article aniseikonia tests: the role of viewing mode ...eprints.ucm.es/35781/1/aniseikonia tests the...

DOI 101167tvst439

Article

Aniseikonia Tests The Role of Viewing Mode ResponseBias and SizendashColor Illusions

Miguel A Garcıa-Perez1 Eli Peli2

1 Departamento de Metodologıa Facultad de Psicologıa Universidad Complutense Campus de Somosaguas Madrid Spain2 The Schepens Eye Research Institute Massachusetts Eye and Ear Department of Ophthalmology Harvard Medical School Boston MAUSA

Correspondence Miguel A Garcıa-Perez Departamento deMetodologıa Facultad de PsicologıaUniversidad Complutense Campusde Somosaguas 28223 MadridSpain e-mail miguelpsiucmes

Received 23 January 2015Accepted 23 April 2015Published 12 June 2015

Keywords aniseikonia eye move-ments size perception vernieracuity sizendashcolor illusion

Citation Garcıa-Perez MA Peli EAniseikonia tests the role of viewingmode response bias and sizendashcolorillusions Tran Vis Sci Tech 20154(3)9 doi101167tvst439

Purpose To identify the factors responsible for the poor validity of the most commonaniseikonia tests which involve size comparisons of redndashgreen stimuli presentedhaploscopically

Methods Aniseikonia was induced by afocal size lenses placed before one eyeObservers compared the sizes of semicircles presented haploscopically via color filtersThe main factor under study was viewing mode (free viewing versus shortpresentations under central fixation) To eliminate response bias a three-responseformat allowed observers to respond if the left the right or neither semicircleappeared larger than the other To control decisional (criterion) bias measurementswere taken with the lens-magnified stimulus placed on the left and on the right Tocontrol for sizendashcolor illusions measurements were made with color filters in botharrangements before the eyes and under binocular vision (without color filters)

Results Free viewing resulted in a systematic underestimation of lens-inducedaniseikonia that was absent with short presentations Significant sizendashcolor illusionsand decisional biases were found that would be mistaken for aniseikonia unlessappropriate action is taken

Conclusions To improve their validity aniseikonia tests should use shortpresentations and include control conditions to prevent contamination fromdecisionalresponse biases If anaglyphs are used presence of sizendashcolor illusionsmust be checked for

Translational relevance We identified optimal conditions for administration ofaniseikonia tests and appropriate action for differential diagnosis of aniseikonia in thepresence of response biases or sizendashcolor illusions Our study has clinical implicationsfor aniseikonia management

Introduction

Aniseikonia is a binocular condition by which theperceived size of an object differs between the eyesoften caused by the retinal images of the objectdiffering in size between the eyes Different retinalimage sizes occur naturally in near vision for objectslocated at meaningfully different distances from eacheye12 but differences in perceived size can also be dueto anatomical optical or neural differences betweenthe eyes3 Within a range optical correction ofmeasured aniseikonia is feasible Early instrumentsdesigned to measure aniseikonia were the Ophthalmo-Eikonometer (OE)45 and the Space Eikonometer

(SE)6 which are complex instruments including ahead rest rigidly fixed cells for trial lenses and adisplay device The OE does not seem to have everentered production whereas production of the SEwas discontinued decades ago These instruments arenowadays rarely available in clinical practice

Simpler aniseikonia tests were proposed soonafterward78 whose use is rarely reported and whosevalidity was to our knowledge never established Analternative whose use is often reported is the NewAniseikonia Test (NAT)910 The NAT uses a printedset of red and green semicircles (anaglyphs) ofdifferent relative sizes that patients view haploscopi-cally with redndashgreen glasses to select the pair in whichboth semicircles are perceptually equal in size The

1 TVST j 2015 j Vol 4 j No 3 j Article 9

Downloaded From httptvstarvojournalsorgpdfaccessashxurl=dataJournalsTVST933932 on 10132016

booklet where anaglyphs are printed (at six per page)is held at reading distance and no complex instrumentis required to deliver the stimuli A validity studyshowed that the NAT underestimates natural anisei-konia as well as aniseikonia induced with size lenses(henceforth lens-induced aniseikonia) by a factor ofapproximately 311 The underestimation was con-firmed later1213 but its cause remained unknown Atiny part of it can be attributed to an unnoticed designerror in the NAT the Plus Aniseikonia Series wascreated by reducing the size of the test (green)semicircle by some specified percentages and thesereductions are incorrectly taken directly as theamount of aniseikonia If the pair is selected in whichthe test semicircle is x smaller the amount ofaniseikonia is not x because an image reduced by xmust be enlarged instead by y to regain its originalsize with yfrac14 100x(100 x) The resultant nonlinearunderestimation y x varies with x as x2(100 x)which is less than the reported amount (eganiseikonia measured as 10 with the NAT wouldactually be 1111 which represents underestimationby a factor of only 11)

A more recent alternative is the Aniseikonia Inspec-tor (AI)14 a test based on anaglyphs similar to thosemaking up the NAT but administered via computerThree versions of the AI were developed over the yearsVersion 1 reproduced the video test of McCormack etal11 where unlimited time is given tomanipulate the sizeof one of the semicircles until it matches subjectively thesize of the other In Version 2 semicircles were replacedwith rectangles unlimited inspection time was replacedwith short presentations and the method of adjustmentwas replaced with a task in which observers indicatewhether the left the right or neither rectangle issubjectively larger than the other The measure ofaniseikonia then arises from fitting a two-parameterpsychometric function to the data via treatment oflsquolsquoequalrsquorsquo responses as half lsquolsquoleftrsquorsquo and half lsquolsquorightrsquorsquo FinallyVersion 3 defaults back to unlimited presentation timealthough the option for short presentations is stillavailable with no other major changes

Studies on the validity of the AI offer conflictingresults In principle the video test11 motivating the AIVersion 1 proved valid In the same line de Wit14

reported accurate measurements of lens-induced anis-eikonia in the horizontal vertical and oblique meridiawith the AI Version 1 Analogous results were reportedfor a custom-made computerized test15 that resemblesthe video test11 and the AI Version 1 In contrastRutstein et al16 reported for the AI Version 1 ameaningful underestimation with respect to the expect-

ed amount of lens-induced aniseikonia and also withrespect to measurements obtained with the SE in thehorizontal and vertical meridia in the light and in thedark This underestimation was lower in magnitudethan that reported byMcCormack et al11 for the NATthe slope of the relation of measured to lens-inducedaniseikonia with the AI Version 1 was 067 in thevertical meridian and 058 in the horizontal meridianUnderestimation of lens-induced aniseikonia with theAI Version 1 has been reported in other studies17ndash19

Version 2 on the other hand seemed to overestimateaniseikonia instead18 Finally the only published studyemploying Version 3 has reported acceptably accuratemeasurements of lens-induced aniseikonia in the shortpresentation mode with central fixation20

Misestimation of aniseikonia has been claimed to beclinically irrelevant21ndash23 because further checks can beconducted before or after prescribing iseikonic specta-cle lenses although the process is lengthy and may beexpensive Yet identifying the cause of the underesti-mation in the NAT and the AI Version 1 as well as thecause of the conflicting patterns of misestimationacross versions of the AI is a problem whose solutioncould render improved tests that save time and costs topatients in clinical practice also helping to preventanalogous problems that might arise in the use ofpsychophysical methods for the assessment of othervisual functions Identifying the causes of underesti-mation of the NAT and the AI (and by contrast thereasons that the OE and the SE provide accuratemeasurements) can also contribute to a better under-standing of the perceptual aspects of aniseikonia andthus lead to better treatment options

The goal of this research was to identify the causesof these patterns of misestimation and to identify theconditions (method stimuli viewing mode etc) foraccurate measurements of aniseikonia The protocolfor data collection used in this study is more thoroughthan those in the NAT or the AI and permits theseparation of sensory and decisional influences (crite-rion bias) on performance Also data analyses arecarried out separately for each observer as group datatypically obscure rather than clarify the outcomes24

The next three sections describe the reasons justifyingthe choices made in the design of the study namelyviewing conditions stimulus characteristics and psy-chophysical procedure

Viewing Conditions

Comparative judgments of the size of two objectscan be made using direct perceptual estimates of sizeor using indirect correlates of size A simple demon-


Garcıa-Perez and Peli


stration will highlight some issues that are relevant toour subsequent discussion Looking through an afocalsize lens readers can see for themselves the perceptualeffects described next The demonstration involvesclosing one eye and placing the size lens stably beforethe other eye in such a way that it magnifies only theright side of the scene The picture in Figure 1a showsthe resultant percept when the optical axis of the sizelens projects onto a point near the vertical center ofthe sheet on the wall In these conditions the right sideof the sheet seems vertically larger and importantlymoving the eyes up or down does not alter this perceptas long as the optical axis of the size lens projects ontothe same point In other words changes of visual axis(determined by the position of the eyes) do not alterthe appearance of the percept which is only deter-mined by the optical axis of the (external) size lens Ifthe optical axis of the lens aims instead at the top or atthe bottom edges of the sheet the percept will changeas illustrated in Figures 1b or 1c the top edge or the

bottom edge appears aligned across left and right sidesof the scene whereas the other edge appears separatedEye movements do not alter these percepts either Thisillustration shows that (isotropic) optical magnifica-tion can be described as occurring around a center ofexpansion determined by the projection of the opticalaxis of the magnifier onto the object plane At thecenter of expansion the magnified and unmagnifiedsides of the scene appear perceptually aligned Thecenter of expansion readily identified in this illustra-tion has bearings on size comparisons when magnifiedand unmagnified images are each seen with one eyebut sharing some spatial anchor as in stimuli used incurrent aniseikonia tests

Now consider the conventional haploscopic con-ditions created in the NAT or the AI (in which eacheye sees only one of the objects whose sizes are to becompared) in a situation in which aniseikonia isnonmeridional and caused by optical factors residingin or on the eye (so that optical and visual axes as

Figure 1 Illustration of the center of expansion of a magnifying lens determined by the projection of its optical axis onto the objectplane Viewing is assumed to be monocular with the size lens placed stably before the eye in a way that it magnifies only the right side ofthe scene (a) With the lens aiming at the vertical center of the sheet on the wall (upper sketch not to scale) expansion of the right side ofthe scene occurs outward from that point so that the top and bottom edges of the sheet on the right side appear to extend verticallybeyond the corresponding edges on the left side (b) With the lens tilted so that it aims at the top edge (upper sketch not to scale) the topedge appears aligned on both sides of the scene and the expansion of the right side makes the bottom edge appear to be located fartherdown in comparison with the same edge on the left side (c) With the lens tilted to aim at the bottom edge instead (upper sketch not toscale) the opposite effect occurs and now the bottom edges appear aligned on both sides whereas the top edge is misaligned acrosssides of the scene




defined above move together when the eyes move)Equivalently consider aniseikonia caused by specta-cle lenses placed outside the eyes but in a situation inwhich eye movements are accompanied by adjust-ments of head position so that optical and visual axesagain move together Although optical and visualaxes could indeed be dissociated when aniseikonia iscaused by external lenses (giving rise to the distinctionbetween static and dynamic aniseikonia)25 coordi-nated eyendashhead movements are a natural occurrencepresumably aimed at bringing the eyes back to theprimary position of gaze26

Figure 2 illustrates the effects that changes inbilateral fixation location that also displace the centerof expansion have when the stimuli are rectangles for ahypothetical patient with thorn10 aniseikonia affecting

the haploscopic perception of the right rectangleWhen the rectangles are physically identical (Fig 2a)fixation at the geometrical center of the configurationresults in the right rectangle appearing in peripheralvision to extend beyond the top and bottom edges ofthe left rectangle (Fig 2b) if the patient then gazes tothe top of the configuration to confirm the impressionobtained peripherally both rectangles would appearaligned at their top (Fig 2c) evidence of alignment atthe bottom would also be obtained when the patientgazes to the bottom of the configuration (Fig 2d) Apatient who scans the stimulus to gather the evidenceneeded to make a size judgment will have a hard timereconciling the conflicting perceptual informationobtained peripherally and foveally across fixationlocations the right rectangle seems larger peripherally(Fig 2b) but the top and bottom edges of bothrectangles are foveally perceived to be aligned (Figs2c d) Conflicting perceptual information arises also ina configuration that physically compensates foraniseikonia (Fig 2e) In this case rectangles wouldappear equal in length when fixation is on the center ofthe configuration (Fig 2f) but fixations at theextremes would reveal foveally that the top andbottom edges of the right rectangle are located inwardsrelative to the edges of the left rectangle (Figs 2g h)

The preceding discussion assumed that the per-ceived size of the right rectangle is 10 larger than thatof an identical rectangle on the left Similar consider-ations apply when the right rectangle has instead asmaller perceived size (ie if aniseikonia is 10)except that the strength of the conflicting evidence maydiffer In general such strength varies with the verticalangular subtense of the rectangles If angular subtenseis small peripherally obtained information would memore reliable perceptually and thus fixation at thecenter (Fig 2b) would provide dependable peripheralevidence of a larger size for the right rectangle whereasfixation at the extremes (Figs 2c d) would revealalignment at the fixated extreme along with differencesin size at the other extreme seen farther peripherally Ifthe angular subtense is instead large central fixations(Fig 2b) may simply give a weaker peripheralimpression of differences in size whereas fixations atthe extremes (Figs 2c d) will only provide evidence ofalignment at the foveated extreme The other extremefalls too far out into the periphery to provide usable orreliable evidence of size

It is unclear how observers will handle thisconflicting information to make their judgmentswhether they will use a consistent criterion acrosspresentations or whether all observers will handle it

Figure 2 Perceived image under aniseikonia varies with fixationlocation when changes in fixation also relocate the optical axisThis illustration assumes a 10 magnification altering thehaploscopically perceived size of the right rectangle (a) Samplephysical stimulus in which the two rectangles have the same size(bndashd) Perceived images when fixation location varies as indicatedby the crosses (e) Sample physical stimulus in which the rightrectangle is shorter so as to become equal in length whenmagnified by 10 (fndashh) Perceived images when fixation locationvaries as indicated by the crosses Fixation at the center of theconfiguration conveys size information which must be judgedwith peripheral vision fixation at the top or bottom edges of theconfiguration conveys alignment information and discloses whatthe physical reality of the stimulus is at the foveated points whichcan be translated into size judgments that will contradict thejudgments arising from central fixations At the same time andprovided the physical width of both rectangles is identical themagnified rectangle also has a larger perceived width whichfurther complicates judgments of size




identically If observers relied more on global sizejudgments gathered via peripheral information uponcentral fixations aniseikonia could be measured asaccurately as the testing conditions and psychophys-ical procedure allow yet if they inferred size usingalso alignment information gathered via fovealevidence upon fixations at the edges aniseikoniawould be underestimated because judgments of sizewould be biased toward the physical reality of thestimulus It is interesting to note that underestimationof lens-induced aniseikonia has been reported for thetwo tests (the NAT and the AI Version 1) in whichpresentation time is unlimited and observers areallowed (or requested) to scan the configurationswithout restraining head movements or otherwiseensuring a fixed center of expansion (which wouldcertainly vary with head movements as size lenses inthose validity studies were placed on a trial frameworn by observers) In these conditions test resultswill be greatly affected by whether or not observersrsquoscanning strategies keep a fixed center of expansionThe left-eye and right-eye images are perceptuallymisaligned when the center of expansion is not at thefixation point something that has been incorrectlyreferred to as fixation disparity14 The AI includesfunctionality to shift the left- and right-eye stimulirelative to one another to restore perceived alignmentbut this strategy is effective only until the center ofexpansion is relocated again by head movements

This research tested the conjecture that viewingconditions are a major determinant of the accuracy ofsize comparisons used to measure aniseikonia causedby optical factors Data were thus collected inconditions in which presentation time was very short(so as to prevent observers from scanning the stimulusand changing the center of expansion) or unlimited andwith instructions to scan the stimulus and check foralignment (so as to bring in the effects of changes in thecenter of expansion) It should be noted that measure-ments taken with the OE or the SE are free of theseeffects for two reasons First the center of expansionis fixed in these instruments lenses are placed in cellsthat are rigidly attached to the instrument which inturn keeps a fixed position relative to the stimulusplane Second measurements are not based onjudgments of size but on the consequences of differentimage sizes on perceived alignment (in the OE)45 or ondistortion of binocular space perception (in the SE)627

This may explain why measurements of lens-inducedaniseikonia with the SE are more accurate than thoseobtained with the NAT or with the AI

Stimulus Shape Color and Binocular Fusion

The semicircular stimuli used in the NAT and inthe AI Version 1 have two desirable characteristicsOne is that semicircles of different diameters arescaled in both spatial dimensions Thus overall andisotropic magnification of a small semicircle brings itssize and shape to match exactly those of a largersemicircle The second characteristic is that twosemicircles facing each other do not superimpose bydilation and translation making binocular fusion ofthe haploscopic percept impossible

Rectangular stimuli in Version 2 and Version 3 ofthe AI lack these characteristics Magnification of thephysically shorter rectangle will make its length matchthat of the other rectangle but their widths will thendiffer some observers may thus be tempted to maketheir judgments weighing also the perceived widths ofthe rectangles perifixation rather than using onlyperceived lengths judged with peripheral vision Useof this joint criterion would result in overestimationas there would be a bias toward the configuration inwhich rectangle widths are perceptually identical (iefoveal evidence of larger width would not becountered until there is compelling peripheral evi-dence of shorter length) Interestingly a meaningfuland often significant overestimation of aniseikoniahas been reported for the AI Version 218

Two further complications arise from the use ofanaglyphs One of them is the sizendashcolor illusion theperceived size of two physically identical areas varieswith their hue28ndash32 The other is the related phenom-enon that hue and brightness affect perceived depthand indirectly perceived size33ndash35 These two percep-tual phenomena can masquerade as aniseikonia whenredndashgreen anaglyphs are used differences in theperceptual (post retinal) processing of the color withwhich each eye is stimulated will be mistaken fordifferences between the eyes themselves

The research reported here used semicircles as inthe NAT and the AI Version 1 Also the sizendashcolorillusion was assessed in an experimental conditionin which semicircle sizes were compared in fullbinocular vision that is without redndashgreen glassesso that both semicircles were seen with both eyesWith the same purpose data were also collected ina haploscopic condition without lens-induced anis-eikonia that swapped the red and green filtersbetween the eyes a strategy that could reveal othercolor effects affecting measurements (eg lateralchromatic aberrations of the eye color-relateddifferences in focus effects of brightness differencesbetween the two colors etc) For simplicity we will




use the term lsquolsquosizendashcolor illusionrsquorsquo to refer to anyaspect of color vision that affects perceived size

Psychophysical Procedure

The method of adjustment used in Version 1 of theAI and implicitly required by the NAT has beencriticized for being prone to criterion bias3637 Versions2 and 3 of the AI implement alternative methods thathave long been presumed to be criterion-free but theyare identically susceptible to contamination fromresponse and decisional biases38ndash41 Alternative psy-chophysical methods used across aniseikonia tests arethus equally prone to bias Individual biases of suchtype would not explain the systematic and differentialpatterns of misestimation reported in the literature butthe problem of bias needs to be resolved

A comparative study of methods used tomeasure spatial bisection ability for the evaluationof unilateral visual-field defects42 determined thecause of persistent discrepancies across methodsand identified an optimal method via whichresponse biases are eliminated and sensory aspectsare separated out from decisional biases affectingobserved performance Because the psychometricproperties of that method are independent of thesensory modality or attribute (brightness blur sizeetc) subject to judgment this research applied onlythis optimal method namely a comparative taskwith three response options Each trial under thismethod involves a stimulus presentation analogousto those in the AI although the red semicircle ispresented on the left or on the right at random ineach trial also observers report on each trialwhether the two stimuli appear identical in sizeor rather the one on the left or the one on theright appears subjectively larger Finally lsquolsquoequalrsquorsquoresponses are treated in the analysis as a distinctcategory instead of being evenly split as half lsquolsquoleftlargerrsquorsquo and half lsquolsquoright largerrsquorsquo as is done in theAI43 nor are the data from the two presentationarrangements (ie red on the left and red on theright) aggregated into a single data set How theresultant data are analyzed will be fully describedbelow

Methods

This research followed the tenets of the Declara-tion of Helsinki and its protocol was approved by theinstitutional review board Informed consent wasobtained from each subject prior to participation

Observers

Seven male (including one of the authors) andthree female observers participated in the study Theirages ranged from 24 to 77 years They were refractedprior to the study and had normal or corrected-to-normal acuity of 2020 or better in each eye Allobservers were nonstrabismic had normal stereoacuity (70 sec of arc) by the Randot Stereo Test(Precision Vision Lasalle IL) had normal binocularvision (fusion) under dichoptic anaglyphic viewingwith the Worth 4-dot Test and (except for the author)were naıve as to the goals of the study One of theobservers was expected to have meaningful aniseiko-nia due to uniocular cataract extraction and implan-tation of an intraocular lens with an intentionalchange in refractive error to reduce myopia in theoperated eye in anticipation of an analogousadjustment to be made when the fellow eye isoperated Two other observers showed a mildaniseikonia in our measurements that could not beattributed to evident optical causes as they had anonly minimally anisometropic prescription A furtherophthalmic measurement of these observers did notidentify meaningful differences in axial length kera-tometry anterior chamber depth or lens thicknessbetween the eyes so their mild natural aniseikonia didnot seem to have an optical origin

Apparatus and Stimuli

All experimental events were controlled by MAT-LAB scripts under the Psychophysics Toolbox Ver-sion 3 (httppsychtoolboxorg) Responses werecollected via the computer keyboard

Stimuli were displayed at 60 Hz on a SAMSUNGSyncMaster 192N liquid crystal display monitor 375-cm horizontally and 30-cm vertically (SamsungElectronics Co Suwon South Korea) The 1280 3

1024ndashpixel image area subtended 28078 3 22628 ofvisual angle at the viewing distance of 75 cm whichwas secured with a chin-and-forehead rest that couldnot prevent observers from making small changes inhead position that would displace the center ofexpansion The stimuli were semicircles displayed sideby side with a separation of 25 pixels (0558) and withtheir centers symmetrically placed with respect to thecenter of the image area (the sketch in Fig 3) One ofthe semicircles was green and the other was red sothat each was visible only to one eye when color filterswere worn Red and green semicircles differed inluminance but no attempt was made to match thembecause anaglyphic viewing implies that they will both




be seen (monocularly) as equally dark patches on abright background with color becoming apparentonly under binocular vision as a result of corticalprocessing Leakage through the filters was preventedby adjusting the yellowish color of the immediateelliptical background using a matching procedure44

The elliptical area had a major axis of 1179 pixels(26048) and a minor axis of 923 pixels (20398) Theouter red background used in the AI was replacedhere with an analogous black background because apilot study showed that a red background disruptedperceptual stability substantially It also becameobvious in the pilot study that extra binocular lockswere needed and these were provided by black squares121 pixels in side and placed at the centers of the fourquadrants of the image area

The red semicircle was arbitrarily designated as thereference (standard) and had the same size on alltrials Its diameter was 257 pixels (5688) to match therecommended angular subtense in the NAT andapproximately the angular subtense used by Fullardet al18 in the AI Version 1

Black (binocularly visible) fixation aids weredesigned to be analogous to those in the AI andunlike the simple fixation crosses used in the NATThey consisted of a 1-pixel wide cross spanning theentire image area and a patterned central fixationmark with an outer diameter of 43 pixels (0958)Fixation aids were present throughout the sessioneven in the intertrial intervals

With the exceptions described below green and redfilters were respectively placed in front of the left andright eyes Aniseikonia was induced by placing afocalsize lenses (Chadwick Optical Inc Souderton PA)with nominal magnifications of 2 3 and 4 beforethe right eye for negative magnifications the lens wasflipped in front of the same eye Thus the standard (redsemicircle) had the same retinal size in all conditions as

it was viewed with the left eye through the green filterwithout lens-induced aniseikonia The only exceptionwas the condition to control for the sizendashcolor illusionin which color filters were swapped between the eyesSize lenses color filters and observersrsquo prescriptions (ifneeded) were mounted on a trial frame that observerswore during the sessions Identical position andadjustment of the trial frame across sessions with thesame observer could not be guaranteed Observerswearing presbyopic correction in their habitual pre-scription were corrected for display distance using athorn125 diopter (D) add which should minimize changesof magnification arising from the use of afocal sizelenses in near vision45

Based on the tools and variables used in thegrinding of our nominal 2 3 and 4 size lensestheir magnifications were calculated to be 205305 and 407 respectively although the manu-facturing process can only achieve these calculatedvalues to some precision Magnification was assesseddirectly by measuring the change in the deviation oflaser beams passing at different angles through thecenter of the lenses which yielded measured magni-fications of 202 293 and 397 For allpractical purposes we used nominal magnificationvalues in all calculations

Procedure

For each viewing mode (short presentation or freeviewing) measurements were taken in nine condi-tions a full binocular condition (ie without colorfilters) two haploscopic conditions without lens-induced aniseikonia (swapping color filters betweenthe eyes) and six haploscopic conditions with right-eye magnifications of 62 63 and 64 Observ-ers indicated in each trial whether the left the right orneither semicircle was perceptually larger than theother In the short presentation mode they wereinstructed to keep their gaze steady on the fixationmark and judge overall size in the free viewing modethey were asked to scan the configuration and checkfor alignment at the top and at the bottom to maketheir judgment (recall that the goal of this study is notto find out whether observers would have done thisnaturally but to determine the consequences of doingit) Assurance that free viewing had the intendedeffects was obtained during the breaks needed tochange size lenses in which observers were informallydebriefed about their experiences and difficulties withthe task With the exception of observer 10 whoadmitted to scanning the configuration only rarely allobservers reported spontaneously that the two semi-

Figure 3 Sample stimulus peripheral fusion locks and fixationaids drawn to scale




circles appeared to slide against each other in oppositedirections as they gazed to the top and to the bottoma clear indication that eye movements were accom-panied by head movements and relocation of thecenter of expansion thus providing conflicting fovealand peripheral size information

Data in the free viewing mode were collected first asinstructions given in the short presentation mode (andthe observersrsquo ensuing experiences) might have prompt-ed them to maintain central fixation also in the freeviewingmode Data collection under each viewingmodeconsisted of several sessions starting always with the fullbinocular condition followed by the two haploscopicconditions without size lenses in a random order andcontinuing with the six remaining haploscopic condi-tions also in a random order In short presentationconditions trials consisted of a get-ready period of 500ms followed by the 600-ms stimulus presentation in freeviewing conditions each trial also included a get-readyperiod of 500 ms but the stimulus was not extinguisheduntil observers had given their response

Data in short presentation conditions were collectedin three (occasionally two or four) blocks of 128 trialspreceded by a practice session of at least 30 trials Ineach trial the size of the test semicircle was determinedby adaptive methods optimized for efficient andaccurate estimation of nonmonotonic psychometricfunctions46 The 128 trials in each session arose fromsixteen 8-trial randomly interwoven adaptive staircas-es with four staircases starting at each of four testdiameters In conditions without size lenses startingdiameters were84 4 and 8 pixels longer than thestandard When size lenses were used these values wereshifted according to the induced magnification withadditional shifts introduced to collect data fromobservers suspected of natural aniseikonia The 16staircases represented two otherwise identical sets ofeight differing in that the test target was displayed onthe left in one of the sets and on the right in the otherWhen the observerrsquos response reflected that the testwas subjectively larger (shorter) test size was reduced(increased) by one step for the next trial along thatstaircase which was not necessarily the next trial in thesession responses reflecting that test and standard sizeswere perceptually equal changed the size of the test intwo steps in a direction decided at random withequiprobability Step size was 2 pixels the leastpossible change in diameter without shifting the centerof the semicircle

With free viewing two consecutive 96-trial blockswere used consisting of twelve 8-trial staircases two ateach of three starting test diameters (6 0 and 6

pixels longer than the standard diameter also shiftedaccording to lens-induced or natural aniseikonia) foreach of the two possible test positions Step size wasalso 2 pixels and staircase rules were identical to thosedescribed above

Data collection proceeded at the observerrsquos paceOnly a response to the current trial triggered the nexttrial In short presentation conditions a further keywas enabled for observers to decline responding if theyhad missed the stimulus (eg due to blinks inadequatefixation etc) but observers were instructed to refrainfrom using this key as a means to give themselves asecond chance with a stimulus they had not missedTrials thus discarded were repeated at a later time

Expected Location of the Point of SubjectiveEquality

The point of subjective equality (PSE) is thephysical magnitude that a test stimulus must havefor its perceived magnitude to match the perceivedmagnitude of a reference (standard) stimulus For-mally if xs is the physical magnitude of the standardls is the function describing how perceived magnitudevaries with physical magnitude for the standard andlt is the corresponding function for the test the PSEis the physical magnitude xPSE at which lt(xPSE) frac14ls(xs) so that xPSE frac14 lt

1(ls[xs])Under full binocular viewing in the absence of a

sizendashcolor illusion ltfrac14 ls and the expectation is thatxPSE frac14 xs A sizendashcolor illusion makes lt bdquo ls andthus significant shifts of the measured PSE withrespect to xs reflect the magnitude of the sizendashcolorillusion In haploscopic conditions without lens-induced (or natural) aniseikonia and without sizendashcolor illusions lt frac14 ls also and the expectation isagain that xPSEfrac14xs Significant shifts of the measuredPSE with respect to xs then reflect either naturalaniseikonia or a sizendashcolor illusion These twopotential causes can be distinguished because our fullbinocular condition provides a measure of themagnitude of a hypothetical sizendashcolor illusion

Finally in haploscopic conditions with only lens-induced aniseikonia affecting perceived size (ie nonatural aniseikonia and no sizendashcolor illusions) xPSEshould be shifted away from xs in quantities that canbe anticipated from the aniseikonia induced by thelens which is what makes lt and ls differ in thesecases With xs frac14 257 pixels the expected xPSE formagnifications of 2 3 and 4 are 25196 24951and 24712 pixels respectively for magnifications of2 3 and 4 the expected xPSE are 2621426471 and 26728 pixels respectively Note that




jxPSE xsj differs for positive and negative magnifi-cations of the same size Without natural aniseikoniaor sizendashcolor illusions shifts of the measured PSEsaway from these expectations will reflect a failure tomeasure lens-induced aniseikonia adequately

Model-Based Analysis

Psychometric functions were fitted to the datafrom each subject in each condition and viewing modeto measure the PSE The theoretical form of thesefunctions was derived from Thurstonian assumptionsabout the sensory and decisional processes involved inthe ternary task Although the application of this typeof model to analogous tasks has been described indetail elsewhere39404247ndash49 the model is sketched inFigure 4 for two sample situations and a briefdescription follows

The psychophysical function l describing therelation between perceived and physical length hasbeen shown to be approximately linear via magnitudeproduction tasks50 and magnitude estimationtasks51ndash53 whereas the standard deviation r ofperceived length increases with physical length50 Thesame holds for perception of circle size50ndash52 Considerhaploscopic conditions with perceived size varyingbetween the eyes due to natural or lens-inducedaniseikonia The perceived size Si of a semicircle ofdiameter x seen with eye i OS OD for left eye (OS)or right eye (OD) is thus a random variable whosemean li and variance r2

i are functions of x given by

liethxTHORN frac14 ai thorn bix eth1aTHORN

r2iethxTHORN frac14 ji thorn cix eth1bTHORN

Without loss of generality assume that observerscompare the sizes of the semicircles on the left andright sides of the screen and base their decision on theperceived difference D frac14 SR SL where subscriptsdenote position on the screen (right or left) and willlater be replaced with references to the eye with whicheach semicircle is seen Observers give a lsquolsquoleft largerrsquorsquoresponse when D d1 they give a lsquolsquoright largerrsquorsquoresponse when D d2 and they give a lsquolsquocanrsquot tellrsquorsquoresponse when d1 D d2 With normally distributedSi the probability of each type of response is

Probethlsquolsquoleft largerrsquorsquoTHORN frac14 U ethd1 ldTHORN=rdfrac12 eth2aTHORN

Probethlsquolsquocanrsquot tellrsquorsquoTHORN frac14 U ethd2 ldTHORN=rdfrac12 U ethd1 ldTHORN=rdfrac12 eth2bTHORN

Probethlsquolsquoright largerrsquorsquoTHORN frac14 1 U ethd2 ldTHORN=rdfrac12 eth2cTHORN

where U is the unit-normal cumulative distributionfunction and ld and rd are the mean and standarddeviation of D which vary with test size x accordingto which semicircle was on each side and which eyewas each semicircle seen with (Appendix A in theSupplementary Material)

By substitution in Equation 2 the psychometricfunctions Wj (for lsquolsquoleft largerrsquorsquo responses) and Yj (forlsquolsquocanrsquot tellrsquorsquo responses) when the test (green) semicircleis displayed at position j (with j L R for left andright position) and the green filter is before the left eyeare (Figs 4b c)

WLethxTHORN frac14 Ud1 ethlOSethxsTHORN lODethxTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2ODethxTHORN thorn r2

OSethxsTHORNp

eth3aTHORN

WRethxTHORN frac14 Ud1 ethlODethxTHORN lOSethxsTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


OSethxsTHORNp

eth3bTHORN

YLethxTHORN frac14 Ud2 ethlOSethxsTHORN lODethxTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


OSethxsTHORNp

Ud1 ethlOSethxsTHORN lODethxTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


OSethxsTHORNp

eth3cTHORN

YRethxTHORN frac14 Ud2 ethlODethxTHORN lOSethxsTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


OSethxsTHORNp

Ud1 ethlODethxTHORN lOSethxsTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


OSethxsTHORNp

eth3dTHORN

Equations for the reverse arrangement of filters(red before the left eye) are analogous except thatsubscripts OS and OD are swapped in the fullbinocular condition equations are also analogousexcept that subscripts OS and OD are respectivelyreplaced with subscripts lsquolsquoredrsquorsquo and lsquolsquogreenrsquorsquo (Appen-dix A in the Supplementary Material) Note that d1frac14 d2 renders YL(x) frac14 YR(x) frac14 0 and hence yieldsobservers who never give lsquolsquocanrsquot tellrsquorsquo responses Onthe other hand d1frac14d2 renders YL(x)frac14YR(x) andthus yields observers without decisional bias Thenthe lateral separation between psychometric func-tions for each arrangement (test presented on theleft or on the right) is a pure indicator of decisionalbias

Substituting Equation 1 into Equation 3 thearguments of U become




Figure 4 Model psychometric functions for an observer whose right retinal image is 2 larger than the left retinal image Color filtersfor haploscopic viewing without additional lens-induced aniseikonia are assumed to be placed before the eyes in opposite arrangements

in each column as indicated at the top (a) Psychophysical functions describing perceived size with each eye lOS(x)frac14 64x and lOD(x)frac146528x (red and green lines) The perceived size of the standard (red) semicircle of size xfrac14 200 (arbitrary units) varies according to the eyewith which it is seen The scale of the vertical axis is irrelevant and has been omitted The variance of perceived size is assumed to be

equal with both eyes and given by equation 1b with jOSfrac14 jODfrac14 0 and cOSfrac14 cODfrac14 016 (b) Distribution of the decision variable D whenthe test stimulus has the same size as the standard and the test is presented on the left side of the screen Vertical lines indicate how thedecision space is partitioned and mapped onto judgments The horizontal scale is irrelevant and has been omitted (c) Identical to theprevious except that the test stimulus is presented on the right side (d) Resultant psychometric functions in the ternary task Each curve

refers to the response (left larger canrsquot tell or right larger) and condition (test on the left or test on the right) indicated by thecorresponding color in (b) and (c) The vertical dashed line indicates the point of objective equality (POE) the solid vertical line indicatesthe PSE which differs from the POE due to the natural aniseikonia of the observer and whose position varies with the arrangement of

color filters before the eyes The sets of curves for test-on-left-side and test-on-right-side presentations are pushed apart by decisionalbias (ie when d1 bdquo d2)




dk ethlOSethxsTHORN lODethxTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

ODethxTHORN thorn r2OSethxsTHORN

pfrac14 dk ethaOS aOD thorn bOSxs bODxTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jOS thorn jOD thorn cOSxs thorn cODxp eth4aTHORN

dk ethlODethxTHORN lOSethxsTHORNTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2


pfrac14 dk thorn ethaOS aOD thorn bOSxs bODxTHORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jOS thorn jOD thorn cOSxs thorn cODxp eth4bTHORN

for k 1 2 Because our experiments involve asingle standard level xs and a narrow range of testlevels x around it fitting this full model revealed thatthe term cOD x in the argument of the denominator ofEquation 4 makes a negligible contribution and doesnot improve the fit compared with a simplifiedversion of the model in which r2

ODethxTHORN thorn r2OSethxsTHORN is

regarded as a constant We thus fitted a simplifiedmodel in which cOD frac14 cOS frac14 0 so that the argumentsof U in Equation 3 finally become



p frac14 dk a thorn bx eth5aTHORN



p frac14 dk thorn a bx eth5bTHORN

where a frac14 ethaOS aOD thorn bOSxsTHORN=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijOS thorn jODp

b frac14bOD=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijOS thorn jOD

pand d

k frac14 dk=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijOS thorn jOD

pfor k

1 2 are the four parameters to be estimatedAnalogous equations with changes in the subscriptsare obtained for the remaining conditions (AppendixA in the Supplementary Material)

Parameter Estimation and ApproximateNonrejection Regions

Maximum-likelihood estimates for a b d1 and

d2 were obtained from data for each observer in eachcondition using the NAG54 subroutine E04JYF underthe constraints a 0 b 0 and d

1 d2 It has

been shown48 that the PSE is the point at which theensemble of psychometric functions has an axis ofbilateral symmetry which can be obtained fromparameter estimates as xPSE frac14 ab The PSEmeasured in units of x can be readily transformedinto any required measure of aniseikonia either as apercentage or as a factor

Expected and measured PSEs are unlikely tomatch exactly due to sampling error but meaningfuldiscrepancies between expected and measured PSEs

should be identifiable In subject-by-subject analysesin which a single PSE is measured per conditionsampling models for confidence intervals (CIs)around parameter estimates are not applicableNevertheless rejection regions around the expectedPSE can be determined via simulation methodsPrevious studies have shown that the standard errorof estimates of the location of a psychometricfunction for discrimination (the PSE) vary with theadaptive procedure implemented the number of trialsand often other parameters such as the slope of thepsychometric function55ndash61 Appendix B in theSupplementary Material describes the bootstrapprocedure that was used to determine approximatenonrejection regions so as to check for the statisticalsignificance of the deviation of a measured PSE withrespect to the expected PSE

Results

SizendashColor Illusion and Natural Aniseikonia

Figure 5 shows results for each observer in the fullbinocular condition under each viewing mode Withthe exception to be discussed below the sizendashcolorillusion is understandably absent for all observersunder the free viewing mode (ie the measured PSE isvirtually at the standard level) because size wasdetermined indirectly by checking for alignment atthe extremes in what becomes a foveal vernier taskThe exception was observer 10 who reportedcompliance with instructions to scan the configura-tion lsquolsquoonly when needed to make a judgmentrsquorsquo addingthat such need did not arise very often In contrast inthe short presentation mode where observers mustperform a size discrimination task at a glimpse of thestimulus traces of a sizendashcolor illusion (ie the PSE is1ndash3 pixels above xs) are apparent for all observersexcept 3 4 8 and 9 with the red semicircle appearingperceptually larger than the green semicircle

All observers gave lsquolsquocanrsquot tellrsquorsquo responses relativelyfrequently under both viewing modes with theexception of observer 3 In general the psychometricfunctions for lsquolsquocanrsquot tellrsquorsquo responses (black and graycurves) were narrower and taller in the free viewingmode than in the short presentation mode suggestingmore ability to discriminate small differences in sizeunder free viewing Finally under the free viewingmode all observers except 5 and 10 showed little orno decisional bias (lateral displacement of psycho-metric functions across test positions darker versuspaler data points and curves) compared with the




larger decisional bias that they generally exhibitedunder the short presentation mode (recall thatobserver 10 reportedly used fixation under bothviewing modes so the only difference for him wasthe unlimited versus short presentation duration)

Figure 6 shows results for each observer andviewing mode in the haploscopic conditions withoutsize lenses and color filters in two arrangementsbefore the eyes Observers display decisional biasesand patterns of usage of the lsquolsquocanrsquot tellrsquorsquo optionanalogous to those that they displayed in the fullbinocular condition As illustrated in Figure 4natural aniseikonia without sizendashcolor illusions showsas ensembles of psychometric functions that are

shifted away from xs by the same amount but inopposite directions when color filters are swappedbetween the eyes In line with this argument in thefree viewing mode (left half of Fig 6) observer 9shows weak traces of natural aniseikonia (about 1)in the form of a larger retinal image in the left eye andobserver 5 shows evidence of a larger aniseikonia(about 4) in the opposite form These measures mayhave been underestimated because free viewing allowsperforming a foveal vernier task only disrupted byconflicting peripheral information about overall sizeIn fact data from the short presentation mode inwhich observers perform a size discrimination task(right half of Fig 6) suggest larger natural aniseikonia

Figure 5 Psychometric functions in the full binocular condition under each viewing mode In each panel the dashed vertical line givesthe expected location of the PSE if perceived size does not vary with color the solid vertical line indicates the actual location of the PSEBecause both semicircles are seen with both eyes in these conditions any meaningful lateral shift of the PSE away from its expectedlocation reflects differences in perceived size associated to color and independent of any natural aniseikonia that observers might havethe gray area around the expected PSE denotes the nonrejection region




Figure 6 Psychometric functions for each observer and viewing mode in the two haploscopic conditions without lens-inducedaniseikonia providing a further test of sizendashcolor illusion and natural aniseikonia Dashed and solid vertical lines and gray areas representexpected and actual PSE and nonrejection regions as in Figure 4 where the expectation reflects no sizendashcolor illusion and no aniseikoniaNote that the horizontal range is broader for observers 4 6 9 and 10 and even broader for observer 5




for these observers 2 or 4 for observer 9(depending on which arrangement of color filters isconsidered) and 8 for observer 5 The shortpresentation mode also revealed a weak naturalaniseikonia (about 15) for observer 4 which wentundetected in the free viewing mode

A sizendashcolor illusion without natural aniseikoniahas a different manifestation the measured PSEwould be at the same point for both arrangementsof color filters before the eyes the reason being thatthe red and green lines in Figure 4a (describingperceived size with each eye) would still followdifferent paths within each panel due to differencesin perceived size according to color but their pathswould be identical in the two panels because the eyewith which each semicircle is seen does not make anydifference in the absence of natural aniseikonia As aresult the ensembles of psychometric functions will beshifted away from xs by the same amount and in thesame direction when color filters are swapped betweenthe eyes No observerrsquos data showed evidence to thiseffect

Finally and by the same argument joint effects ofnatural aniseikonia and sizendashcolor illusion show asensembles of psychometric functions displaced awayfrom xs by different amounts and possibly in differentdirections when color filters are swapped between theeyes (recall that many color-related issues mayproduce these asymmetric displacements and that werefer to them as sizendashcolor illusions for simplicity)Data from some observers (eg 1 2 and 7) showtraces of this asymmetry with short presentations thatare consistent with the traces of a sizendashcolor illusionthat were identified for them in the full binocularcondition also under the short presentation mode(Fig 5) On the other hand observers 3 6 9 and 10show stronger signs of a mixture of natural aniseiko-nia and sizendashcolor illusion in the short presentationmode although the lack of evidence of a sizendashcolorillusion for observers 3 and 9 in the full binocularcondition suggests other causes

Lens-Induced Aniseikonia

Results for haploscopic conditions with size lensesare shown in Figure 7 for two sample observers undereach viewing mode revealing features that can onlybe extracted with a ternary task Consider observer 2first (left half of Fig 7) This observer gave lsquolsquocanrsquottellrsquorsquo responses more often in the free viewing mode(left column) than in the short presentation mode(right column) At all levels of lens-induced aniseiko-nia the psychometric functions for lsquolsquocanrsquot tellrsquorsquo

responses (black and gray curves) are taller underfree viewing than with short presentations inagreement with an analogous behavior displayed bythis observer without size lenses (Fig 6) Secondlyand also in agreement with results obtained withoutsize lenses psychometric functions are steeper underfree viewing than with short presentations Thirdlydecisional bias is larger in the short presentationmode than under free viewing and its form isconsistent across levels of lens-induced aniseikonia(darker data and curves are always shifted to the leftof lighter data and curves as was the case also in Fig6) Finally and more importantly underestimation oflens-induced aniseikonia as indicated by a measuredPSE (solid vertical line in each panel) that is displacedtoward 0 relative to the expected PSE (dashed verticalline in each panel) is generally larger under freeviewing than with short presentations

In comparison observer 3 (right half of Fig 7)gave very few lsquolsquocanrsquot tellrsquorsquo responses (and only underfree viewing) has psychometric functions that onlydiffer slightly in slope across viewing modes rarelydisplays any decisional bias seems more immune tolens-induced aniseikonia for negative magnificationsunder free viewing and seems instead about equallyprone to lens-induced aniseikonia under both viewingmodes for positive magnifications

This range of individual differences in usage of thelsquolsquocanrsquot tellrsquorsquo response option slope of the psychometricfunctions decisional bias and susceptibility to lens-induced aniseikonia is representative of analogousfeatures consistently displayed by the data from otherobservers across conditions (including the conditionwithout size lenses) Other observers also showedsystematic differences in these respects between thefree viewing and short presentation modes Some ofthese features (specifically differences in usage of thelsquolsquocanrsquot tellrsquorsquo response option decisional bias and slopeof psychometric functions across viewing modes) wereclearly apparent in Figures 5 and 6 also

For a compact display of results about suscepti-bility to lens-induced aniseikonia Figure 8 plotsmeasured aniseikonia against expected lens-inducedaniseikonia for each observer under each viewingmode including also data from the null condition(ie haploscopic presentation without size lenses)Observer 5 shows systematic signs of a relativelystrong natural aniseikonia measured at about 4 inthe free viewing mode but at about 8 in the shortpresentation mode Observer 10 also shows traces of aweaker aniseikonia of the opposite sign measured atabout 3 under both viewing modes (recall that this




observer used fixation also under the free viewingmode) and observer 9 shows also traces of an evenweaker aniseikonia of the same sign and measured atabout 1 in the free viewing mode but at about 2 inthe short presentation mode This natural aniseikoniadid not seem to affect the accuracy with which theseobservers make size comparisons in the full binocularcondition under free viewing (Fig 5) Data fromobservers 5 and 10 in the short presentation modewith full binocular vision were only noisier andrevealed broader psychometric functions suggesting

a sizendashcolor illusion in observer 10 and less decisionalbias than in the free viewing mode for observer 5 datafrom the also aniseikonic observer 9 showed none ofthese characteristics under full binocular viewingwhere data and psychometric functions were indistin-guishable from those of iseikonic observers Leavingthe aniseikonic observers 5 9 and 10 aside Figure 8reveals some additional interesting features undereach viewing mode and across them

Consider first the free viewing mode (left panel inFig 8) Data points fall everywhere under the

Figure 7 Full set of data and fitted psychometric functions for two observers (left and right halves of the figure) across viewing modes(left and right column within each half) and levels of lens-induced aniseikonia (rows)




diagonal in the first quadrant indicating thatmeasured aniseikonia can be anywhere between nulland the expected value when size lenses are flipped toproduce negative magnifications This space is cov-ered in such form because data from differentobservers follow distinctly different paths suggestingthat they carry out the task in different ways thatproduce different outcomes under the free viewingmode A similar scatter occurs above the diagonal inthe third quadrant (when size lenses are placed toproduce positive magnifications) except that the areanear the horizontal axis is less covered

Results under the short presentation mode differremarkably (right panel of Fig 8) Data fromobservers 3 4 and 6 describe paths parallel to thediagonal but meaningfully above or below it Thesepatterns were not observed for them under the freeviewing mode and they are consistent with a naturalaniseikonia in which either the right-eye image

(observers 3 and 6) or the left-eye image (observer4) is larger than the other eyersquos image It is alsoremarkable that the apparent aniseikonia of observers3 and 6 might have gone unnoticed if color filters hadbeen placed in reverse before the eyes due to theasymmetry described earlier on discussing resultsshown on the right half of Figure 6 The naturalaniseikonia of observer 4 on the other hand did showupon swapping color filters between the eyes as thisobserverrsquos measured PSEs in such cases were sym-metrically placed with respect to the expected PSE(the right half of Fig 6)

With or without natural aniseikonia data from allobservers under the short presentation mode describea path that is very approximately parallel to thediagonal and thus indicate absence of any systematicunderestimation Although these data paths speak forthemselves we conducted the conventional regressionanalysis to check for (overall) unit slope in each

Figure 8 Relation between expected and measured aniseikonia Expected and measured aniseikonia are given as the increment size ofthe diameter of the test semicircle (in pixels) for its perceived size to match that of the standard semicircle and also as a percentage Theleft panel shows results under the free viewing mode the right panel shows results under the short presentation mode Data from eachindividual observer are identified by the color described in the legend inside the left panel




viewing condition For free viewing the regressionslope was 0697 which was significantly differentfrom unity (t68frac14443 P 0001) and thus indicatesunderestimation of lens-induced aniseikonia (95 CIfor the slope [0560 0833]) in the short presentationmode the regression slope was 0937 which was notsignificantly different from unity (t68 frac14 056) andthus indicates adequate estimation of lens-inducedaniseikonia (95 CI for the slope [0712 1162])Note also that under free viewing the slope at positivemagnifications seems closer to unity whereas the slopeat negative magnifications seems farther from unityThis reflects the role of contradictory perceptualinformation discussed earlier which has differentstrengths for positive and negative magnificationsunder the free viewing mode In sum compared withsignificant underestimation of lens-induced aniseiko-nia in the free viewing mode the short presentationmode does not result in underestimation and allowsdetecting weak natural aniseikonia that is notapparent under free viewing

Additional Conditions

The results discussed thus far prompted us tocollect data from some observers under additionalconditions to look for the origin of some of thefeatures discussed earlier Because those data addressside issues their description and discussion are placedin Appendix C in the Supplementary Material

Discussion

This study investigated factors causing the under-estimation reported in the literature when aniseikoniais measured with the NAT or the AI We includedcontrol conditions to check for contaminating factorssuch as sizendashcolor illusions or decisional biases whichmight masquerade as aniseikonia (or mask an existinganiseikonia) under the conventional protocol Ourresults are summarized next followed by commentaryon their implications for the clinical measurement ofaniseikonia for the empirical validation of aniseikoniatests and for the clinical management of aniseikonia

Annotated Summary of Results

Viewing mode seems to be the main factorcontributing to the underestimation of lens-inducedaniseikonia (Fig 8) With size lenses mounted on atrial frame worn by observers free viewing withunlimited time to give a response allows observers torelocate the center of expansion and gather contra-

dictory foveal and peripheral perceptual informationabout size The resolution of the conflict appears toinvolve weighing the two sources of contradictoryinformation which can only favor underestimationThe strength of conflicting information decreaseswhen size lenses are placed to produce negativeinstead of positive magnifications In contrast shortpresentations eliminate changes in the center ofexpansion while the stimulus is available for inspec-tion although such changes may occur across trialsBut even with short presentations use of anaglyphsto measure aniseikonia brings in other contaminantswhose role we also investigated

We collected data in a full binocular condition inwhich both semicircles are seen with both eyes (Fig 5)so as to check for a sizendashcolor illusion that might bemistaken for aniseikonia when each color semicircle isseen with one eye We found no traces of a sizendashcolorillusion under free viewing surely due to the fact thatobservers carried out a vernier task instead of a sizediscrimination task Yet we found significant tracesof a sizendashcolor illusion in 6 of 10 observers under theshort presentation mode the green semicircle (whichwas brighter than the red semicircle when seenwithout filters) needed to be up to about 1 largerthan the red semicircle for both semicircles to beperceived as having the same size Although the sizendashcolor illusion seems small in magnitude (whenpresent) in our sample it will only contribute anadditive error to measurements of aniseikonia Theerror has bearings on clinical practice although sucherrors would cause a systematic misestimation of lens-induced aniseikonia not the multiplicative underesti-mation reported in the literature11ndash1316ndash19

In all experimental conditions data were collectedwith the red and green semicircles displayed in bothpresentation arrangements (red on the left and greenon the right or vice versa) to check for decisionalbias We found strong evidence of decisional bias inmost observers which manifests as psychometricfunctions that are laterally displaced for eachpresentation arrangement The magnitude and formof the bias varied across observers and across viewingmodes for some observers but it was otherwiseconstant for each observer across levels of lens-induced aniseikonia (Fig 7) The presence of deci-sional bias also has bearings on clinical practice as itrepresents a strong contaminant when data arecollected using a single arrangement (ie greenalways on the right or on the left) Consider the datafor observer 2 under the short presentation mode inthe haploscopic condition without size lenses and with




red and green filters placed before the left and righteyes respectively (rightmost panel in the second rowof Fig 6) If the clinical test arbitrarily places thegreen (test) semicircle only on the left of the screenonly the darker symbols and curves would beobtained so that observer 2 would be taken toperceive a larger image with the left eye if the testsemicircle were placed instead on the right of thescreen only the lighter symbols and curves would beobtained so that observer 2 would instead be taken toperceive a smaller image with the left eye Thesecontradictory outcomes are only the result ofdecisional bias whose presence can only be unveiledby using both presentation arrangements and analyz-ing the data separately In any case decisional biasintroduces additive errors of clinical relevance thatwould cause a systematic misestimation of lens-induced aniseikonia but again not the multiplicativeunderestimation reported in the literature11ndash1316ndash19

Our haploscopic conditions also included mea-surements of aniseikonia without size lenses but withcolor filters placed in both possible arrangements (iegreen filter before the left eye and red filter before theright eye and vice versa) to distinguish trueaniseikonia from sizendashcolor illusions In most caseswe found the expected symmetric effects of naturalaniseikonia (or lack thereof) but data from three often observers under the short presentation moderevealed what appeared to be a mild but significantaniseikonia in one of the filter arrangements and noaniseikonia whatsoever in the other (see the results forobservers 3 and 6 in the right half of Fig 6) Datafrom two other observers in the same situationreflected a mild aniseikonia in one of the arrange-ments of color filters and a stronger aniseikonia in theother (results for observers 9 and 10 in the right halfof Fig 6) These asymmetries were also apparentwhen we measured lens-induced aniseikonia underboth filter arrangements for three of these observers(Fig C2 in Appendix C of the SupplementaryMaterial) suggesting a combination of color effectsand aniseikonia Although these asymmetries cannotcause the underestimation reported in the litera-ture11ndash1316ndash19 they indicate that the test protocolshould include both arrangements of color filters

Development and Validation of Clinical Testsfor Aniseikonia

Our results have implications for the developmentadministration and validation of aniseikonia tests ofthe NAT type (ie those based on direct comparisons

of the perceived size of haploscopically presentedstimuli) and also on the use of color in such testsThese are summarized and commented on next

Firstly short presentations and viewing undercentral fixation are advisable to ensure that observerstruly perform a size discrimination task Shortpresentations preclude major changes in the centerof expansion during the short time in which thestimulus is perceptually available although the centerof expansion may nevertheless change across trials ifthe observer does not maintain a steady headposition These changes across trials introduce noisein the measurements by hampering size comparisonson trials in which the two stimuli are perceived to bevertically displaced in opposite directions It isimportant to stress that these problems were entirelyabsent in the OE and the SE not only because thoseinstruments did not assess size perception but becausethe instruments themselves ensured a fixed center ofexpansion Preferably the AI and analogous testsshould be administered with a phoropter or a similardevice that can be rigidly placed and aligned with thestimulus plane to ensure a fixed and appropriatelylocated center of expansion just as the OE and SE didIn these conditions the free viewing mode could besafely used as observersrsquo scanning strategies would beinconsequential

Secondly stimulus shape should be such thatbinocular fusion of the two haploscopically presentedstimuli is prevented The semicircles originally used inthe NAT or in Version 1 of the AI are thus moreappropriate than the rectangles used in versions 2 and3 of the AI

Thirdly a test using color stimuli for haploscopicpresentations should clearly differentiate aniseikoniafrom sizendashcolor illusions Although such illusion wasnot very strong in our sample and it was not apparentfor all observers it will be mistaken for aniseikoniawhen each color is seen with one eye Adding a fullbinocular condition to assess the magnitude of suchillusion should be part of the protocol

Fourthly sizendashcolor illusions or other aspects ofcolor perception may produce different measures ofaniseikonia according to which color is presented toeach eye Therefore the test protocol should includeseparate measurements with color filters in the twopossible arrangements before the eyes

Fifthly decisional biases produce lateral shifts ofthe psychometric functions according to whichpositions test and standard stimuli occupy on thescreen For each arrangement these shifts haveopposite directions away from the PSE (the measure




of aniseikonia) and they can only be differentiatedfrom true aniseikonia if the protocol includes separatemeasurements and analyses under both positionarrangements

All of these recommendations are supported byour results and render a test protocol that includes theconditions and analyses underlying the results plottedin Figures 5 and 6 under the short presentation mode(or with free viewing if a fixed center of expansion isensured) Note that some of these aspects of the testprotocol are motivated by the use of color stimuli forhaploscopic presentations and the need to differenti-ate aniseikonia from other color-related issues Afeasible alternative would involve haploscopic pre-sentation using a polarized three-dimensional displaysystem (or liquid crystal shutters in a temporalhaploscopic system) so that each stimulus is againseen with only one eye Both stimuli could thus bepresented in black on a gray background rendering atest that is free of color vision Research on thefeasibility and use of aniseikonia tests based on thoseprinciples is still needed but they would definitelyeliminate from the test protocol the need for a fullbinocular condition and perhaps the need for a swapof polarizers between the eyes Decisional biasesshould still need to be checked for by measuringaniseikonia under each position arrangement forstandard and test stimuli as decisional bias affectsall psychophysical measurements

All tests must also be validated and the validationof aniseikonia tests requires a proof that lens-inducedaniseikonia is adequately estimated Because placingsize lenses on a trial frame brings about contamina-tion from changes in the center of expansion acrosstrials with the same size lens and across conditionsinvolving different size lenses (even under the shortpresentation mode) using a method that ensures afixed center of expansion is highly advisable Thismight be easily achieved by using a phoropter insteadof a trial frame Phoropters already incorporate colorfilters polarized lenses and all that is needed to fixeach observerrsquos prescription although placing sizelenses on them may require a custom-made attach-ment

Management of Aniseikonia

We have shown that the location of the center ofexpansion has important consequences on the accu-racy with which aniseikonia can be measured Thisobservation in turn suggests some perceptual conse-quences when natural viewing conditions are in-volved that is when both eyes see the same object

When the cause of aniseikonia is in or on the eyechanges of fixation in natural viewing conditions alsorelocate the center of expansion so that the differentperceived sizes manifest mostly in the peripheralvisual field while the foveated area on the stimulusplane always stays aligned in both eyes across eyemovements In contrast when aniseikonia is causedby external lenses (eg on glasses) changes offixation by simply moving the eyes behind the lenseshave distinct consequences because the center ofexpansion remains at a location on the stimulus planethat is only determined by the position of the externallens moving the eyes brings to the foveae images ofdifferent sizes and in disparate positions requiringfusional efforts Our limited ability to accomplish thisespecially in the vertical direction may be responsiblefor the visual discomfort and symptoms that are oftenreported (mainly upon reading) by patients whoseaniseikonia is caused by spectacle lenses Remole25

seemed to refer to these two situations when hepointed out the difference between static and dynamicaniseikonia showing also that dynamic aniseikoniaproduces more visual discomfort than static anisei-konia particularly in the vertical direction

Such distinction does not apply under the condi-tions of our measurements where the images seenwith each eye do not need to (and cannot) be fusedinto a single percept Nevertheless our observationssuggest a simple way to treat dynamic aniseikoniaprompt patients to tilt their head (and not just tomove their eyes) as needed when they change fixationThe fact that some aniseikonic patients do not reportsymptoms or visual discomfort may reflect that theylearned to do so by themselves Moving the head toaccompany changes in fixation is a strategy that usersof progressive addition spectacle lenses implement tolook through the part of the lens that is appropriatefor the distance using a similar strategy should not bedifficult for prepresbiopic aniseikonic patients re-moving entirely the extra discomfort produced bydynamic aniseikonia Of course this strategy may beunfeasible for presbyopic users of bifocal or multifo-cal lenses With bifocals reading with anisometropiaresults in dynamic aniseikonia or near-point prismaticeffect and requires prism slab off to assist fusionOther options that may make head tilt useful formanaging aniseikonia include the recourse to separatereading glasses clip-on frames with a full lens nearadd in the magnetic or swing-away clip or deform-able adjustable spectacle lenses62 Various designs ofadjustable lenses have been recently commercial-ized63 An empirical study on the efficacy of these




management options is worth carrying out but suchstudy is beyond the scope of this paper

Acknowledgments

The authors thank Azma Rehman for assistancewith data collection and Jae-Hyun Jung for mea-surements of the magnification of size lenses Parts ofthe computations were carried out on EOLO theMECD- and MICINN-funded HPC of ClimateChange at Moncloa Campus of International Excel-lence Universidad Complutense

Supported by Grant PSI2012-32903 from Minis-terio de Economıa y Competitividad (MAGP) and byNational Institutes of Health Grants R01EY05957and R01EY023385 (EP)

Disclosure MA Garcıa-Perez None E Peli

None

References

1 Ogle KN Relative sizes of ocular images of thetwo eyes in asymmetric convergence Arch Oph-thalmol 1939221046ndash1067

2 Peli E Optometric and perceptual issues withhead-mounted displays In Mouroulis P edVisual Instrumentation Optical Design and Engi-neering Principles New York NY McGraw-Hill1999205ndash276

3 Bradley A Rabin J Freeman RD Nonopticaldeterminants of aniseikonia Invest OphthalmolVis Sci 198324507ndash512

4 Berens C A study of 288 patients examined withthe ophthalmo-eikonometer Br J Ophthalmol193721132ndash142

5 Berens C Loutfallah M Aniseikonia a study of836 patients examined with the ophthalmo-eikonometer Trans Am Ophthalmol Soc 193836234ndash267

6 Ames A The space eikonometer test for anisei-konia Am J Ophthalmol 194528248ndash262

7 Brecher GA A new method for measuringaniseikonia Am J Ophthalmol 1951341016ndash1021

8 Brecher GA Winters DM Townsend CA Imagealternation for aniseikonia determination Am JOphthalmol 195845253ndash258

9 Awaya S New Aniseikonia Tests Tokio Han-daya Co Ltd 1957

10 Awaya S Sugawara M Horibe F Torii F ThelsquolsquoNew Aniseikonia Testsrsquorsquo and its clinical applica-tion [In Japanese] Nippon Ganka Gakkai Zasshi198286217ndash222

11 McCormack G Peli E Stone P Differences intests of aniseikonia Invest Ophthalmol Vis Sci1992332063ndash2067

12 Yoshida M Sato M Awaya S Evaluation of theclinical usefulness of the New Aniseikonia Tests[In Japanese] Nippon Ganka Gakkai Zasshi 1997101718ndash722

13 Antona B Barra F Barrio A Gonzalez ESanchez I The validity and repeatability of theNew Aniseikonia Test Optom Vis Sci 200683903ndash909

14 de Wit GC Evaluation of a new direct-compar-ison aniseikonia test Binocul Vis Strabismus Q20031887ndash94

15 Ugarte M Williamson TH Aniseikonia associ-ated with epiretinal membranes Br J Ophthalmol2005891576ndash1580

16 Rutstein RP Corliss DA Fullard RJ Compar-ison of aniseikonia as measured by the aniseiko-nia inspector and the space eikonometer OptomVis Sci 200683836ndash842

17 Antona B Barra F Barrio A Gonzalez ESanchez I Validity and repeatability of a newtest for aniseikonia Invest Ophthalmol Vis Sci20074858ndash62

18 Fullard RJ Rutstein RP Corliss DA Theevaluation of two new computer-based tests formeasurement of aniseikonia Optom Vis Sci2007841093ndash1100

19 Weise KK Marsh-Tootle W Corliss D Evalua-tion of computer-based testing for aniseikonia inchildren Optom Vis Sci 201087883ndash889

20 Kehler LAF Fraine L Lu P Evaluation of theAniseikonia Inspector Version 3 in school-agedchildren Optom Vis Sci 201491528ndash532

21 de Wit GC Comparison of aniseikonia asmeasured by the Aniseikonia Inspector and theSpace Eikonometer Optom Vis Sci 200784E535ndashE536

22 de Wit GC Validity and repeatability of a newtest for aniseikonia discussion Invest OphthalmolVis Sci 200748eLetter 3620

23 Currie D Partial correction of irregular anisei-konia secondary to retinal traction Optom VisSci 2012891081ndash1086




24 Estes WK The problem of inference from curvesbased on group data Psychol Bull 195653134ndash140

25 Remole A Dynamic versus static aniseikoniaAust J Optom 198467108ndash113

26 Stahl JS Amplitude of human head movementsassociated with horizontal saccades Exp BrainRes 199912641ndash54

27 Ogle KN Theory of the space-eikonometer JOpt Soc Am 19463620ndash32

28 Oyama T Nanri R The effects of hue andbrightness on the size perception Jpn PsycholRes 1960213ndash20

29 Tedford WH Bergquist SL Flynn WE The size-color illusion J Gen Psychol 197797145ndash149

30 Cleveland WS McGill R A color-caused opticalillusion on a statistical graph Am Stat 198337101ndash105

31 Nakano M Tanabe S Mori Y Ikegami B FujitaI Expansive and contractive size perception withcolor patches J Vis 20055(8)1027

32 Yoo HS Smith-Jackson TL Colour size illusionon liquid crystal displays and design guidelinesfor bioinformatics tools Behav Inform Tech201130775ndash785

33 Edwards AS Effect of color on visual depthperception J Gen Psychol 195552331ndash333

34 Egusa H Effects of brightness hue and satura-tion on perceived depth between adjacent regionsin the visual field Perception 198312167ndash175

35 Vos JJ Some new aspects of color stereoscopy JOpt Soc Am 196050785ndash790

36 Burbeck CA Criterion-free pattern and flickerthresholds J Opt Soc Am 1981711343ndash1350

37 Heckmann T Schor CM Panumrsquos fusional areaestimated with a criterion-free technique PerceptPsychophys 198945297ndash306

38 Alcala-Quintana R Garcıa-Perez MA A modelfor the time-order error in contrast discrimina-tion Q J Exp Psychol 2011641221ndash1248

39 Garcıa-Perez MA Alcala-Quintana R The dif-ference model with guessing explains interval biasin two-alternative forced-choice detection proce-dures J Sens Stud 201025876ndash898

40 Garcıa-Perez MA Alcala-Quintana R Intervalbias in 2AFC detection tasks sorting out theartifacts Atten Percept Psycho 2011732332ndash2352

41 Garcıa-Perez MA Alcala-Quintana R Shifts ofthe psychometric function distinguishing biasfrom perceptual effects Q J Exp Psychol 201366319ndash337

42 Garcıa-Perez MA Peli E The bisection pointacross variants of the task Atten Percept Psycho2014761671ndash1697

43 Optical Diagnostics Aniseikonia Inspector Ver-sion 3 Userrsquos Manual Available at httpwwwopticaldiagnosticscomproductsaiai_manualpdf Accessed 19 September 2014

44 Mulligan JB Optimizing stereo separation incolor television anaglyphs Perception 19861527ndash36

45 Wang G-J Principles and designs of the iseikoniclenses for near vision Optom Vis Sci 200683E930ndashE937

46 Garcıa-Perez MA Adaptive psychophysicalmethods for nonmonotonic psychometric func-tions Atten Percept Psycho 201476621ndash641

47 Garcıa-Perez MA Alcala-Quintana R On thediscrepant results in synchrony judgment andtemporal-order judgment tasks a quantitativemodel Psychon B Rev 201219820ndash846

48 Garcıa-Perez MA Does time ever fly or slowdown The difficult interpretation of psychophys-ical data on time perception Front HumanNeurosci 20148415

49 Sridharan D Steinmetz NA Moore T KnudsenEI Distinguishing bias from sensitivity effects inmultialternative detection tasks J Vis 201414(9)16

50 Seizova-Caji T Size perception by vision andkinesthesia Percept Psychophys 199860705ndash718

51 Rule SJ Subject differences in exponents ofpsychophysical power functions Percept MotorSkill 1966231125ndash1126

52 Rule SJ Subject difference in exponents fromcircle size numerousness and line length Psy-chon Sci 196915284ndash285

53 Armstrong L Marks LE Differential effects ofstimulus context on perceived length implicationsfor the horizontalndashvertical illusion Percept Psy-chophys 1997591200ndash1213

54 Numerical Algorithms Group NAG FortranLibrary Manual Mark 19 Oxford 1999

55 Garcıa-Perez MA Alcala-Quintana R Samplingplans for fitting the psychometric function SpanJ Psychol 20058256ndash289

56 Alcala-Quintana R Garcıa-Perez MA A com-parison of fixed-step-size and Bayesian staircasesfor sensory threshold estimation Spatial Vis200720197ndash218

57 Garcıa-Perez MA Alcala-Quintana R Bayesianadaptive estimation of arbitrary points on a




psychometric function Brit J Math Stat Psy

200760147ndash174

58 Garcıa-Perez MA Alcala-Quintana R Empirical

performance of optimal Bayesian adaptive esti-

mation Span J Psychol 2009123ndash11

59 Kershaw CD Statistical properties of staircase

estimates from two interval forced choice exper-

iments Brit J Math Stat Psy 19853835ndash43

60 Leek MR Hanna TE Marshall L Estimation of

psychometric functions from adaptive tracking

procedures Percept Psychophys 199251247ndash256

61 Maloney LT Confidence intervals for the pa-rameters of psychometric functions PerceptPsychophys 199047127ndash134

62 Peli E Rabinovich J Barnea D A new deform-able adjustable lens for presbyopia Optom VisSci 199269(Suppl)115

63 Variable Focus Eyeglasses Available at httpwwwallaboutvisioncomlensesvariable-focushtm Accessed March 25 2015




Introduction

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booklet where anaglyphs are printed (at six per page)is held at reading distance and no complex instrumentis required to deliver the stimuli A validity studyshowed that the NAT underestimates natural anisei-konia as well as aniseikonia induced with size lenses(henceforth lens-induced aniseikonia) by a factor ofapproximately 311 The underestimation was con-firmed later1213 but its cause remained unknown Atiny part of it can be attributed to an unnoticed designerror in the NAT the Plus Aniseikonia Series wascreated by reducing the size of the test (green)semicircle by some specified percentages and thesereductions are incorrectly taken directly as theamount of aniseikonia If the pair is selected in whichthe test semicircle is x smaller the amount ofaniseikonia is not x because an image reduced by xmust be enlarged instead by y to regain its originalsize with yfrac14 100x(100 x) The resultant nonlinearunderestimation y x varies with x as x2(100 x)which is less than the reported amount (eganiseikonia measured as 10 with the NAT wouldactually be 1111 which represents underestimationby a factor of only 11)

A more recent alternative is the Aniseikonia Inspec-tor (AI)14 a test based on anaglyphs similar to thosemaking up the NAT but administered via computerThree versions of the AI were developed over the yearsVersion 1 reproduced the video test of McCormack etal11 where unlimited time is given tomanipulate the sizeof one of the semicircles until it matches subjectively thesize of the other In Version 2 semicircles were replacedwith rectangles unlimited inspection time was replacedwith short presentations and the method of adjustmentwas replaced with a task in which observers indicatewhether the left the right or neither rectangle issubjectively larger than the other The measure ofaniseikonia then arises from fitting a two-parameterpsychometric function to the data via treatment oflsquolsquoequalrsquorsquo responses as half lsquolsquoleftrsquorsquo and half lsquolsquorightrsquorsquo FinallyVersion 3 defaults back to unlimited presentation timealthough the option for short presentations is stillavailable with no other major changes

Studies on the validity of the AI offer conflictingresults In principle the video test11 motivating the AIVersion 1 proved valid In the same line de Wit14

reported accurate measurements of lens-induced anis-eikonia in the horizontal vertical and oblique meridiawith the AI Version 1 Analogous results were reportedfor a custom-made computerized test15 that resemblesthe video test11 and the AI Version 1 In contrastRutstein et al16 reported for the AI Version 1 ameaningful underestimation with respect to the expect-

ed amount of lens-induced aniseikonia and also withrespect to measurements obtained with the SE in thehorizontal and vertical meridia in the light and in thedark This underestimation was lower in magnitudethan that reported byMcCormack et al11 for the NATthe slope of the relation of measured to lens-inducedaniseikonia with the AI Version 1 was 067 in thevertical meridian and 058 in the horizontal meridianUnderestimation of lens-induced aniseikonia with theAI Version 1 has been reported in other studies17ndash19

Version 2 on the other hand seemed to overestimateaniseikonia instead18 Finally the only published studyemploying Version 3 has reported acceptably accuratemeasurements of lens-induced aniseikonia in the shortpresentation mode with central fixation20

Misestimation of aniseikonia has been claimed to beclinically irrelevant21ndash23 because further checks can beconducted before or after prescribing iseikonic specta-cle lenses although the process is lengthy and may beexpensive Yet identifying the cause of the underesti-mation in the NAT and the AI Version 1 as well as thecause of the conflicting patterns of misestimationacross versions of the AI is a problem whose solutioncould render improved tests that save time and costs topatients in clinical practice also helping to preventanalogous problems that might arise in the use ofpsychophysical methods for the assessment of othervisual functions Identifying the causes of underesti-mation of the NAT and the AI (and by contrast thereasons that the OE and the SE provide accuratemeasurements) can also contribute to a better under-standing of the perceptual aspects of aniseikonia andthus lead to better treatment options

The goal of this research was to identify the causesof these patterns of misestimation and to identify theconditions (method stimuli viewing mode etc) foraccurate measurements of aniseikonia The protocolfor data collection used in this study is more thoroughthan those in the NAT or the AI and permits theseparation of sensory and decisional influences (crite-rion bias) on performance Also data analyses arecarried out separately for each observer as group datatypically obscure rather than clarify the outcomes24

The next three sections describe the reasons justifyingthe choices made in the design of the study namelyviewing conditions stimulus characteristics and psy-chophysical procedure

Viewing Conditions

Comparative judgments of the size of two objectscan be made using direct perceptual estimates of sizeor using indirect correlates of size A simple demon-



































Methods


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Methods


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Procedure



































OSethxsTHORNp

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b frac14bOD=


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Introduction

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Procedure



































OSethxsTHORNp

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b frac14bOD=


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Introduction

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Introduction

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article aniseikonia tests: the role of viewing mode ...eprints.ucm.es/35781/1/aniseikonia tests the...

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