Modeling the contrast-sensitivity functions of older adults

Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. A 1591

Modeling the contrast-sensitivity functions of older adults

Ann Marie Rohaly* and Cynthia Owsley

Department of Ophthalmology, School of Medicine, Eye Foundation Hospital,University of Alabama at Birmingham, Birmingham, Alabama 35294-0009

Received July 8, 1992; revised manuscript received November 23, 1992; accepted January 11, 1993

To determine whether a parabolic template is a good description of the contrast-sensitivity functions (CSF's)exhibited by older adults, the curve-fitting method of Pelli et al. [J. Opt. Soc. Am. A 3(13), P56 (1986)] was ap-plied to contrast-sensitivity data from 100 older subjects (ages 53-85 years). Although the method resulted inreasonable fits for most subjects, closer inspection revealed that this technique may be problematic. A signifi-cant number of observers had functions that were nonparabolic, and for many subjects the error tended to beconcentrated at the peak of the CSF. In addition, in contrast to the study of Pelli et al., the peak contrast sen-sitivities of the subjects were only weakly related to Pelli-Robson contrast sensitivity and letter acuity. Thedata were also fitted with an asymmetric function of variable shape. Whereas this function provided a betterfit to the nonparabolic CSF's, it resulted in inferior fits to most of the remaining data. These results demon-strate that the spatial CSF's of older adults cannot be described by a single parametric curve such as a parabolaor a function of an exponential and that Pelli-Robson contrast sensitivity and letter acuity are not adequatepredictors of their peak constrast sensitivities.

INTRODUCTION

Many older adults experience visual problems that inter-fere with their daily lives."2 Unlike the standard clinicalmeasure of Snellen acuity, spatial contrast sensitivityhas been shown to be highly correlated with performancein such routine daily activities as reading, mobility, anddriving.3 5 Unfortunately, in a typical clinical setting itis not practical to measure a complete spatial contrast-sensitivity function (CSF) for every patient. The standardtwo-alternative forced-choice psychophysical procedure istime consuming and requires a high degree of trainingand motivation on the part of the subject. Thus it is notideal for the clinical environment.

Because of the predictive power of peak contrast sensi-tivity and the well-documented age-related changes inspatial contrast sensitivity, 6

-8 it would be beneficial to

have a convenient description of CSF's that could be usedin a clinical setting to characterize the eye health of olderadults. At first glance this would seem to be an unrealis-tic goal given the wealth of data in the literature docu-menting a variety of changes in CSF shape with disease,9-"age,6-8 and experimental conditions. 2 -4 A priori it doesnot seem possible to describe an entire CSF parsimo-niously; spatial contrast sensitivity is a function of manyparameters. As testimony to this complexity, a numberof different mathematical descriptions of the spatial CSFhave been proposed in the literature on various theoreticaland empirical bases, including differences of Gaussians' 6"6

and an assortment of functions involving exponentials.7 .2 4

Recently, however, Pelli et al.2 5 reported that a singleparabolic template with only two free parameters was ableto fit the CSF's of 30 low-vision observers exhibiting awide range of visual disorders. In addition, the locationof the peak of the CSF could be predicted from measuresof Pelli-Robson contrast sensitivity and letter acuity,making their technique a prime candidate for the quickmeasurement of spatial contrast sensitivity in a clinical

environment. For this reason the abstract of Pelli et al.25

has generated a great deal of excitement in the clinicalvision community and is frequently cited as validation ofthe claim that the Pelli-Robson chart measures peak con-trast sensitivity.26 The purpose of the present study wasto test whether the results of Pelli et al. with low-visionobservers can be generalized to another group of subjects.Because the elderly constitute as much as 70% of the low-vision population 2 and the majority of eye-clinic visits,we investigated whether a parabolic curve is also a gooddescription of the CSF's exhibited by older adults andwhether their peak contrast sensitivities can be predictedfrom simple clinical measures.

METHODS

SubjectsTwo groups of subjects were tested. Younger adults(n = 57) ranged in age from 18 to 45 years old (mean22 years) and were recruited from the undergraduatesubject pool in the Department of Psychology, Universityof Alabama at Birmingham. Letter acuity for youngersubjects averaged 20/13 (range 20/10 to 20/24), and eyehealth was normal as indicated by an interview on eye-health history. Older adults (n = 100) ranged in age from53 to 85 years old (mean 69 years) and were recruited fromthe Primary Care Clinic of the University of Alabama atBirmingham School of Optometry. Their average letteracuity was 20/25 (range 20/11 to 20/115). All older sub-jects had undergone a standard eye-health examinationwithin 6 months of psychophysical testing that includeddilated ophthalmoscopy, biomicroscopy, applanationtonometry, refraction for near and far distances, ocular-motility assessment, tangent screen, and external ex-amination. The eye-health characteristics of the oldersubjects are listed in Table 1. A range of diagnoses wasrepresented in our older sample; the majority of subjects

0740-3232/93/071591-09$06.00 C 1993 Optical Society of America

A. M. Rohaly and C. Owsley

1592 J. Opt. Soc. Am. A/Vol. 10, No. 7/July 1993

Table 1. Summary of Eye-Health Classificationsfor Older Adult Subjects

Indentifying Description Number of Subjects

Normal ocular health: visual acuity20/25 or better 44

Early cataract: visual acuity worse than20/25 but better than 20/40 16

Cataract: visual acuity 20/40 or worse 13Retinal changes: includes retinal vascu-

lar changes, drusen, pigment changes,age-related macular degeneration,macular edema 9

Background diabetic retinopathy 1Aphakia 1Pseudophakia 1Elevated interocular pressure 1Vision loss after stroke 1Strabismic amblyopia 1Optic atrophy 1Unexplained visual-acuity loss 11

had either normal eye health or some degree of cataract orretinal disease (not atypical for this age group2 8 ).

Procedures

Measurement of the CSF, LetterAcuity, and Pelli-RobsonContrast SensitivityContrast thresholds were measured for vertical sinusoidalgratings generated by a microprocessor and displayed ona television monitor. Spatial frequencies of 0.5, 1.0, 3.0,6.0, 11.4 and 22.8 cycles per degree (c/deg) were tested inorder from the lowest to the highest frequency. Themean luminance of the gratings remained at 100 cd/M2

despite changes in contrast, and the luminance of thesurround was 5 cd/m 2 . Maximum contrast was 0.7 asspecified by the standard Michelson contrast formula(Lmax - Lmin)/(Lmax + Lmin). The gratings subtended5.5 deg (height) X 4.2 deg (width). Subjects viewed thegratings binocularly from a distance of 3.0 m while wear-ing their best distance correction.

Thresholds were determined under computer control bya method of increasing contrast. To reduce stimulusuncertainty, before actual testing for a given spatial fre-quency the subject viewed a suprathreshold preview of thegrating. After the preview the grating was presentedat a randomly determined, near-zero, subthreshold con-trast, and the contrast was gradually increased. Thesubject pressed a button when the grating first becamevisible, and then the contrast returned to a randomly de-termined subthreshold level and a new trial began. Con-strast threshold for a given spatial frequency was definedas the geometric mean of 8-16 threshold determinations(depending on the subject). Before beginning actual datacollection all subjects received practice in this task withthe 0.5-c/deg grating. All data on each subject were col-lected during a single testing session. The duration ofthe entire procedure, including instructions and practice,was 20-30 min.

Letter acuity was measured with the Bailey-Lovie let-ter chart, whose letter size decreases from row to row in0.1-log-unit steps.29 The chart was viewed at a distanceof 4.2 m. The Pelli-Robson chart 26 was used to measure

contrast sensitivity for letters subtending 2.86 deg of vi-sual angle and was viewed at a distance of 1.0 m. Bothcharts had a luminance of 100 cd/M2, as measured fromtheir white backgrounds. Subjects viewed the chartsbinocularly while wearing their best corrections for thetest distance. The duration of testing with the lettercharts was -5 min. The presentation order of the threetasks, CSF, acuity, and Pelli-Robson contrast sensitivity,was counterbalanced across subjects.

Curve FittingThe CSF data of the 100 older subjects were fitted withparabolas according to the method of Pelli et al.2 5 Theirmethod uses linear regression to fit CSF data with a tem-plate curve of the form

S - So = -k(F - FO)2, (1)

where S is log contrast sensitivity, F is log spatial fre-quency, (Fo, So) are the coordinates of the peak, and k isa constant that determines the curvature (shape) of theparabola. Because Eq. (1) is meant to be a fixed template,the parameter k is set to a predetermined value, definingthe curvature of the template, and linear regression isthen used to find the values of Fo and So that minimizethe mean-square error of the fit. In other words, themethod of Pelli et al. is equivalent to sliding a single fixedparabolic curve along the log contrast sensitivity and logspatial-frequency axes to achieve the best fit to the CSFdata for each subject.

In their study, Pelli et al.2 5 determined the value of k byfitting quadratic equations of the form

S = -kiF2 + aF + b, (2)

to the CSF data of normally sighted subjects, using poly-nomial regression and then averaging the individual ki.(Remember that their test subjects were low-vision ob-servers.) Using this method, we determined k from anindependent group of 57 younger subjects. The resultingvalue, 1.309 ( = ±0.314), was used in Eq. (1) to fit theCSF data of the 100 older subjects. The value used byPelli et al., 1.15, corresponds to a parabola of similar cur-vature. Note that, if the assertion of Pelli et al. that allindividuals have the same characteristic CSF shape is true(i.e., the same k), we may obtain the value of k from anysample of subjects. As a test of this conjecture, we ob-tained a second estimate of k by repeating the above proce-dure on the data 'of the 44 older subjects given eye-healthdiagnoses of normal (see Table 1). For this subset of sub-jects the average value of k was 1.230 (vr = ±0.370). Thevalues of Fo and So generated by the parabola-fittingprocedure for the two values of k were not significantlydifferent. Therefore only the results obtained with theindependently determined k, 1.309 (from the younger sub-jects' data), are reported below.

To test the validity of the two-parameter description ofthe CSF given by Pelli et al.,2 5 the data of the older popu-lation were also fitted with two different three-parameterfunctions. The first function is simply Eq. (1) with k as afree parameter, and the second function is

s(f) = Af' exp(-pf), (3)



(a)

. . . . . . .0.2 0.3

rms error (log units)0.4

1 2 -

, 1 0-

no 8-00

'0

6-0.0

E 4-z

2-

0.0

exponential (C)

. . . 1

0.1 0.2 0.3rms error (log units)

0.4

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0.3 0.4

where s is contrast sensitivity, f is spatial frequency, andA, n, and p are parameters determined by curve fitting.This description of the CSF was originally proposed in atheoretical context by Kelly7 and has been shown to pro-vide reasonable fits to CSF data from both humans2 2 -

24

and monkeys.2 4 This particular function was chosen be-cause, unlike the parabola, it is inherently asymmetric andits shape may be varied from the characteristic inverted Uto a curve that is flat at low to mid frequencies (low pass).Both three-parameter functions were fitted to the data ofthe older subjects by an iterative nonlinear regression al-gorithm3 0 that minimized the mean-square error betweenthe fit and the data on log-log coordinates.

RESULTS

For both the parabolic [Eq. (1)] and the exponential[Eq. (3)] curves, we assessed the goodness of fit for eachsubject by computing the rms error between the functiongenerated by the methods described above and the CSFdata, according to the formula

1 = d E[Si - S(F )]2 (4)

where Si is the measured mean log contrast sensitivity,S(Fi) is the value of the fitted function, n is the number ofdata points, and d is the number of degrees of freedom in

Fig. 1. Distributions of rms errors resulting from fitting para-bolic and exponential curves to the CSF data of older observers.(a) Two-parameter parabolic fit from the method of Pelli et al.2 5

[Eq. (1), with k = 1.309]. The mean error is 0.130 log unit (cr =

±0.075), with a range from 0.022 to 0.354 log unit. The blackbars represent the errors for subjects given the eye-health diagno-sis of normal. (b) Three-parameter parabolic fit [Eq. (1), with kas a free parameter]. The mean error in this case is 0.073 logunit (a- = ±0.041), with a range from 0.008 to 0.216 log unit.Thus, by adding another parameter, we reduced the mean error ofthe fit by a factor of 1.78 and the range of error by a factor of 1.60.(c) Three-parameter exponential fit [Eq. (3)]. The mean error is0.174 log unit (oa = ±0.068), with a range from 0.060 to 0.421 logunit. The black bars on the graph represent the errors for the20 subjects with poor two-parameter parabola fits [(a), rms er-ror > 0.200 log unit]. Comparison with (a) reveals that, evenwith an added parameter, the error distribution for the exponen-tial curve is shifted toward larger rms errors.

the curve fit. The results of the two-parameter parabolic-fitting procedure of Pelli et al.2 5 are discussed first. Thehistogram in Fig. 1(a) shows the distribution of rms errorsfor the parabolic fits. The mean error across subjects was0.130 log unit (a = ±0.075), with a range from 0.022 to0.354 log unit. To illustrate the relationship between themagnitude of the rms error and the goodness of fit, thebest- and worst-fitting parabolas are plotted in Fig. 2.

By normalizing the peak coordinates of each observer'sdata, Pelli et al.25 calculated an overall rms error of 0.2 logunit, indicating the goodness of fit of a single standardparabola to the pooled data from all 30 of their low-visionobservers. In our population 80% of the fits had rmserrors below 0.200 log unit [Fig. 1(a)]. The magnitudes ofthese errors can be put into perspective by the fact that,over a spatial-frequency range similar to that of the pres-ent study Rubin3

1 found the average rms error associatedwith test-retest reliability of CSF data measured with aforced-choice psychophysical procedure to be 0.176 logunit.3 2 Although we did not use forced choice there wasrelatively little variability in the data; the standard errorsassociated with each contrast-sensitivity measurementwere almost constant across spatial frequency, averaging0.050 log unit.

In addition, although all the data were measured in asingle session, Gilmore et al.3 showed that the method ofincreasing contrast produces mean contrast sensitivities

hvo-parameter parabola1 2 -

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Fig. 2. Best- and worst-fitting parabolas. In this and all subse-quent figures, error bars represent ±1 standard error of the meanand are shown when they exceed the size of the symbol. Thebest parabolic fit, with an rms error of 0.022 log unit, is shown inthe left-hand graph. The right-hand graph shows the worstparabolic fit, with an rms error of 0.354 log unit. The low-passnature of subject 40's data (constant sensitivity between 0.5 and11.4 c/deg) was responsible for the failure of the parabola to pro-vide an adequate description of the data.

and standard errors that are stable across time, with amean test-retest reliability across spatial frequency of0.70. The test-retest reliability found by Rubin"' for aforced-choice procedure (0.77; Ref. 34) is similar. Thusin test-retest reliability the method of increasing contrastused in this study is comparable with forced-choice meth-ods, and for the majority of subjects the error associatedwith the parabolic fit was within the rest-retest reliabilityof the CSF data itself, as reported by Rubin. Therefore,on the basis of the magnitudes of the rms errors, the pa-rabola provided a reasonable description of the CSF's ofmost older adults.

For the 20 subjects whose ms error was >0.200 logunit, there was no single factor or group of factors distin-guishing them from the remainder of the subject sample.For example, one might expect the oldest subjects or thosewith the worst acuity or most severe vision problems to

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exhibit characteristic changes in CSF shape and thereforeto have the worst-fitting parabolas. We tested this hy-pothesis by performing a multiple-regression analysis inwhich the rms error of the parabolic fit was regressedagainst subject age, acuity, Pelli-Robson contrast sensi-tivity, and eye-health diagnosis. None of these indepen-dent variables was found to be significantly related tothe rms error; all thep values were >0.2, with one as highas 0.99.

To illustrate the lack of correlation among the variousmeasures, the rms errors of the 44 subjects classified ashaving normal eye health (see Table 1) were plotted withblack bars in Fig. 1(a). It can be seen that the errors forthese subjects are evenly distributed along the entirerange of error. In fact, the three worst-fitting parabolaswere for normal subjects. Thus eye health was not re-lated to the goodness of fit of the parabolic template.Likewise, Fig. 3 demonstrates that both age and letteracuity35 were uncorrelated with the magnitude of the rmserror of the fit.

As stated above, because the data of 80% of the subjectscould be fitted with rms errors of <0.200 log unit, onemight conclude that a parabolic template provides a rea-sonable description of the CSF's of most older adults.Closer inspection of the fits, however, revealed that themethod of Pelli et al. may be problematic. A significantnumber of older observers (15%) had functions that werenot accurately represented by a parabola. Examples ofsuch nonparabolic data are shown in Figs. 2 and 4. Thesedata had a notch (Fig. 4, subject 82), were flat over a sig-nificant frequency range (Fig. 2, subject 40), or wereasymmetric about the peak of the CSF (Fig. 4, subject 83).The same patterns of CSF shapes occurred in the youngersample with approximately the same frequency. Thusit seems that these types of CSF simply illustrate therange of different shapes to be expected in the populationat large.

A more disturbing finding, however, is that most of theerror in the fits tended to be concentrated at the peak ofthe CSF. The maximum discrepancy in contrast sensi-tivity between the data and the fitted parabola occurredat the peak of the data for 19% of the 80 subjects with

u-r i-1 ** I I.u . I .

50 60 70 80 90 -0.4 -0.2 0.0

Age (years) Letter

0.2 0.4 0.6 0.8

xcuity (log arcmin)

Fig. 3. Relationships between rms error of the parabolic fit and subject age and between rms error and letter acuity. It is apparentfrom the r2 values of the fitted lines that neither age nor letter acuity35 is a strong predictor of the goodness of fit (rms error) of theparabolic template.



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Fig. 4. Examples of poor fits resulting from nonparabolic data.In both cases shown, the rms error of the parabolic fit is>0.200 log unit. The CSF of subject 82 (left-hand graph) has anotch at 3 c/deg (0.477 log spatial frequency), whereas the CSF ofsubject 83 (right-hand graph) is asymmetric about the peak.

reasonable fits (rms error < 0.200 log unit). In addition,for 25% of these subjects the spatial frequency of the peakof the parabola differed from that of the data by a factor of1.5-2.0. Thus, for these subjects, although the rms errorof the fit was low, implying a reasonable description of thedata, the fit was actually unsatisfactory because of themisplacement of the peak of the CSE Figure 5 shows twoexamples in which the overall error of the fit was accept-able and the parabola fitted the data well at all points ex-cept the peak. For subject 62 (left-hand graph), the peakof the parabolic fit was at the proper spatial frequency butthe peak sensitivity was underestimated by a factor of1.61 (0.207 log unit). In contrast, the peak of subject 51'sdata was incorrectly predicted along both axes. Thespatial frequency was underestimated by a factor of 1.71(0.234 log unit), whereas the sensitivity of the peak wasunderestimated by a factor of 1.28 (0.107 log unit).

In their study, Pelli et al.25 also reported that Pelli-Robson contrast sensitivity was highly predictive of thepeak contrast sensitivity of the parabolic fit on a log-logscale. Figure 6 shows the peak sensitivity of the para-bolic fits as a function of Pelli-Robson contrast sensitivity.In the left-hand graph, Pelli-Robson contrast sensitivitywas obtained with the scoring method proposed by Pelliet al. 2 6 ; contrast sensitivity was taken as the lowestcontrast at which two of three letters in a triplet werecorrectly identified. The sensitivities plotted in theright-hand graph were obtained with the more recentscoring method of Elliott et al.36 in which credit was givenfor each individual letter. Elliott et al. argued that theirscoring method is more reliable and more sensitive thanthe former.

Figure 6 demonstrates that, for our subjects, neitherscoring method resulted in a high degree of correlationbetween Pelli-Robson contrast sensitivity and peak con-trast sensitivity of the fitted parabolic function. Forboth scoring methods r2 values for the best-fitting linesare -0.20, which is significantly different from zero(p < 0.001). For comparison, however, Pelli et al.25 founda linear relationship between these two variables with anr2 of 0.81, a value that is significantly higher (p < 0.001).

Pelli et al. also reported a linear relationship (on a log-logscale) between letter acuity and the spatial frequency ofthe peak of the parabolic fit with an r of 0.54. For oursubjects, the relationship between the two variables hasan r2 of 0.33 (Fig. 7), which is not significantly different(p > 0.05) from the value reported in the previous study.

In an attempt to reconcile the results of these two stud-ies, we discarded the data from the poor-fitting parabolas(rms error > 0.200 log unit) and recomputed the relation-ships between Pelli-Robson contrast sensitivity and peaksensitivity and between letter acuity and peak frequency.Omitting the data from the poor-fitting parabolas didnot increase the r2 values of the fitted lines; in fact, ther2 values decreased slightly for two of the three data sets.Thus we conclude that, for older adults, the sensitivityof the peak of the CSF is not adequately predicted fromPelli-Robson contrast sensitivity, nor is the spatial fre-quency of the peak adequately predicted from letter acuity.

Note, however, that it does not make sense to try to pre-dict the position of the peak of the CSF from measures ofPelli-Robson contrast sensitivity and letter acuity unlessthese two quantities are relatively independent. The r2

for the best-fitting line relating Pelli-Robson contrastsensitivity to letter acuity is 0.36 for the scoring methodof Pelli et al. and 0.26 for the method of Elliott et al. ThusPelli-Robson contrast sensitivity and letter acuity arenot totally independent, but they are somewhat dissoci-able. This result agrees with the findings of Rubin et al.,37

who used the same two charts to measure Pelli-Robsoncontrast sensitivity and letter acuity in a similar groupof subjects.

In light of the problems associated with a two-parameterdescription of the CSF as explained above, we examinedwhether another functional form might provide a betterfit to the CSF data of older adults. The obvious alterna-tive is simply a parabola with three degrees of freedom-in other words, Eq. (1) with k as a free parameter. The

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Fig. 5. Examples of problematic parabolic fits. For both ofthese subjects, the overall rms error of fit is low (<0.200 log unit),but inspection of the fit reveals that the error is concentrated atpeak of the CSF. For subject 62 (left-hand graph), the sensitivityat the peak of the parabola is 0.207 log unit or a factor of 1.6 lowerthan the data. Likewise, the peak of the parabolic fit forsubject 51 (right-hand graph) is 0.107 log unit (a factor of 1.3) toolow in sensitivity and 0.234 log unit (a factor of 1.7) too low inspatial frequency.

biect #82s error = 0.287


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log Pelli-Robson Contrast SensitivityFig. 6. Relationship between Pelli-Robson contrast sensitivity and the peak sensitivity of the parabolic fit. Plots are shown for two dif-ferent scoring methods for the Pelli-Robson chart. The scoring method of Pelli et al.2 6 defines contrast sensitivity as the lowest contrastat which two of three letters of a triplet are correctly identified. In contrast, the method of Elliott et al.36 gives credit for each individualletter that is identified correctly. Neither scoring method results in a strong relationship between Pelli-Robson contrast sensitivity andthe peak sensitivity of the parabolic fit.

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Letter Acuity (log arcmin)Fig. 7. Relationship between letter acuity and the frequencyof the peak of the parabolic fit. For this population of subjectsletter acuity is not highly predictive of the peak frequency.

histogram in Fig. 1(b) shows the distribution of rms errorsfor the three-parameter parabolic fits. The mean erroracross subjects was 0.073 log unit (ur = +0.041), with arange from 0.008 to 0.216 log unit. Thus, by permittingthe curvature of the parabola to vary across subjects, wereduced the mean ms error by a factor of 1.78 and therange of error by a factor of 1.60. An F test comparingthe two- and three-parameter parabolic fits showed thatthe three-parameter fit was significantly better at the0.01 level [F(99,300) = 4.46]. This result contradicts theclaim of Pelli et al.2 5 that the CSF has a single stereotypedshape because the best-fitting k ranged from 0.260 to2.226 for our sample of subjects.

As a second alternative description of the CSf the expo-nential function given in Eq. (3) was particularly attrac-tie in that it has certain features that the parabola lacks.For example, it is inherently asymmetric, and its shape maybe varied from the characteristic inverted U to a curve

that is flat at low to mid spatial frequencies (low pass).Inspection of the data for which the two-parameter para-bolic fits were poor indicated that these features of the ex-ponential function (asymmetry and variable shape) werenecessary to provide a reasonable description of the data.

The histogram in Fig. 1(c) shows the distribution of rmserrors for the exponential fits. In this case the meanerror across subjects was 0.174 log unit (- = ±0.068), witha range from 0.060 to 0.421 log unit. Comparing this dis-tribution of errors with that for the parabolic fits inFigs. 1(a) and 1(b), one can see that the distribution forthe exponential function is shifted toward larger errors.In fact, even with an extra degree of freedom the ex-ponential fit was superior to that of the two-parameterparabola for only 27 of the 100 subjects. Of these 27,however, ten corresponded to data sets previously labeledas nonparabolic and 13 corresponded to data sets forwhich the rms error of the parabolic fit was >0.200 logunit. The errors for all 20 subjects with parabolic-fiterrors >0.200 log unit have been plotted with black barsin Fig. 1(c).

Two examples of nonparabolic data for which the expo-nential function gave an excellent fit are plotted in Fig. 8.Subject 40 (left-hand graph) had the worst-fitting two-parameter parabola, with an rms error of 0.354 log unit, asshown in the right-hand graph of Fig. 2. The exponentialfit, on the other hand, had an rms error of only 0.069 logunit. The superior fit was a direct consequence of thefunction's ability to assume a low-pass shape. Likewise,the improvement in describing the data of subject 83,shown in the right-hand graphs of Figs. 4 and 8, was dueto the asymmetry in the exponential curve.

As noted above, however, for our sample of subjects as awhole the parabola resulted in more-accurate descriptionsof the data, based on the magnitude of the rms errors.Some insight into the reasons behind this finding may begained by examining the data and fits for subject 54, plot-ted in Fig. 9. The two-parameter parabolic fit for thissubject, shown in the left-hand graph of the figure, was

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very good, with an rms error of 0.068 log unit. The expo-nential fit, on the other hand, shown in the right-handgraph, was considerably worse, with an rms error of0.207 log unit. As can be seen from the figure, it wasthe asymmetry of the exponential curve that caused thepoorer fit.

DISCUSSION

We have found that, although a two-parameter parabolictemplate provided reasonable fits to the CSF data of mostolder adults, a significant number of subjects had non-parabolic data, exhibiting notches, flat regions, or asym-metry about the peak. Obviously, this range of featurescannot be accurately represented by a fixed template.These patterns of data are not associated with any par-ticular subject characteristic. Rather, they represent therange of CSF shapes that were present in the populationat large. In addition, the parabolic template's failure toprovide a reasonable description of an individual CSF wasnot predictable on the basis of the age, letter acuity, or eyehealth of the subject (Figs. 1 and 3) as determined bymultiple-regression analysis.

We also found that the curve-fitting method of Pelliet al.25 tends to concentrate the error of the two-parameterparabolic fit at the peak of the CSF This finding is moreproblematic in the sense that the rms error of the fit canbe low, implying a reasonable description of the data,whereas the predicted coordinates of the peak of the CSFcan be off by a factor of 1.5 or more (Fig. 5), makingthe fit unacceptable. The fact that errors tend to concen-trate at the peak limits the practical usefulness of a para-bolic CSF template. Rubin and Legge3 and Marron andBailey4 have argued that, compared with other points onthe CSF, the peak is the strongest predictor of perfor-mance on everyday tasks. Without an accurate estimateof the position of the peak this predictive power is lost.

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0

.0-

0

0.5-


S

0.0 0 I I. I I I

-0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 1.5

log spatial frequency (log c/deg)Fig. 8. Examples of data for which the exponential curve pro-vided a superior fit as compared with the parabola. Subject 40(left-hand graph) had the worst-fitting parabola, with an rms er-ror of 0.354 log unit (Fig. 2, right-hand graph). Because the ex-ponential curve is able to assume a low-pass shape, it fits the datawith an rms error of only 0.069 log unit. Similarly, the parabolicfit to the data of subject 83 has an rms error of 0.247 log unit(Fig. 4, right-hand graph). The error for the exponential fit(right-hand graph), however, is only 0.081 log unit because of theinherent asymmetry of the function.

2.5-parabolic fit 2.5- exponential fitrms error = 0.068 rms error = 0.207

2.0 2.0-

3

-0.5 1.5

cr p

w.0e 0.00.50 '0 1. 0.5 0 . .

log spatial frequency (log C/deg')

Fig. 9. Illustration of a typical subject for which the exponentialcurve provides a poorer fit than does the parabola. The data ofsubject 54 are plotted along with the parabolic fit on the left andthe exponential fit on the right. The parabola provides a good fitto the data, with an rms error of 0.068 log unit. The error for theexponential fit, however, is significantly larger, 0.207 log unit,because the function is more asymmetric than are the data.

A parabolic function whose curvature was permitted tovary from subject to subject was found to provide a signifi-cantly better description of the data. This result contra-dicts the claim of Pelli et al.25 that the CSF has a singlestereotyped shape. The best-fitting k value, denoting thecurvature of the parabola, was found to range across oursubject sample by almost a factor of 10.0. Although therms errors of the fits with the three-parameter parabolaare all quite low [Fig. 1(b)], this description of the CSF isalso inadequate. Theoretically, the spatial frequency forwhich contrast sensitivity is 1.0 (i.e., log contrast sensitiv-ity is 0.0) should be related to an individual's high-contrastletter acuity. When these frequencies were calculatedfrom the equations of fitted parabolas, however, 47% wereabove 60 c/deg. In fact, many were above 100 c/deg, witha few between 350 and 500 c/deg. The predictions werenot uniformly high, however; there was no correlation be-tween predicted and measured letter acuity (r2 = 0.04).In other words, these fitted curves are valid only in theneighborhood of the CSF data used to generate them (0.5-22.8 c/deg); they cannot be extrapolated to imply other as-pects of the subjects' visual systems, thereby limiting theclinical applicability of this type of analysis.

Unfortunately, contrary to our expectations, the expo-nential function proposed by Kelly'7 did not provide a bet-ter overall description of the older adults' CSF data. Thisfunction seemed promising in that it is asymmetric andits shape can vary from low pass to bandpass. These arefeatures that the parabola lacks, and they seemed to benecessary for describing a number of data sets. Whereasthe use of this function resulted in dramatic improvementsin fitting data sets that were obviously nonparabolic inshape (Fig. 8), the function was too asymmetric to provideadequate fits to the majority of the remaining data sets(Fig. 9). Together the above results imply that the CSF'sof older adults cannot be represented by a single para-metric curve of the form given by Eq. (1) or (3).

Furthermore, in contrast with the results of Pelli et al.,25

Pelli-Robson contrast sensitivity and letter acuity werenot highly predictive of the coordinates of the peaks of the

2.5- Subject #83~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Subject #83rms error = 0.081

2.5 -

2.0-

1.5 -

1.0-

0.5 -



parabolic fits. Pelli et al. reported an r of 0.81 for therelationship between Pelli-Robson contrast sensitivityand sensitivity at the peak of the parabola. We found anr2 of 0.20 (Fig. 6), a value that is significantly lower thanthe r2 of Pelli et al.

This result is not surprising in light of the discrepancybetween the peaks of the parabolas and the original CSFdata. If the true peak of the CSF is not accurately repre-sented by the parabola, there is no reason to expect thatPelli-Robson contrast sensitivity will predict the peak sen-sitivity of the parabolic fit. The more relevant questionthen is, "How much correlation should we expect?" As 96of our 100 subjects had peak contrast sensitivities at eitherthe 3.0- or 6.0-c/deg points, correlations were determinedfor regression lines fitted to Pelli-Robson contrast sensi-tivity and the actual CSF data at these two spatial fre-quencies. For contrast sensitivity at 3.0 c/deg, r2 is 0.23,and for 6.0 c/deg it is 0.25. Thus the same degree of cor-relation exists between the peak contrast sensitivity of thefitted parabola and Pelli-Robson contrast sensitivity asbetween the raw data itself and this measure. This resultis clearly contrary to the claim of Peli et al.26 that the mea-sure of visual function obtained from the Pelli-Robsonchart is strongly related to (although not equal to) peakcontrast sensitivity measured with sinusoidal gratings.

Another explanation for the difference in correlation be-tween Pelli-Robson contrast sensitivity and peak contrastsensitivity is that our low correlation is due to the rela-tively small range of Pelli-Robson contrast sensitivityspanned by our subjects. The range of values reported byPelli et al.25 is -1.0 log unit greater than that reportedhere. This greater range would tend to increase the valueof r. To correct for this effect, the r for the fitted lineof Pelli et al. was recalculated for a restricted range ofPelli-Robson contrast-sensitivity values comparable to therange spanned by our older subjects.3' For this restrictedrange, r2 is 0.56, a value that is still significantly higherthan the value of 0.20 found in the present study(p < 0.03). However, this statistical test assumes thatboth subject samples are drawn from the same underlyingnormal distribution. Mean Pelli-Robson contrast sensi-tivity for the subject sample of Pelli et al. (on the re-stricted range) is 1.37 but is 1.65 for our sample. Thusthe difference in correlation may simply be due to the factthat the subjects from the two studies do not belong to thesame underlying population.

In summary, the results of the mathematical modelingpresented above demonstrate that the spatial CSF's ofolder adults cannot be described by a parabola or a func-tion of an exponential and that Pelli-Robson contrast sen-sitivity and letter acuity cannot be used to predict theposition of the peak of the CSF in a significant number ofolder observers.

ACKNOWLEDGMENTS

We thank Tiffany Threlkeld and Pam Alverson for collect-ing the data and the University of Alabama at BirminghamSchool of Optometry for assisting with subject recruit-ment. We also thank Gary Rubin and Denis Pelli for help-ful discussions and for providing unpublished details oftheir research and Al Ahumada for invaluable assistancein performing the statistical analyses. A. M. Rohaly was

supported by a Helen Keller Eye Research FoundationPostdoctoral Fellowship. Additional support for thisproject was provided by National Institutes of Healthgrant AG04212 and a Research to Prevent Blindnessdepartment development grant. Tiffany Threlkeld wassupported by the National Institutes of Health MinorityHigh School Student Research Apprenticeship Program(RRO3001-09).

*Present address, U.S. Army Research Laboratory,Human Research and Engineering Directorate, ATTN:AMSRL-HR-SD, Aberdeen Proving Ground, Maryland21005.

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8. K. E. Higgins, M. J. Jaffe, R. C. Caruso, and F deMonasterio,"Spatial contrast sensitivity: effects of age, test-retest, andpsychophysical method," J. Opt. Soc. Am. A 5, 2173-2180(1988).

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11. A. Atkin, I. Bodis-Wollner, M. Wolkstein, A. Moss, andS. Podos, 'Abnormalities of central contrast sensitivity inglaucoma," Am. J. Ophthalmol. 88, 205-211 (1979).

12. F L. van Nes and M. A Bouman, "Spatial modulation transferfunction in the human eye," J. Opt. Soc. Am. 57, 401-406(1967).

13. J. G. Robson and N. Graham, "Probability summation andregional variation in contrast sensitivity across the visualfield," Vision Res. 21, 409-418 (1979).

14. F W Campbell and D. G. Green, "Optical and retinal factorsaffecting visual resolution," J. Physiol. (London) 181, 576-593 (1965).

15. A. M. Rohaly and G. Buchsbaum, "Inference of global spatio-chromatic mechanisms from contrast sensitivity functions,"J. Opt. Soc. Am. A 5, 572-576 (1988).

16. A. M. Rohaly and G. Buchsbaum, "Global spatiochromaticmechanism accounting for luminance variations in contrastsensitivity functions," J. Opt. Soc. Am. A 6, 312-317 (1989).

17. D. H. Kelly, "Spatial frequency selectivity in the retina,"Vision Res. 15, 665-672 (1975).

18. C. A. Burbeck and D. H. Kelly, "Spatiotemporal characteris-tics of visual mechanisms: excitatory-inhibitory model,"J. Opt. Soc. Am. 70, 1121-1126 (1980).



19. D. H. Kelly, "Spatiotemporal variation of chromatic and ach-romatic contrast thresholds," J. Opt. Soc. Am. 73, 742-750(1983).

20. F. W Campbell, R. H. S. Carpenter, and J. Z. Levinson, "Visi-bility of aperiodic patterns compared with that of sinusoidalgratings," J. Physiol. (London) 204, 283-298 (1969).

21. D. J. Ulrich, E. A. Essock, and S. Lehmkuhle, "Cross-speciescorrespondence of spatial contrast sensitivity functions,"Behav. Brain Res. 2, 291-299 (1981).

22. H. R. Wilson and S. C. Giese, "Threshold visibility of fre-quency gradient patterns," Vision Res. 17, 1177-1190 (1977).

23. H. R. Wilson, "Quantitative prediction of line spread func-tion measurements: implications for channel bandwidths,"Vision Res. 18, 493-496 (1978).

24. J. A. Movshon and L. Kiorpes, 'Analysis of the development ofspatial contrast sensitivity in monkey and human infants,"J. Opt. Soc. Am. A 5, 2166-2172 (1988).

25. D. G. Pelli, G. S. Rubin, and G. E. Legge, "Predicting the con-trast sensitivity of low-vision observers," J. Opt. Soc. Am. A3(13), P56 (1986).

26. D. G. Pelli, J. G. Robson, and A. J. Wilkins, "The design ofa new letter chart for measuring contrast sensitivity," Clin.Vision Sci. 2, 187-199 (1988).

27. C. Kirchner and R. Peterson, "The latest data on visual dis-ability from NCHS," J. Vis. Impair. Blind. 73, 151-153 (1979).

28. H. M. Leibowitz, D. E. Krueger, L. R. Maunder, R. C. Milton,M. M. Kini, H. A. Kahn, R. J. Nickerson, J. Pool, T. L. Colton,J. P. Ganley, J. I. Loewenstein, and T. R. Dawber, "The Fram-ingham Eye Study monograph," Surv. Ophthalmol. Suppl. 24,335-610 (1980).

29. F L. Ferris III, A. Kassoff, G. H. Bresnick, and I. L. Bailey,"New letter acuity charts for clinical research," Am. J. Oph-thalmol. 94, 91-96 (1982).

30. "Igor" (WaveMetrics, Lake Oswego, Ore., 1990). Igor uses theLevenberg-Marquardt algorithm to search the multidimen-sional error space.

31. G. S. Rubin, "Reliability and sensitivity of clinical contrastsensitivity tests," Clin. Vision Sci. 2, 169-177 (1988).

32. We calculated this value by averaging the square roots of themean-square errors for all frequencies and subjects reportedin the column labeled CRT-based test, MSw in Table 2 ofRef. 31.

33. G. C. Gilmore, C. G. Andrist, and F L. Royer, "Comparison oftwo methods of contrast sensitivity assessment with youngand elderly adults," Opt. Vis. Sci. 68, 104-109 (1991).

34. We calculated this value by averaging the intraclass correla-tion coefficients for all frequencies and subjects reported inthe column labeled CRT-based test, ri in Table 2 of Ref. 31.

35. In this and subsequent figures involving letter acuity or Pelli-Robson contrast sensitivity, the number of subjects listed (n)is one less than the total number of subjects tested. One ofthe subjects (#51) had a visual agnosia for letters that pre-cluded measurement of both letter acuity and Pelli-Robsoncontrast sensitivity. The agnosia, however, did not preventus from obtaining a CSF and fitting those data with a parab-ola (Fig. 5) and an exponential curve.

36. D. B. Elliott, M. A. Bullimore, and I. L. Bailey, "Improvingthe reliability of the Pelli-Robson contrast sensitivity test,"Clin. Vision Sci. 6, 471-475 (1991).

37. G. S. Rubin, I. A. Adamsons, and W J. Stark, "Comparison ofacuity, contrast sensitivity and disability glare before andafter cataract surgery," Arch. Ophthalmol. 111, 56-61 (1993).

38. Q. McNemar, Psychological Statistics, 4th ed. (Wiley, NewYork, 1969), Chap. 10, p. 162.


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Modeling the contrast-sensitivity functions of older adults

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