService Fellowship (number 1 F1 MH-19, 820-01) from the Na-tional Institute of Mental Health, Public Health Service.

t Now at California State College, Hayward, California 94542.1 C. H. Graham and P. Ratoosh in Psychology: a study of a

Science, edited by S. Koch (McGraw-Hill Book Co., Inc., NewYork, 1962), Vol. IV, pp. 483-514.

functions comparable to those obtained with the methodof single stimuli, where the functions obtained corre-spond to a constant-response probability. In particular,category judgments were obtained for stimuli of varyingluminance and duration. The threshold constant-response function for these variables is well-known:Bloch's law (Lt= C) holds for all stimulus durations lessthan some critical duration, t.; when the duration ofthe test flash exceeds the critical duration, threshold isdetermined solely by luminance. Bloch's law also holdsfor constant effects above threshold, but the well-knownBroca-Sulzer effect is obtained if a comparison, or con-trast-inducing, stimulus is present. Only one study byRaab,2 using Stevens' estimation techniques,3 indicates

2 D. H. Raab, Science 135, 42 (1962).3S. S. Stevens, Am. J. Psychol. 69, 1 (1956).

1655