Factors underlying individual differences in the color matches of ...psy2.ucsd.edu/~dmacleod/publications/44WebsterMacLeod1988.pdf · served variables are due to the common influence

1722 J. Opt. Soc. Am. A/Vol. 5, No. 10/October 1988

Factors underlying individual differences in the colormatches of normal observers

Michael A. Webster*

Department of Psychology, University of California, Berkeley, Berkeley, California 94720

Donald I. A. MacLeod

Department of Psychology, University of California, San Diego, La Jolla, California 92093

Received March 13, 1987; accepted May 31, 1988

We have used a factor analysis of the Stiles-Burch [Opt. Acta 6,1 (1959)] 100 field color matches to examine the ba-sis of individual differences in the color matches made by observers with normal color vision. The differences in thematches are primarily due to interobserver variations in the macular-pigment density [with a standard deviation (a)of 0.12 at 460 nm]; the lens-pigment density (a = 0.18 at 400 nm); the spectral position of the long-wavelength-sensitive (a = 50.3 cm-'), medium-wavelength sensitive (vi = 31.9 cm-'), and short-wavelength-sensitive (a = 45.3cm-') photopigments; the covarying densities of the three photopigments (a = 0.045); and the degree of rodintrusion. Variations in the different factors appear to be uncorrelated. Comparable estimates of the sources andrange of interobserver differences in color matching were obtained from a similar analysis of the Stiles-Burch 20color matches [Opt. Acta 2, 168 (1955)].

INTRODUCTION

Observers with normal color vision often make substantiallydifferent color matches. One well-known reason for thesedifferences is that individuals vary widely in the amounts ofthe prereceptoral screening pigments present, such as thelens and the macular pigments. For example, the lens pig-ment absorbs a large fraction of the short-wavelength lightentering the eye but little of the longer wavelengths. Thusan observer with a higher than average concentration of thispigment will require a less than average intensity of a redprimary light when matching a blue test light, simply be-cause the blue light has been selectively more attenuated.By expressing the color matches in a form such as the WDWcoordinate system, the effect of any inert pigment screeningall the receptors is eliminated.' Yet substantial individualdifferences in the matches remain, suggesting that the pho-topigments and/or receptor mechanisms also vary amongobservers.

It is thus clear that the interobserver differences in colormatching are due to a large number of factors. However,exactly what this set of factors is, and how much each con-tributes to the total observed variability, is not definitelyknown. In this study we have examined these questions byparceling out and identifying the sources of variability in aset of normal color matches with the technique of factoranalysis. The analysis not only allows us to determinewhich potential factors influence the matches and the rangeover which they vary but also allows us to examine suchproperties of the factors as the spectral absorption charac-teristics of the pigments involved.

The different steps of our analysis are organized as fol-lows. First, we use standard factor analytic techniques toestimate the underlying patterns of individual variation inthe Stiles-Burch 100 field color matches. 2 We then identify

the variable physiological characteristics to which these ob-served factors correspond and determine how much thesecharacteristics vary across observers. We also examinewhether the variations in the different factors are indepen-dent or correlated. Next, we discuss a theoretically guidedfactor analysis, which was designed to improve our estimatesof the pattern of variability due to each factor, by takingadvantage of some established properties of color matching.Following this, we derive separate estimates of the range ofinterobserver differences in the identified factors, by findingthe amount of variation in each predicted factor that pro-vides the best fit to the observed correlations between thedifferent color matches. Finally, we perform a similar anal-ysis of the Stiles-Burch 20 color-matching data.3 In theaccompanying paper,4 we use individual differences in thecolor matches, and independent data on the changes in colormatches at high light levels, as a way to estimate directly theabsorption spectra of the cone photopigments.5

METHODS

In the standard color-matching experiment the stimulusconsists of a circular split field. On one side a test light of aselected wavelength and fixed radiance is presented, andthis is matched by a mixture of three primary lights ofsuitably chosen, fixed wavelengths. The match is achievedby adjusting the radiances of the three primaries until thetwo half-fields appear equal in both color and brightness.For spectral test lights this is generally possible only byadding one of the primaries to the test half-field, in whichcase the sign of the radiance of that primary in the matchingmixture is considered to be negative.

The particular data that we chose to examine were the 10°field color matches of Stiles and Burch for 49 observers. 2

These represent the largest comprehensive study of normal

0740-3232/88/101722-14$02.00 © 1988 Optical Society of America

M. A. Webster and D. I. A. MacLeod

Vol. 5, No. 10/October 1988/J. Opt. Soc. Am. A 1723

color matching. Measurements were made for 32 mono-chromatic test lights ranging from 392 to 714 nm, in equal-wave-number steps of 250 cm-' between 455 and 625 nm andin steps of 500 cm-' outside this range.6 The results werereported in terms of red-, green-, and blue-matching prima-ries at 645, 526, and 444 nm, respectively. 7

For our purposes, one can think of this experiment as themeasurement for each observer of 96 variables, namely, thelogs of the radiances of each of the three primaries requiredto match each of the 32 test wavelengths. Not surprisingly,many of these variables are redundant. For example, theradiances of the 444-nm primary required to match testlights of 400 and 408 nm are highly correlated with eachother-an individual who requires a lower than average radi-ance to match one will require less than average for theother. Presumably these relationships between the ob-served variables are due to the common influence of a small-er set of underlying variables, or factors. Thus the amountsof the 444-nm primary in the 400- and 408-nm matches arenot each measures of two separate determinants of the colormatches but rather are assumed to be largely two measuresof the same influences.

Given this assumption, the goal of factor analysis is to usemeasurements of the observed variables in a set of individ-uals to define the underlying variables and their values foreach individual. This is done by examining the correlationsamong the observed variables. Correlation among a groupof variables implies the existence of one or more commonunderlying factors. The degree to which any single ob-served variable is influenced by a factor is measured by thecorrelation between them, termed the factor loading. If thefactor loading for some variable on a factor is 1.0, then thetwo are perfectly correlated and all the variable's variance isdue to variations in that factor alone. More typically,though, values of an observed variable will reflect the influ-ence of more than one factor and will thus have loadings ofless than 1.0 on two or more factors. In standardized form,each measured value zjk of an observed variable j in anindividual k is regarded as the result of contributions fromeach of the n underlying common factors, and these contri-butions in turn depend on both the value Fik for factor i inthe individual case and the factor loading aji expressing theinfluence of factor i on variable j:

zJk= ajlFlk + aj2F2k + * ajnFnk + aj Ujk

Ujk is a factor corresponding to any variance that is uniqueto j. Note that the factors do not provide any informationabout the actual values of the color-matching functions butdepend only on the variations in those functions across ob-servers, as reflected in the correlation matrix. The squaredloading of any variable on a factor represents the proportionof the total variance of the variable that is due to variationsin that factor. Thus the contribution of a factor to the totalstandardized variance in the data set is given by the sum ofits squared loadings. For the single variable j, the sum ofthe squared loadings across the n common factors,

n

hj E aji

is termed the communality and is equal to the proportion of

observed variance accounted for by the set of common fac-tors.8

The dependence of an observed variable on the commonfactors can be visualized by representing it as a point in amultidimensional space, where the axes of the space corre-spond to the different factors and the point's coordinates areits factor loadings. Mathematically, this is simply an alter-native representation of the original correlation matrix-within this multidimensional space the correlation betweenany two observed variables is given by the cosine of the anglethat they subtend at the origin (times the product of the twovector lengths from the origin, which represent the commun-alities for each variable). However, to the extent that thenumber of factors is small relative to the number of observedvariables, this new representation has the advantage of de-scribing the data much more parsimoniously, in terms of asmall set of dimensions. In our case these dimensionsshould represent those properties of the visual system thatunderlie the individual differences in the color-matchingfunctions defined by the 96 measurements on our 49 sub-jects, and the method thus allows us to specify what thoseproperties are and over what range they vary. The proce-dure is particularly well suited to the analysis of color match-es, both because of the high reliability with which the match-es can be made2 and because the significant underlying vari-ables are probably few in number and well defined by theobserved measurements.

A serious problem with this representation, however, isthat it is not unique. Rotating any of the axes through anyarbitrary angle produces a mathematically equivalent solu-tion, while resulting in a radically different pattern of factorloadings-one that might lead to a quite different theoreti-cal interpretation. Suppose, for instance, that two uncorre-lated variables in a two-dimensional factor space had load-ings of (0, 1.0) and (1.0, 0). This would suggest that each ofthe variables is influenced by only one of the factors. How-ever, we could just as well have described the relationshipbetween the observed variables by rotating the axes 450, inwhich case the angle between the variables would remain 900but the loadings would change to (0.707,0.707) and (-0.707,0.707). Yet this representation would suggest that the un-derlying dimensions have the property that both influenceboth variables to the same extent (though for one of thefactors in opposite ways). Obviously the identity of thefactors inferred from these two rotations would be very dif-ferent.

The interpretation of the extracted factors thus dependscritically on the orientation of the axes, and some assump-tions must therefore be brought to the analysis to constrainthe rotation. Initially we relied on the general Varimaxrotation criterion. This makes no specific assumptionsabout the nature of the underlying factors but assumes onlythat different variables tend to be influenced by differentfactors and that the underlying factors are orthogonal. Therotation is then found that best satisfies these criteria withinthe constraints of the data; specifically, this is done by find-ing the solution that maximizes the variance in the squaredloadings for each factor. (Note that in the example abovethe squared loadings for the first rotation have the maxi-mum possible variance, while for the second it is at a mini-mum; the Varimax rotation would therefore converge on thefirst.) As we shall see, this one general assumption was for



the most part sufficient to guide our analysis to the theoreti-cally correct rotation.

RESULTS

Initial Factor AnalysisThe basic factor analysis was performed with the StatisticalPackage for the Social Sciences (SPSS) subprogramFACTOR 9 by inputting 97 items for each subject (the 96primary settings plus the subject's age). Initial factors wereextracted from the item correlation matrix by using a proce-dure that iteratively estimates the communalities, whichdetermine how much of the variance is to be partitioned intocommon factors, and then derives factors based on thoseestimates.'0 The number of factors determined in this waywas originally limited to 20, with each successive factor ac-counting for a smaller proportion of the variance. However,only a subset of these presumably corresponds to real prop-erties of the visual system, as opposed to random noise in themeasurements, and these could be qualitatively identifiedby a pattern of loadings that varied systematically withwavelength. The first 10 factors clearly met this criterion,and the first 11 were selected for rotation. These 11 ac-counted for 85% of the variance in the items.

Later we quantitatively assessed the statistical signifi-cance of the unrotated factors by finding the correlationbetween the loadings of items for adjacent wavelengths(with each item included in only one pair). For factors thatare continuous functions of wavelength the loadings onneighboring items should in general be similar, while pairs ofrandomly varying loadings will be uncorrelated. Of the 20factors, 12 yielded significant correlations (the first 10, the12th, and the 18th), but the magnitude of the loadings forthe 18th was low. Thus only the first 10 to 12 are likely to beof theoretical interest, in agreement with our initial esti-mate.

After the Varimax rotation the first six factors were imme-diately physiologically interpretable; together these ac-counted for 70% of the total variance." However, if only themost reliable measurements were considered (such as theamounts of the 526- and 645-nm primaries required tomatch yellow test lights), then the amount of variance ex-plained rose to over 90%. The communalities based onthese six factors are shown in Table 1 for each of the items.

The factor accounting for the most variance (16.9%) isshown in Fig. 1, where the observed loadings (unconnectedsymbols) are plotted as a function of the test light wave-length for each of the three primaries. (The abscissa islinear with wave number, reflecting the equal-wave-numberintervals sampled by Stiles and Burch.) Note that the signsof the loadings change at the primary wavelengths (dashedvertical lines). This generally happens because at thesewavelengths two of the primaries change sign as they aretransferred from one half of the matching field to the otherin order to permit the match. In this particular case it alsoreflects the change in whether the test light or the primary ismore strongly absorbed by the pigment as the primary wave-length is crossed.

This factor was qualitatively identified as representingindividual differences in macular-pigment density. Its clearimportance as a source of variability in these data is some-

what surprising given the fact that the field size was 100 andsubjects were instructed to ignore the central macular regiondefined by Maxwell's spot when matching the lights. Con-sistent with the absorption spectrum of the macular pig-ment, which peaks near 460 nm and is negligible at longwavelengths, the loadings for the red and green primaryitems for this factor are highest in the blue-green region ofthe spectrum and fall to near zero for yellow or red testlights. The exact relationship between variations in thepigment density and the resultant loadings is quite compli-cated. However, in broad outline it can be understood bynoting that neither the 526- and 645-nm primaries nor thelonger test wavelengths are absorbed to a great degree by themacular pigment, so that variations in the density cannotcontribute much to the variability in the amounts of theseprimaries in matches to the longer-wavelength test lights.In contrast, at shorter wavelengths the pigment will selec-tively absorb the test lights, decreasing the logs of theireffective radiances at the cone level (relative to the 526- and645-nm primaries) in direct proportion to the amount of

Table 1. Communalities (Proportion of VarianceAccounted for) Based on the 6 Identified Factors forthe 97 Items Included in the Initial Factor Analysisa

Test LightWave Wave-

Number(cm-')

1400014500150001600016250165001675017000172501750017750180001825018500187501925019500197502000020250205002075021000212502150022000230002350024000245002500025500

length(nm)

714.3689.7666.7625.0615.4606.1597.0588.2579.7571.4563.4555.6547.9540.5533.3519.5512.8506.3500.0493.8487.8481.9476.2470.6465.1454.5434.8425.5416.7408.2400.0392.2

Age

R (645.2 nm)

0.3958*0.57410.3224*0.4880*0.86380.91810.88490.94140.84030.92030.79570.90090.76780.76560.7050*0.5725*0.74780.83290.75230.89920.88380.90630.88600.87970.80580.4773*0.3945*0.52590.76870.9203*0.95920.9155*

PrimariesG (526.3 nm)

0.6519*0.79090.6886*0.7742*0.94190.93820.90160.94660.86000.39500.71680.24180.27690.11640.3549*0.4449*0.62750.85080.90700.91250.89680.93800.93820.83060.78840.4729*0.1950*0.24580.66010.8150*0.92300.8845*

B (444.4 nm)

0.1531*0.28910.2299*0.3898*0.6013*0.79700.7372*0.82030.8179*0.72760.7529*0.82800.7580*0.7374*0.7166*0.7184*0.71800.85660.85070.84580.90090.92340.73330.51610.43690.3137*0.1988*0.60440.65250.9474*0.96400.9504*

0.5545

I Asterisks indicate items that were deleted from the a priori rotationanalysis to make room for the theoretically zero-loading items.



0.8

0

0

0.4

0

-0.4

645 526 444observed o a Mpredicted * A "

II

NNIN

o 1

A _ I NR

l3;

-0.8,, I II

400 500 600 700

TEST WAVELENGTH (n m

Fig. 1. Factor loadings for a factor corresponding to variations inmacular-pigment density. Observed loadings (unconnected, opensymbols) are plotted for each of the three primaries as a function ofthe matched test wavelength. Connected, filled symbols show thepredicted loadings for macular density variations, with a standarddeviation of 0.12 at 460 nm.

pigment present. Thus in this region differences in thedensity are a major source of the observed differences amongsubjects. The 444-nm primary behaves differently. In theblue-green region the loadings are near zero because boththe test light and this primary are absorbed to a similarextent-a difference in density will not produce a relativedifference in their radiances. It is only at longer test wave-lengths, where the 444-nm primary is now selectively attenu-ated, that the influences of the pigment variations on thisprimary show up. At still longer wavelengths and at theshorter end of the spectrum, other factors become muchmore important as sources of variability in the blue primary(rod intrusion and lens pigment, respectively; see below) sothat the proportion of variance explained by macular-pig-ment differences again diminishes. This final point illus-trates that, while the derived factors are orthogonal (as arequirement of the Varimax rotation), it cannot be consid-ered in isolation.

The polarity of the loadings is essentially arbitrary; rotat-ing the relevant axes 1800 changes their sign but not theirmagnitude. In this case the polarity is such that a lowerdensity corresponds to a higher factor score. For example,an individual with a higher score on this factor would requiremore of the 526-nm primary and more negative amounts ofthe 645-nm primary (added to the test) to match the blue-green tests, as indicated by the positive and negative load-ings, respectively, for these primaries at these wavelengths,and this results because a lower density makes the individualmore sensitive to these tests.

The identification of this factor was confirmed by compar-ing the observed loadings with quantitative predictions ofthe effects of macular-pigment variations. As we notedabove, the square of the factor loading for a given itemequals the variance due to the factor divided by the totalvariance for the item. The loading itself is therefore thesquare root of this ratio, or the ratio of the respective stan-dard deviations. Thus, if the effects of a known factor i on

the measurements are theoretically determinable (and thevariability among individuals in factor i is known), then thepredicted loading of factor i on measurement j is

aii = ofji/oj,

where aj is the observed standard deviation of item j and ojiis what the standard deviation of item i would be if all thevariations in it were due to the ith factor. (An alternativewould be to define o-j as the square root of the sum of thevariances due to each of the predicted factors. We rejectedthis alternative because it would assume that all the factorsinfluencing the matches were known and included.) Now,for a given test wavelength X, oji is equal to the change inprimary radiance, px, per unit change in Xik (a random vari-able representing the value of factor i for some individualk-for example, the macular-pigment density), multipliedby ati, the standard deviation of Xik for the set of observers.Thus

Iiii = oXi(aPx/OXd)/1 j-

Finally, we chose to base all our predictions on variationsin the logs of the primary radiances rather than on the linearradiances, so that our loadings were calculated from

aji = oxi(a log1PI/OXd)/r0Iog~j1 (1)

with the sign of the primaries restored after the logs of theabsolute values were taken. The reason for using the logs ofthe radiances was that this allowed for predictions that wereunaffected by assumptions about the particular spectra ofother prereceptoral filters or the photopigments. Theseassumptions would affect the mean predicted primary ra-diances and their absolute variances but not, to a first ap-proximation, the variance of their logs. Consider, for exam-ple, the effect of different lens-pigment spectra on the colormatches calculated for a high or low density of the macularpigment. Any arbitrary change in the lens absorption willalter the pair of predicted primary radiances (for the highand low macular density), and also the difference betweenthem, by the same multiplicative factor. However, the logsof the primaries will be changed by the same added constant,which cancels out when their difference is taken (and thuscancels from the calculated variance). Thus when the loga-rithms of the primary radiances are adopted as the observedvariables, the predicted loadings are comparatively indepen-dent of any fixed but unknown scaling effects of other fac-tors on the matching radiances.

In the case of the macular pigment, we chose to representthe value of the factor, Xik, by the pigment density possessedby an individual at 460 nm, which is close to the wavelengthof maximum absorption. An increase in Xik changes therequired log primary radiance loglpxl both by attenuatingthe test light and (in the opposite direction) by attenuatingthe primary stimulus itself. The net effect on the log match-ing intensity is proportional to the difference in the macular-pigment density at the test and primary wavelengths (rela-tive to its density at 460 nm). Thus

a loglpxl/0xi = DX/D460 -DpD460.

The density values were obtained from the macular-pigmentabsorption spectrum tabulated by Wyszecki and Stiles.'2



This equation, when divided by the observedations in the log primary radiances, completthe predicted factor loadings for a factor relvidual variation in macular-pigment densilfree parameter only a multiplicative constanto ur', the standard deviation of the macularthe observers. The IMSL subroutine zxssvary rxi to obtain a least-squares fit of the prEto the observed loadings. This provided anfor the standard deviation of the macular-pat 460 nm among the observers of Stiles anloadings do not provide information aboutment density.)

The resultant predicted loadings are showrconnected symbols and are clearly in closethe observed loadings. This not only confirof the factor but also indicates that, at leastVarimax criterion was in fact sufficient to cappropriate rotation without requiring any kthe actual nature of the factors.

The second prereceptoral factor that we iisponded to individual differences in the derpigment and accounted for 12.1% of the totthis case the loadings were characterized byall three primaries at the shortest test wavelenear zero at longer wavelengths (Fig. 2). Tomation this parallels the absorption spectrpigment, as one would again predict from c(the effects of variations in its density. Howtive predictions based on the Wyszecki-Stithe lens-absorption spectrum yielded mode]the amounts of the blue and green primaries <

ly long wavelengths, in part because of tinegligible density of the lens pigment for tFor example, when the original predictions (rfigure) were scaled to coincide with the obse]short wavelengths, the predicted loadings figreen primaries matched to yellow tests w(

0

0

obsei

p red i

-0.41

-0.8

400 500

TEST WAVELENGTH (nm)

Fig. 2. Loadings for a factor identified as varinpigment density. In this case the predicted loamfilled symbols) were first rotated with other predicthe Varimax rotation. The best-fitting standardlens density is 0.18 at 400 nm.

I standard devi-ely specifies all)resenting indi-ty, leaving as at correspondingr density amongQ13 was used toedicted loadingsestimate of 0.12pigment densityd Burch. (Thethe mean pig-

n in Fig. 1 by theagreement withms the identityin this case, theconverge on thenowledge about

dentified corre-isity of the lens

645 526 4440 a M

0.8 -

0.4 .

0

0

1

-0.

-0.

observ

I A

ed I

I N

I NN CI A M H aN N

N NN . NN N

A., . 0! _ -CI -A

6 N 0 N N A 0 00 ju it 8A , W a ° ; 0-0,4 ; °_~eA.4 I | N' I a A A

A l I N1 0 N I A&

I I I

I ~ ~ ~ ~ l I I8 I I II~

400 500 600 700


Fig. 3. Observed loadings for a factor corresponding to variation inthe degree of rod intrusion, tentatively identified by the high load-ings on the 444-nm primary matched to long test wavelengths. Nopredictions were calculated for this factor.

al variance. In compared with less than 0.2 for the observed values. Thishigh values for pattern of loadings, which changes only gradually acrossngths, falling to much of the spectrum, was probably discouraged by thea first approxi- Varimax solution because of the low variance in the magni-um of the lens tudes of the loadings. To find out whether the rotation)nsiderations of might in this case have been inappropriate, we performed aiever, quantita- Varimax rotation of the entire set of predicted factor load-les'2 values for ings. This did in fact change the lens-pigment predictions-ate loadings on so as to bring them into close agreement with the set ofeven at relative- observed loadings (as shown by the connected symbols inie low but not Fig. 2). The changes in the rotated loadings for the lens-hese primaries. pigment predictions obviously meant that other predictediot shown in the factors had absorbed some of the variance that was due towed loadings at lens variations. Nevertheless, except where noted below,or the blue and the predictions for these other factors were not substantially-re roughly 0.4, affected by the rotation. The standard deviation of the

lens-pigment density was estimated by a least-squares fit tobe 0.18 at the reference wavelength of 400 nm (at which the

645 526 444 mean lens density is 1.2 for the Wyszecki-Stiles tabulated,ved 0 a N 1)

cted * A N values'2 ).In Fig. 3 the observed loadings have been plotted for a

third factor, which accounts for 6.4% of the variance. Itloads mainly on the amounts of the blue primary required tomatch long-wavelength tests, and for that reason we believe

NA ' XP that it corresponds to the degree of rod intrusion. In colora 2 matches with large field sizes and intensities that are not

extremely high, the rods may contribute differentially to thematches at longer wavelengths.'4 To a first approximationthis results in a desaturation of the colors, requiring a com-pensating adjustment in the radiance of the blue primary.Differences in the extent of rod intrusion should thus ac-count for much of the variability in the 444-nm primary atthe longer wavelengths. Because of the large number ofassumptions that it would require, we did not try to calculatepredicted loadings for this factor.

The final three factors identified by our initial analysisltion in the lens- appeared to represent variations in the long-, medium-, andted factors, using short-wavelength-sensitive photopigments (hereafter re-d deviation of the ferred to as L, M, and S, respectively) and accounted for

15.9, 10.5, and 8.5% of the total variance, respectively. One



possibility that we investigated was that these variationscorresponded to variations in the density of the photopig-ments. An increase in the density alters the absorptionspectrum (and hence changes the color matches) because itproduces a relatively large increase in sensitivity in the tailsof the spectra, with little gain at the peaks (because at thepeaks the probability of photon capture is already high).'5

To examine whether this was the basis for the observedfactors, we calculated predicted loadings for density varia-tions using the photopigment spectral sensitivity estimatesof Smith et al.

16 For a photopigment of sensitivity sx (nor-malized to a peak of 1.0) and initial density dj, the normal-ized sensitivity at density d2 can be calculatedl 5 from

Sd,, = (1-o10)/(1 - ),

where

x = (d2/d1)flog[1l - (1 - lod)(sd,)]I

Starting with the sensitivities of Smith et al., new sensitiv-ities were calculated for a +0.05 change in the density of oneof the pigments, and, by using these, two sets of color match-es were derived. The difference in the logs of the matching

645 526 444o A N observed I I

* A N predicted lI 0 I

I 0000 I0.8 |I 00 ° I

01

0.4 - I I A

0 N0 0n a a \o ̂ 1 N

I4 AA

I I ~~~~~~AA

400 500 600 700

TEST WAVELENGTH Inm)(a)

400 500

TEST WAVELENGTH in)m(b)


primary radiances at the high and low densities, divided bythe difference in density, provided an estimate of the deriva-tive in Eq. (1). Finally, ZXSSQ was again used to find thestandard deviation of density that yielded the best fit of thepredicted to the observed loadings.

In Fig. 4(a) the predictions for density changes in the Lpigment (connected symbols) are compared with the appro-priate set of observed loadings, and it is obvious that the fit ispoor owing to the negligible influence on the matches pre-dicted for long test wavelengths. The fit of individual densi-ty variations in the M photopigment (to the loadings for thefactor best described by that hypothesis) is still worse, asshown in Fig. 4(b). In this case the predictions fail largelybecause of the incorrect polarity predicted for the red prima-ry matched to blue-green tests. The predictions for the Spigment turn out to be reasonably close to the correspondingS factor [Fig. 4(c)], but another interpretation of this factoris possible (see below). We also constructed a single theo-retical factor based on a common density change in all threepigments, which might be more likely if density differenceswere due to individual differences in the lengths'7 or orienta-tion18 of all the cone outer segments. Yet here again the fitof this factor (shown in Fig. 7 below) to any of the three sets

645 526 4440 a M observed I

* A N predicted l

0.8 I

Z 0.4 - I

O~~~~~Of 7~~~~~~~~~~~~0

I I0 0 AA,0 I ANM

A A N 0

400 500 600 700


(C)

Fig. 4. (a) Unconnected, open symbols: observed loadings for aJ4 y: factor thought to reflect variations in the L cones. Connected, filledsymbols: predicted loadings for density variations in the L pig-

, ment. The predictions are substantially different from the ob-served loadings at long test wavelengths, where they fail to predict

A I the large observed influence on the matches. (b) Observed loadingsAAA I for the M cone factor (unconnected, open symbols) compared with

predictions for density variations in the M pigment. PredictionsC I, ,, for the red primary matched to blue-green tests have the wrong

600 700 polarity. (c) Observed loadings for the S cone factor (unconnected,open symbols) compared with predictions for density variations inthe S pigment (connected, filled symbols).

0.8

0.

0

10

-0.4

-0.8


of observed loadings was poor. Thus, with the possibleexception of the S-pigment factor (further discussed andrejected below), none of these factors appears to representindividual differences in photopigment density.

Instead, the observed factor loadings for all three factorscorrespond closely to predictions based on the hypothesisthat the pigment variation takes the form of slight shifts inthe absorption spectrum from one individual to another.To test this hypothesis, predicted loadings were constructedby shifting the spectra +250 cm-' along the wave-numberaxis without changing their shape and calculating the resul-tant changes in the color matches. As shown in Figs. 5(a),5(b), and 5(c) for shifts in the L, M, and S pigments, the fit ofthese predictions to the observed loadings is sufficientlyclose to argue strongly for Xmax variations as the correctfactor identity. The required standard deviation of thepeak was estimated by least-squares fits to be 50.3 cm-' forthe L pigment, 31.9 for M, and 45.3 for S. (Earlier informalestimates5 differed slightly from these values.) Predictedloadings for a common shift factor, in which all three absorp-tion spectra were displaced together in the same direction,did not resemble any of the three observed factors.

In the case of the S pigment the predictions for Xmrax and

645 526O A

* A

444

-0.4

-0.8

TEST WAVELENGTH tn)m

(a)

645 526O A

* A

density variations are roughly similar, so that it is difficult todiscriminate between these alternatives on the basis of thepredicted loadings of Figs. 4(c) and 5(c). However, theidentification in a later analysis of a factor resembling com-mon density variations in all three cone types (see below)argues against a separate and independent S-pigment densi-ty variation as the basis of the present factor.

Orthogonality of FactorsAs we noted above, one property of the Varimax rotation isthat the factors produced are all orthogonal to one another.However, the actual physiological factors need not necessar-ily be independent. For example, if variation in the threephotopigment factors reflected some common influence,such as a general change in receptor morphology, then onewould expect the factors to be correlated. In that case theVarimax rotation and the associated factor loadings wouldbe in error. Alternatively, if the Xmax variation were due todifferences in the actual photopigments, for example, be-cause the genes encoding the pigments'9 varied slightly, thenthe corresponding factors would be expected to vary inde-pendently.

To examine whether the factors were truly orthogonal, we

645 526 444-O a

* A N

400 500 600 700

TEST WAVELENGTH )nrm)

(C)

444

ED

2c]

0.8

0.4

0

-c

-0.4

-0.8

400 500

TEST WAVELENGTH tnm)

(b)

NN ,, /,Fig. 5. (a) Loadings for the L cone factor (unconnected, opensymbols), this time compared with predictions for variation in the

/ | spectral position of the L pigment (connected, filled symbols). TheI , predictions are reasonably close to the observed loadings, suggesting

^AAA that Xmax variations is the correct factor identity. The standarddeviation in X ax for these predictions is 50.3 cm-'. (b) Loadings for

| the M cone factor compared with predictions for Xmax variations inC C @ | the M pigment. The predictions (for a standard deviation of 31.9

600 700 cm-') closely follow the observed loadings. (c) Loadings for the Scone factor compared with predictions for Xmax variations in the Spigment (with a standard deviation of 45.3 cm-').

0.8

0.40

0

-0.

-0.

0.81

0.4



performed the analysis using the SPSS OBLIQUE rotation.In this rotation the parameter DELTA can be varied to con-trol the degree of correlation among the factors. By settingit to 0.0 a solution with moderate interfactor correlationswas actually encouraged. Nevertheless, the resulting corre-lations were all low and in most cases insignificant [with theexceptions of 0.38 for macular density and Xmax(S) and of 0.28for lens density and Xmax(S)]. We also used the same obliquerotation with the set of theoretical factors (macular density,lens density, and Xmax for each cone type). These wereassumed to be orthogonal, yet an oblique rotation consistentwith the predicted correlation matrix yielded interfactorcorrelations that were larger than those that we obtainedusing the empirical correlation matrix. (For the predictedfactors the rms correlation was 0.33, compared with 0.16 forthe empirical factors.) Thus the actual factors influencingthe matches do indeed appear to vary independently. Thisimplies, for instance, that there is no tendency for the densi-ties of the lens and the macular pigments to covary (see alsoFig. 4 of Ref. 2). Similarly, it suggests that the three photo-pigment factors do reflect variations that are restricted toeach individual cone type.

Guided Factor RotationBy relying on general statistical criteria to perform the fac-tor rotation, we obviously assumed nothing about the specif-ic mechanisms determining color matching or about whatfactors might be present. A better approach would be totake advantage of known properties of these mechanisms inorder to help to guide the rotation. To do this we construct-ed a new set of variables by combining the original color-matching functions in ways that were designed to producezero loadings on different types of factors. These zeros thenserved as targets to help to direct the rotation to a possiblymore appropriate solution. Of course, if Varimax had ar-rived at the correct solution then the two procedures wouldhave yielded similar results.

The derived variables were defined as follows. Eighteenitems were created by taking, for each individual, the ratio ofthe radiance of two primaries at one test wavelength dividedby the same ratio at a second wavelength. Such items areunaffected by differences in the densities of the prerecep-toral pigments 20 and should therefore have zero loadings onthe lens- and macular-pigment factors. For the rod intru-sion factor, zero-loading items can be obtained simply by

choosing the primary settings for test wavelengths at whichthe rod system is presumably driven to saturation. Avoid-ing the B primary settings where the rod intrusion is mainlyreflected, we used 30 of the original R or G primary radiancesmatched to test wavelengths of 580 nm or less.14

To derive items that should be unaffected by variations inthe photopigments, we relied on the fact that the colormatch can be regarded as being established independentlywithin each of the cone types, when for each photopigmentthe absorptions due to the test light and the sum of the threeprimaries is the same.21 For the Stiles-Burch data thematch is thus defined by the following three equations:

SLQX = SL444Q444 + SL526Q526 + SL645 Q6 45~SMQ\ = SM Q4 44 + SM26Q52 6 + SMQ 6 4 5

S X= SS444Q44 4 + SS526Q52 6 + sS645Q6 45, (2)

where SLY is the sensitivity of. the L pigment to test wave-length X and Qx is the radiance of X, etc. Now, suppose thattwo observers have the same S and M photopigments butdiffer in the sensitivity of their L pigments. Because SL 1

differs for the two, the red, green, and blue primary ra-diances that they choose to match a particular test light willgenerally differ. However, for each of the pigments held incommon, the chosen mixture of primaries must duplicatethe excitation produced by the test light in order for a matchto be achieved. Thus the different matches made by differ-ent observers will preserve equality of excitation for anypigments that the observers hold in common: increasedamounts of one primary must be counterbalanced by de-creased amounts of another, so that the excitation of thecommon pigment [determined as on the right-hand sides ofEqs. (2) by weighting each primary radiance by the sensitiv-ity of the common pigment to that primary and summing theresult for all three primaries] remains constant.2 2 A factorthat represents variation of any kind (for instance, in Xmmax,

density, or shape) in only one pigment should therefore haveno influence (and zero loading) on such a measure of excita-tion of the unaffected pigments. (Of course, any two sub-jects may differ in any of or all their photopigment sensitiv-ities. For the items isolating the contribution of one pig-ment to the match to have zero loadings on factors for theother pigments requires that the three photopigments varyindependently, as our results suggest. See above.)

Eighteen of these new observed variables were calculatedto represent excitation of either the L or the M photopig-ment as hypothetical common pigments; factors represent-ing variations in the S pigment should have zero loadings onboth of these sets. The items were constructed by weightingthe three primaries matched to chosen test wavelengths bythe L or M sensitivity of Smith et al., adjusted to reflect thepresumed effective optical density for 100 fields.' 7 Alongwith the variables described above for prereceptoral factors,these were then entered into the standard analysis withVarimax rotation. Because of limitations in the total num-ber of items permitted, the inclusion of these items necessi-tated deleting some of the original variables. Those re-moved have been indicated by asterisks in Table 1.

With the new Varimax factors as a starting point, theloadings were next modified by using an iterative proce-dure23 in which we successively rotated the factor axes inpairs to try to line up the full matrix of observed loadingswith the partial target matrix of theoretically defined zeroloadings. The two factors of a pair were always rotatedthrough a common angle to keep them orthogonal. Initiallythe angle was chosen to minimize the squared loadings onthe particular set of targeted items (prereceptoral, photopig-ment, or rod intrusion) that were determined to be appropri-ate for each factor. However, this occasionally allowed oneof the factors to be- rotated into a plane in which all itsloadings, and notji t those for the theoretically zero-loadingitems, were near z- ,. To correct this, we therefore calculat-ed the angle for each pair that minimized the ratio of the sumof the squared loadings on the targeted items to the totalsum of the squared loadings for each factor. The procedurewas repeated for each of the identified factors (paired withall other factors within a space of as many as 12 of theVarimax factors) until the rotation converged.

Despite the use of this entirely different rotation criterion,



645 526o a

* A

-0.48

C AN

400 500 600 700

TEST WAVELENGTH inm)

Fig. 6. Initial Varimax factor loadings for the L cone variations[unconnected, open symbols, same as Fig. 5(a)], compared with therevised loadings following the theoretically guided rotation of thefactors (connected, filled symbols). There is remarkably littlechange in the loadings, indicating that the Varimax rotation for thisfactor is substantially correct and that the factor does representvariations in the L cones.

the loadings for most of the factors remained remarkablysimilar to the original Varimax estimates, indicating that theVarimax rotation was largely correct. This supports theidentification of the different observed factors with varia-tions in the prereceptoral pigments, the photopigments, andprobably the participation of rods in the matches, all ofwhich was based on the Varimax rotation. An example ofthe revised loadings is shown in Fig. 6, where the new Xmax(L)

loadings are plotted along with the Varimax loadings of Fig.5(a). The two sets of loadings are similar throughout thespectrum. Thus the slight discrepancies between the ob-served pattern of loadings and the predicted loadings forXmax(L) variations remain [see Fig. 5(a)]. The source of thesediscrepancies is considered in the following paper. 4

While this refined rotation using a priori assumptionsabout the nature of the operative factors largely confirmedour initial analysis, one significant modification in the re-sults was the emergence of a factor resembling a commondensity variation in all the photopigments. The loadings forthis factor are shown in Fig. 7, along with the predictions forcommon density variations as calculated above. The stan-dard deviation of the density difference, which was heldequal for each cone type, was estimated by a least-squares fitto be 0.045.

The loadings for this factor tend to vary little across muchof the spectrum. As we have noted above, such a pattern isdiscouraged in the Varimax rotation, which probably ob-scured the presence of density differences in the initial anal-ysis by allowing some of the variability that was due to themto be absorbed into the loadings for other factors. As a testof this, we examined whether our theoretical common densi-ty factor could survive a Varimax rotation in combinationwith the predicted factors for Xmax and prereceptoral differ-ences. Several rotations were performed for a range of mag-nitudes of the initial density loadings (set by varying thestandard deviation of density) to control the salience of thedensity factor. As expected, the effect of the rotation was to

redistribute part of the variance due to density among theother factors, so that a pattern of loadings characteristic ofdensity variations largely disappeared when the standarddeviation in density was set to 0.03 or less. Again, however,the resulting change in the loadings for the other factors wasslight. Thus, as the similarity between the initial and re-fined loadings of Fig. 6 suggests, the initial Varimax factorsprobably correspond closely to the actual physiological vari-ations.

Predicting the Correlation MatrixThus far we have assessed the variance in the differenthypothesized factors by fitting the loadings predicted bythem to our set of observed factors. Yet, as we have repeat-edly discussed, an incorrect rotation of the observed factorscan introduce an error into the loadings by taking the vari-ance that is really due to one factor and attributing it toanother. If this happens then both the factor variance andhow much the factor contributes to the total variance in thematches-as determined by the loadings-may be miscalcu-lated.

One way around this problem is to choose the factor'sstandard deviations to fit directly the correlation matrix ofthe observed items rather than the particular set of loadingsextracted from the matrix. In this case the question ofwhether the observed covariances are correctly partitionednever arises. (On the other hand, addressing the more prac-tical problem of trying to identify the sources of variabilityby fitting the correlations would be problematic, as predic-tions for most potential factors would explain some of thevariance.) The predicted correlation between two items, jand k, based on a set of n factors, is

n

Pik = ajzaki,

where 6ji and aek are the predicted loadings of factor i onitems j and k, respectively. The observed correlation thenequals the predicted correlation plus a residual error term:

645 526 444o A N observed I

* A w predicted |

0.8- I

M 0~~~~~

0 M AAA

0~~~~~~A A

400 500 600 700

TEST WAVELENGTH (nm

Fig. 7. Open, unconnected symbols: loadings for a factor thatemerged from the theoretically guided rotation and qualitativelyresembles predictions for a common density variation (standarddeviation, 0.045) in all three pigments (connected, filled symbols).



Table 2. Estimated Interobserver Standard Deviations in Prereceptoral and Photopigment Density and inPhotopigment Xmax (Wave Number)

Macular Lens XL XM XS Density

Fits to factor loadings100 Field color matches 0.12 (at 460 nm) 0.18 (at 400 nm) 50.3 31.9 45.3 0.0452° Field color matches 0.18 0.15 47.2 25.0 41.3

Fits to correlation matrix100 Field color matches

Independent Xmax 0.120 0.156 42.2 18.1 40.5 0.0458Equal Xmax 0.121 0.153 32.6 32.6 32.6 0.0420

20 Field color matchesIndependent Amax 0.184 0.231 48.0 25.1 24.1 0.0222

rjk = Pjk + ejk-

For our set of factors we included predictions for macularand lens pigment differences, Xmax variation in the L, M, andS cones, and a change in photopigment density common toall three cones. Rod intrusion was not included because ofthe difficulties in making unambiguous predictions for thisfactor. The loadings for each factor were calculated as de-scribed above, using the standard deviation of each factor asa free parameter. ZXSSQ was then used to find the values ofthe standard deviations that would minimize the mean-squared error in the predictions.

The results of this procedure are shown in Table 2, wherethe best-fitting standard deviation for each of the identifiedfactors is listed. For comparison, the standard deviationsobtained by fitting the predicted loadings to the set of ob-served factor loadings are included in the first two rows.Also shown are the results of fitting either the correlationmatrix or the factor pattern derived from an analysis of theStiles-Burch 20 color matches, 3 which we discuss in thefollowing subsection. Finally, for the fits to the correlationmatrix for the 10° data two estimates of the variability of thepredicted factors are shown, depending on whether the spec-tral peak factors were all constrained to have equal standarddeviations or were allowed to vary independently.

Despite the fact that they did not show up among theinitial set of factors, the present approach indicates thatphotopigment density variations do contribute significantlyto the interobserver differences in the matches (as the guid-ed rotation of the initial factors also suggested; see Fig. 7).The standard deviation in density required to fit the correla-tion matrix is roughly 0.045 for each pigment, a value close tothe standard deviation suggested by the observed loadings ofFig. 7. Again, some of the variance due to density differ-ences was probably absorbed into other factors in the Vari-max solution in the way described above. Perhaps partlyreflecting this, the standard deviations for the spectral peakfactors fitted to the correlation matrix are somewhat lowerthan the estimates obtained by fitting to the observed fac-tors. In particular, the standard deviation for the M pig-ment is less than 20 cm-' when it is allowed to vary indepen-dently. The standard deviations for the S and L pigmentsare similar to each other but are almost double that of the Mpigment. In a further analysis (also documented in Table 2)the three spectral peak factors were forced to have identicalstandard deviations, partly to test whether the value for theM pigment was due to competition with the density predic-tions for the same variance. However, this left the estimatefor the density variations virtually unaffected, while produc-

ing a single intermediate value for the standard deviation ofXmax. The assumption of equal variability in Xrmax acrosspigments is therefore not strongly supported.

The rms observed correlation in the original matrix was0.39. After the variance due to the fitted predicted loadingswas parceled out, the rms of the residuals, ejk, dropped to0.16. If the only relationships among the variables were dueto the set of factors that we removed, then these residualsshould be distributed in roughly the same way as correla-tions in samples from a population in which the correlationis zero. For 49 subjects the expected standard deviation ofthe zero correlation is 1/49- 1, or 0.144, which is slightlyless than the obtained value for the residuals of 0.16. Thissuggests that some real linkages between the variables stillexist. As a final attempt at identifying potential sources ofvariability in the matches, we performed a factor analysis ofthe residual correlation matrix, following the procedure forour initial analysis above. One factor that we obviouslyexpected to reappear was rod intrusion, and, in fact, a possi-ble rod intrusion factor was again obtained, though it dif-fered from the original by containing moderate loadings onlong-test-wavelength items for the G primary as well as forthe B primary. (A similar change in the rod intrusion factorwas obtained in the guided rotation of the factors.)

Several other factors had loadings that varied systemati-cally with wavelength and therefore potentially reflect somesubstrate of the color-matching process (if not systematicerrors in some of our predicted loadings). However, we wereunable to identify any of these. Among the possibilities thatwe considered was that of density variations in a filter withthe absorption spectrum of the macular pigment, but screen-ing only the L cones, which was postulated by Smith andPokorny24 to account for fits to Konig-type fundamentalsand to explain individual variability in color matching.'6' 2 5

The significant loadings predicted by this factor are restrict-ed to the red primary items at mid to short test wavelengths.However, none of the observed factors from the residualmatrix or from the initial set of 11 resembled this pattern.In any case, these unidentified factors account for only asmall percentage of the variance in the Stiles-Burch data.Thus, whatever their possible theoretical significance, theircontribution to the interobserver differences in these colormatches is relatively minor.

Analysis of the Stiles-Burch 20 Color MatchesAs we noted, Stiles and Burch' also measured color-match-ing functions for 20 fields. The data for the individualsubjects were recently recovered and published by Tre-



zona.26 We were interested in applying a similar analysis tothe 20 data, both because these data provide an additionalset of measures with which to examine individual differencesand because they more closely reflect the color-matchingproperties of central vision. However, they have the draw-back that the measures were made for only 10 observers, asopposed to 49 for the 100 fields.

For the 20 data, the correlation matrix was partitionedinto nine components (one fewer than the number of observ-ers), using principal-components analysis (which differsfrom factor analysis in that values of unity rather than esti-mated communalities are used in the diagonal elements).2 7

As before, the factors were identified after Varimax rotationby comparing their loadings with the predicted loadings for agiven theoretical factor.

The second row of Table 2 lists the best-fitting factorstandard deviations that we obtained for each of the identi-fied components. As before, we found factors representingmacular- and lens-pigment variations and variations in thespectral peaks of the L, M, and S cones. There was noevidence of a rod intrusion factor, nor was there a clearphotopigment density factor. A new far-red sensitivity fac-tor, not clearly present in the analysis of the 100 matches,was associated with a need for greater energies of each of thethree primaries matched to long-wavelength (X > 650-nm)test lights. This could possibly be linked to individual vari-ation in receptor morphology or refractive index, alteringthe waveguide properties of the receptors and thus the effec-tive sensitivities of the cones for wavelengths that are largerelative to the cone diameter. Variations in such a factormay be more apparent in the 2° data because of the smallerdiameters of the foveal cones.

Probably because the observed loadings were based on sofew subjects, the fit of our predictions was generally poorerthan for the corresponding 100 field factors (particularly forthe factors representing M and S cone variations). Never-theless, the identity of the factors was clear (or at leastsuggestive for the M and S factors), and the estimated stan-dard deviations were similar to the estimates based on the10° data. The largest discrepancy, in the standard devi-ation of macular-pigment density, was in fact close to whatwe would have predicted from the effective mean density ofthe macular pigment for the two field sizes (see below).2 8

As with the 100 data, we also tried to estimate the factorstandard deviations by fitting the correlation matrix for the20 matches. The results are shown at the bottom of Table 2.These values are again in reasonable agreement with ourother estimates, although if they are correct the lower vari-ance in photopigment density would be surprising, given thegreater length of the foveal cones.

DISCUSSION

To summarize, we have identified the following sources ofinterobserver variability in the Stiles-Burch 100 colormatches: differences in the lens- and the macular-pigmentdensities, in the degree or form of rod intrusion, and in thedensity and spectral peak of the photopigments. All thesefactors have been known or previously postulated to under-lie the observed variability in color matching. Numerousother physiological factors no doubt exist, but it is unlikelythat the overall contribution of any of them is large, because

the variance of the majority of items is largely accounted forby the set of identified factors. For the most reliable match-es the communalities based on these factors are near 0.90,providing close to a complete description of the intersubjectdifferences. Thus, for example, if one were to rank thesubjects for these items (according to how much of the pri-mary they would require to match the test) simply on thebasis of their combined factor scores, then the predictedrank would correlate 0.90, or 0.95, with the observed rank.

For our initial set of factors we noted the percent of thetotal variance that was accounted for by each. However, itshould be kept in mind that this number is dependent onwhich variables are included. For instance, if the item setcontained more short-wavelength tests, then the proportionof variance due to the inert pigments would obviously in-crease. As we noted above, the Stiles-Burch data includetwice as many wavelengths from the midspectral region asfrom either extreme. Recall, moreover, that the percent ofthe total variance, as commonly defined, reflects the sum ofthe standardized variances for each item. However, thematches to some wavelengths have much more absolute vari-ability associated with them than others. The factor load-ings contain no information about what the absolute vari-ance is, indicating only how much of it is due to the factor.The actual variations in primary radiance that are attribut-able to the factor can be found by multiplying the loadingsby the observed standard deviations. As an example, Table3 shows the standard deviations (expressed as percentages ofthe mean settings) in the radiance of the primaries matchedto 588 and 482 nm that would be created by the estimatedvariation in each of our factors from the initial Varimaxrotation. (The complete list of observed standard devi-ations can be found in Ref. 2.)

Our results suggest a standard deviation of 0.12 for themacular-pigment density at 460 nm and from 0.15 to 0.18 forthe lens pigment at 400 nm (depending on whether thecorrelation matrix or the factor loadings were predicted).These values are fairly comparable with previous estimates.For example, Bone and Sparrock 29 reported a standard devi-ation of 0.15 to 0.20 for the macular density of 49 subjectsmeasured with 20 fields. For a 20 matching field the meandensity at 460 nm is roughly 0.5.12 However, Stiles andWyszecki estimate that the effective density for 100 fields isroughly 0.6 times that for 20.28 Our estimate of 0.12 mighttherefore suggest a standard deviation of 0.20 for the smallerfield size and thus is very close to the value of 0.18 weactually obtained for the 20 matches (see Table 2).

Van Norren and Vos30 calculated presumed lens-densityspectra from the scotopic sensitivity data of Crawford andfound that 95% of the observers (between 17 and 30 years ofage) fell within +25% of the mean density. This is consis-tent with our estimate of -0.15 as the standard deviation indensity at 400 nm if we use the Wyszecki-Stiles'2 meandensity of 1.2 at 400 nm; it suggests a slightly smaller rangeof +20% if the van Norren-Vos higher estimate of 1.45 isused. Both of these mean values may underestimate theaverage value for the Stiles-Burch subjects. Pokorny etal.

3 ' have suggested that the Wyszecki-Stiles density func-tion should be multiplied by 1.33 to adjust it for the average

age of this sample, which would reduce our estimate of thepercentage change accordingly. Note that the standard de-viation of 0.15 is 12.5% of 1.20 (our lowest mean density

M. A. Webster and D. 1. A. MacLeod


Table 3. Standard Deviations (a) in the Color-Matching Functions That Would Be Generated by the EstimatedVariation in the Factors Identified in the Initial Analysisa

Test Total RodLight Primary Observed Macular Lens XL XM XS Intrusion

588 nm R 9.1 0.41 1.3 8.5 1.2 0.15 1.7G 6.5 0.43 0.68 1.9 5.9 1.1 0.31B 22 4.7 0.31 4.4 0.86 4.1 18

482 nm R 15 10.4 1.8 7.8 4.2 3.4 0.96G 10 9.5 0.61 0.39 0.23 1.4 1.3B 9.4 1.1 1.6 1.4 2.0 8.5 0.40

a The values are expressed as percentages of the mean matching intensities.

estimate), while that of 0.12 for the macular density is 40% ofthe presumed 100 mean of 0.3. Thus the relative variabilityin the macular-pigment density is probably at least threetimes as large.

There is considerable evidence that the density of the lenspigment increases with age.32 However, subject age did notsignificantly correlate with our lens-density factor (r = 0.24)and was only weakly correlated with other factors. [Thehighest correlations were 0.44, 0.40, and 0.34 for Xmax(S),

Xmax(M), and macular pigment, respectively.] Stiles andBurch similarly reported only a weak relationship betweenlens density and age for these data. 2 This is perhaps be-cause there were few very young or old observers in thissample (mean age, 31.8; standard deviation, 10.3), and dif-ferences within age groups were large. Moreover, we havemodeled the individual differences in lens pigment by uni-formly scaling the density of the Wyszecki-Stiles absorptionspectrum. However, it has been suggested that this func-tion might be composed of two separate components, onlyone of which shows significant changes over the age range ofthis sample.3"

The fitted predictions for optical density variations in thephotopigments indicate a standard deviation of roughly0.045. Only a single density factor was suggested, reflectinga correlated variation in the pigments. In contrast, threeAmax factors were obtained, which represent independentvariations in the three cone types. A somewhat relatedresult with respect to density was found by Smith et al.' 6

They evaluated the range over which the density and thespectral peak of the L and M cones could vary and stillproduce color matches that were within the range of individ-ual differences of the Stiles-Burch 2° matches.3 While theacceptable Xmax values for one cone type depended little onthe other, the fit to the observed variations required that thedensities of the two pigments covary. A common densityvariation could possibly reflect a general variation in the sizeof all the cone outer segments. The standard deviation of0.045 would correspond to roughly a 3-,Mm change in theouter segment lengths.33

In Table 4 we have reproduced our estimates of the stan-dard deviations in each of the spectral peak factors, this timeexpressed in wavelengths (at Xmax). Also shown are severalother psychophysical and physiological measures of the vari-ability in Xmax. Alpern34 has calculated presumed Amax dis-tributions for a population of protanopes and deuteranopes,based on the anomolascope settings made by Alpern andWake35 and Bastian.36 From these we have estimated thatthe standard deviation in the L and M peaks is roughly 1.4

and 2.0 nm, respectively. Using flicker photometry on col-ored adapting backgrounds to try to isolate the cone sensi-tivities, Eisner and MacLeod37 found that their group ofseven normals clustered about peak L cone sensitivities 85cm'1 (2.7 nm) apart, suggesting a standard deviation ofabout half of this amount. Finally, Smith et al.' 6 calculatedthat deviations covering a range of 300 cm-' (9.3 nm) in theL cones and 200 cm-' (5.7 nm) in the M cones yieldedpredicted matches that were within the extremes of theStiles-Burch 20 matches (though they also argued that vari-ation in density with no variation in Xmax provided an equallygood account of the data that they considered).

Baylor et al.3 8 have measured the spectral sensitivities ofthe cones by directly recording the outer segment photocur-rents. The standard deviations that they obtained for theirsample of L, M, and S cones were 1.0, 1.3, and 1.4 nm,respectively. Another approach has been to measure thelight absorption of individual cones by using microspectro-photometry (MSP). The MSP data shown in Table 4 arethe standard deviations in Xmax found from estimates of thesensitivities of both human39 and Old World Monkeycones.40'4' These values, which range from 2 to 5 nm, tend tobe larger than the corresponding psychophysical estimatesas well as those based on recording the cone responses. Ofcourse, the psychophysical estimates reflect the differences

Table 4. Variability (Standard Deviation inNanometers) in the Estimated Spectral Peaks of thePhotopigments Reported by Several Psychophysical

and Physiological Studies

Variability (nm)Study L M S

Psychophysical EstimatesThis study 1.5 0.9 0.8Alpern34 1.4 2.0Eisner and MacLeod3 7 1.3Smith et al.16 9-3 5.5a

Physiological estimatesBaylor et al.38 1.0 1.3 1.4

(Macaca fascicularis data)Dartnall et al. 3.6 3.5 5.2

(human data)39Bowmaker et al. 4.8 4.8

(M. mulatta data)40

MacNichol et al. 2 2 3.5(M. fascicularis data)4'

Measures of the full range.

M. A. Webster and D. 1. A. MacLeod


between large populations of cones in different individuals,while the MSP is based on differences in single receptors.In any case, the standard deviations that we have obtainedsuggest an upper limit on the expected interobserver vari-ance. The standard deviations in peak sensitivity couldconceivably be smaller if the Amax factors included somevariance due to other factors, but they are unlikely to besignificantly larger, as this would exceed the observed vari-ability in the matches. On the other hand, large variabilityof the pigments within a single individual is inconsistentwith the known additivity of color matches, because thematches should then break down as different subpopula-tions are selectively adapted. For example, Nagy et al.4 2

have calculated that a standard deviation of greater than 1.6nm is inconsistent with the constant Rayleigh matches thatthey found for different adapting backgrounds.

Our conclusion that there is less jitter in the Amax for the Mcones than for the L cones is consistent with the results ofNeitz and Jacobs43 but differs from the findings of Alpern.34Nathans et al.' 9 report that many individuals have multiplecopies of the M-pigment gene. If differences in the geneswere the basis of the variation, then simultaneous expressionof these copies would reduce variability in the effective (av-erage) Amax of the M cones. But deuteranomalous observerswho possess apparently normal genes among their copies ofthe M-pigment gene (for example, author DIAM)19 seem tolack the normal M pigment, and this makes the hypothesisof simultaneous expression unlikely.

Several studies have suggested that the variations in Amaxare not normally distributed. With regard to human pig-ments, for example, Dartnall et al.39 noted that the MSPresults for the Amax of both the M and L cones appear to formbimodal distributions (though only the L distribution isstatistically nonnormal). As we mentioned above, Eisnerand MacLeod37 found that the L cone peaks of their observ-ers clustered into two groups, and recently Neitz and Ja-cobs43 argued from an analysis of Rayleigh matches that theL cone spectra in males are bimodally distributed (with aseparation of roughly 3 nm in the peaks). The estimateddistributions of Amax for the Stiles-Burch observers areshown in Fig. 8, with the individual values for the 34 maleand 15 female subjects plotted as downward or upward dis-placed lines. These distributions represent the factor scoresfor each of the AmeX factors obtained from a principal-compo-nents analysis of the item set with the theoretically zero-loading items (which again yielded factors similar to theinitial Varimax factors.) The scores have a mean of 0 and astandard deviation of 1. An individual's score is thus arelative measure of how many standard deviations he fallsfrom the mean. The actual presumed distribution of Amaxcan be obtained by multiplying the scores by the estimates ofthe Amax standard deviations for each factor (in Table 2).

Following Dartnall et al.,' 9 we applied the Shapiro-Wilk44

test to determine whether the factor scores for the Amax andother factors are normally distributed, analyzing all scorestogether as well as the male and female scores separately.However, only the distribution of Amax(M) for males is non-normal by this criterion (W = 0.929 for n = 34, p < 0.05), andin this case there is no obvious clustering of the scores. Thusthe factor scores give no real indication of separate subpopu-lations of each pigment type. We also compared the relativespread in the scores for male and female observers. Suppose

F

M

F

! M

M

-3

L PIGMENT

I I I ^ I I 11 11 111 11 1 Gi1I 1111

M PIGMENT

I l I I I Il 11 II I ll II ll I I II I ~ I I 1

S PIGMENT

__ _____ __I __11 1I . I I III 1-2 -I 0 1 2 3

FACTOR SCORE (standard deviations from mean)

Fig. 8. Individual factor scores for the factors representing Xmaxvariations in the L, M, and S pigments. Each score indicates howmany standard deviations the individual's Xmax fall from the meanfor the group, with positive deviations corresponding to longer-wavelength peaks. For each factor the scores for females have beendisplaced upward, the male scores downward.

that the variations in spectral peak were due to differencesin the protein sequences for the genes coding the pigments.Because the L and M genes are X linked, females have twoalleles for each, which are independently inherited and inde-pendently inactivated in each cone.45 Their effective Xmax

should therefore be roughly the average of the two, and thefemale variance in Amax should be half that of the males. Incontrast, the male and female variances for the autosomal Sgene should be similar. The actual ratio of male/femalevariance in the factor scores was 2.57 for L, 1.03 for M, and1.63 for S. While these appear to differ from the predictionsfor a simple genetic basis for the Amax variation, they are infact not inconsistent with it, because the critical F ratio forevaluating them is itself well above 2 (F = 2.41, df = 33, 14).

Our results for the Stiles-Burch 2° color matches arelargely similar to those that we obtained for the 100 matches.As before, we found three factors that appear to reflectindependent variations in the Amax of the three photopig-ments (though only the L pigment factor was clearly de-fined), and the estimated variability in Xmax was close to thevalues that we estimated from the 100 data. Smith et al.' 6

and Estevez 46 have also suggested that Amex variations mightpartly contribute to the individual differences in these data,though they could not distinguish these from possible photo-pigment density variations. None of the extracted factorsthat we found clearly corresponded to differences in photo-pigment density, and only a weak role of density differenceswas suggested by the fits to the correlation matrix. Thus forboth sets of data Xmax variations appear to be the moresalient determinant of individual differences in the matches.

ACKNOWLEDGMENTS

We thank P. J. J. Clarke of the National Physical Laboratoryfor providing us with the individual color-matching datafrom the study of Stiles and Burch. This study was support-ed by National Institutes of Health grant EY01711.


lulI


* Present address, Department of Experiment Psycholo-gy, University of Cambridge, Downing Street, CambridgeCB2 3EB, UK.

Address reprint requests to D. I. A. MacLeod.

REFERENCES AND NOTES

1. W. D. Wright, Researches in Normal and Defective ColourVision (Kimpton, London, 1946).

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4. D. I. A. MacLeod and M. A. Webster, "Direct psychophysicalestimates of the cone-pigment absorption spectra," J. Opt. Soc.Am. A 5, 1736-1743 (1988).

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6. Because of their greater familiarity, we have tried to use mea-sures of wavelength whenever practical. The wave number isthe reciprocal of the wavelength expressed in centimeters.

7. In the actual study a yellow primary at 588 nm was substitutedfor the green for test wavelengths longer than 588 nm. Further,subjects were run in two separate groups, which differed in theintensity levels used and the location of the blue primary (445.4versus 470.6 nm). The correction factors supplied by Stiles andBurch for slit width differences, etc. between the two groupswere applied to the data to make values for the two sets compa-rable.

8. General treatments of factor analysis are available in numeroustexts. We relied mainly on H. H. Harman, Modern FactorAnalysis (U. Chicago Press, Chicago, Ill., 1960).

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