Psychometric functions for detection and discrimination with and ...

Atten Percept Psychophys (2011) 73:829–853DOI 10.3758/s13414-010-0080-8Psychometric functions for detection and discriminationwith and without flankersMiguel A. García-Pérez & Rocío Alcalá-Quintana &Russell L. Woods & Eli PeliPublished online: 13 January 2011# Psychonomic Society, Inc. 2011Abstract Recent studies have reported that flankingstimuli broaden the psychometric function and lowerdetection thresholds. In the present study, we measuredpsychometric functions for detection and discriminationwith and without flankers to investigate whether theseeffects occur throughout the contrast continuum. Ourresults confirm that lower detection thresholds withflankers are accompanied by broader psychometricfunctions. Psychometric functions for discriminationreveal that discrimination thresholds with and withoutflankers are similar across standard levels, and that thebroadening of psychometric functions with flankersdisappears as standard contrast increases, to the point thatpsychometric functions at high standard levels are virtuallyidentical with or without flankers. Threshold-versus-contrast(TvC) curves with flankers only differ from TvC curveswithout flankers in occasional shallower dippers and lowerbranches on the left of the dipper, but they run virtuallysuperimposed at high standard levels. We discuss differencesbetween our results and other results in the literature, and howthey are likely attributed to the differential vulnerability ofalternative psychophysical procedures to the effects ofpresentation order. We show that different models of flankerfacilitation can fit the data equally well, which stresses thatsucceeding at fitting a model does not validate it in any sense.M. A. García-Pérez (*) : R. Alcalá-QuintanaDepartamento de Metodología, Facultad de Psicología,Universidad Complutense,Campus de Somosaguas,28223 Madrid, Spaine-mail: miguel@psi.ucm.esR. L. Woods : E. PeliSchepens Eye Research Institute, Harvard Medical School,Boston, MA, USAKeywords Detection threshold . Discrimination threshold .TvC curve . Psychometric function . Lateral interactions .Facilitation . Order effects . 2AFCStudies on lateral interactions have shown that the contrastthreshold for detection of a target varies when it ismeasured in the presence or in the absence of flankingabove-threshold stimuli. The magnitude and sign of thisvariation are, above all else, subject dependent, but theyalso depend on spatial frequency, visual eccentricity,presentation duration, or on the precise location, size,orientation, contrast, or onset asynchrony of the flankers(Adini, Sagi, & Tsodyks, 1997; Cass & Spehar, 2005;Giorgi, Soong, Woods, & Peli, 2004; Polat, 2009; Polat &Sagi, 1993, 1994a, 1994b, 2006; Shani & Sagi, 2005;Solomon & Morgan, 2000; Solomon, Watson, & Morgan,1999; Tanaka & Sagi, 1998; Williams & Hess, 1998;Woods, Nugent, & Peli, 2002), and also on the paradigmused in the experiments (García-Pérez, Giorgi, Woods, &Peli, 2005).Although contrast detection threshold was traditionallythe only parameter of concern in these studies, Petrov,Verghese, and McKee, (2006) and Shani and Sagi (2006)reported that lower thresholds in foveal presentations withflankers are accompanied by a flattening of the psychometricfunction. Other studies on the peripheral vision ofpatients with central field loss and age-matched normalshave also shown a consistent flattening of psychometricfunctions without strong effects on detection thresholds(Alcalá-Quintana, Woods, Giorgi, & Peli, 2010). Interestingly,this flattening can also be observed in results reportedby authors who failed to notice the outcome. For instance,García-Pérez et al. (2005) estimated psychometric functionswith and without flankers in 16 different conditions foreach of four observers at 4 deg of eccentricity, and their

830 Atten Percept Psychophys (2011) 73:829–853analysis focused as usual on a comparison of thresholdestimates. Yet, their Fig. 2 showed sample data and fittedpsychometric functions that seem to corroborate the claimsof Petrov et al. (2006) and Shani and Sagi (2006). Thebroader picture is shown here in Fig. 1, which shows therelation between threshold and support (a measure ofspread that is an inverse function of the usual slopeparameter) with and without flankers in the data ofGarcía-Pérez et al. (2005). Because the effect of flankersvaried with spatial and temporal paradigms, data from eachcondition are plotted separately in Fig. 1a, b. The left panelsshow thresholds with flankers against thresholds withoutflankers, where it can be noted that facilitation is notobserved in spatial 2AFC but is prevalent in temporal2AFC. The center panels show that the effect of flankers isconsistent across paradigms in producing a flattening of thepsychometric function (i.e., an increase in its support);finally, the right panels show that lower thresholds in thepresence of flankers are associated with flatter psychometricfunctions only under temporal 2AFC.ThedatapresentedinFig.1 concur with the results ofPetrov et al. (2006), Shani and Sagi (2006), and Alcalá-Quintana et al. (2010) in suggesting that the presence offlankers affects support more than threshold. Actually,Fig. 1 reveals that the only unique, strong, and consistenteffect of the presence of flankers is a flattening of thepsychometric function, but all these data reveal the effectof flankers on threshold and support only at thresholdcontrasts. Direct evidence on whether this flattening ofpsychometric functions occurs also at suprathresholdlevels is lacking. Some studies have measured discriminationthresholds with and without flankers with an eyetoward obtaining threshold-versus-contrast curves (TvCcurves; see Adini & Sagi, 2001; Chen & Tyler, 2001,2002, 2008; Zenger-Landolt & Koch, 2001), but theyprovide only indirect evidence that the psychometricfunctions for discrimination have a larger support (i.e.,they are flatter) in the presence of flankers. Nonetheless, itis worth discussing what the data reported in those studiesindicate, however indirectly, about the support of thepsychometric functions at high contrast levels with andwithout flankers.Interestingly enough, the TvC curves reported in thesefew studies show a large diversity of patterns as to howcontrast discrimination thresholds vary with standardcontrast (i.e., with the contrast of the standard stimulus)with and without flankers. For instance, Zenger-Landoltand Koch (2001; see their Fig. 3) and Adini and Sagi(2001; see their Fig. 2b) reported TvC curves in thepresence of flankers that were substantially above thosefor the target alone, and a large fraction of the curves had asecond dipper in the high-contrast region. These resultsthus suggest that psychometric functions with flankers areflatter, 1 although they are not uniformly so along thecontrast continuum. Yet, a different pattern was reported byChen and Tyler (2001, 2002, 2008). In this case, TvCcurves with flankers were generally merely shifted to theleft of TvC functions without flankers so that the TvCfunction with flankers lay below that without flankers inthe low-contrast region and then crossed over and layabove that with flankers in the high-contrast region.Chen and Tyler (2008) interpreted this pattern asindicating that flankers produce facilitation in the lowcontrastregion and inhibition (threshold elevation) in thehigh-contrast region. But these characteristics are alsoconsistent with the notion that, with flankers, psychometricfunctions are steeper with low-contrast standards andflatter with high-contrast standards.Because of these discrepant results and also because theevidence that they provide about the effect of flankers onthe support of the psychometric function is only indirect,the aim of the work described in the present article was togather direct evidence on this issue. Thus, psychometricfunctions with and without flankers were estimated fordetection and for discrimination at standard contrastsranging from near detection threshold to well abovedetection threshold. And, in doing so, we paid particularattention to circumventing a problem of 2AFC methods thatis often overlooked and that affects the estimated support(or flatness) of the estimated psychometric functions. Useof 2AFC tasks in contrast discrimination studies ispotentially affected by so-called “order effects,” whichshow that the results differ according to whether thestandard stimulus is presented in the first or secondintervals of each 2AFC trial. Actually, psychometricfunctions separately fitted to data from either type of trialare generally laterally shifted away from the standard leveland in opposite directions, with the consequence that apsychometric function fitted to aggregated data from bothtypes of trials is flatter than either of its components1 The link between a higher point in a TvC function and a flatterpsychometric function is easy to establish from the conventionalassumptions implied in the estimation of TvC functions. Indeed, theselatter functions represent the increment in contrast that is needed toachieve a certain performance level in a 2AFC discrimination task, asa function of pedestal contrast. Thus, each point on the TvC curveindicates the location of a percentage point on the underlyingpsychometric function for discrimination at the applicable pedestalcontrast. A second point on the same psychometric function is givenby the location of the point of subjective equality, which in TvCmeasurements is implicitly assumed to occur when standard andcomparison stimuli have the same contrast. Thus, given that thepsychometric function for discrimination is assumed anchored so thatits 50% point occurs at the pedestal contrast, the increment thresholdgiven by the height of the TvC curve at the pedestal contrastnecessarily determines how flat or steep the psychometric functionis. This issue will be further illustrated in Fig. 3, and a more thoroughtreatment is deferred to the Discussion section.

Atten Percept Psychophys (2011) 73:829–853 833lnð99Þ 9:19b. Alternatively, when d ¼ 37:754ð1 l gÞ=100, Equation 5 reduces to σ = β so that βcan be interpreted as the width of the central 24.49% spanof Ψ. These characteristics are illustrated in Fig. 2 for alogistic Ψ with γ = .5 (for 2AFC detection tasks), 1 =.02,β = 0.2 (yielding σ = 1.083 with δ =.03),andθ = –1.5(yielding x T = –1.32 relative to π T =.84).ExperimentsThe experiments for this investigation required estimatingthe psychometric function for detection of a targetin the presence and also in the absence of abovethresholdflankers. Also required was a contrast discriminationexperiment involving a number of differentstandard contrasts—an experiment that also needs to beconducted in the presence and in the absence of thesame flankers that were used in the detection experiment.These experiments were carried out in the twolabs in which the authors work, using identical softwarebut slightly different hardware. The details and resultsof these experiments are described next.Apparatus and StimuliAll experiments were controlled by PCs equipped withVisionWorks (Swift, Panish, & Hippensteel, 1997). InMadrid, stimuli were displayed on a 20-in. ClintonMonoray (Richardson Electronics Ltd., LaFox, IL) monochromemonitor (model M20ECD5RE, DP104 phosphor)with a spatial resolution of 1,024 × 768 pixels (horizontal ×vertical), a luminance resolution of 2 15 gray levels, and aframe rate of 127 Hz. The image area spanned 36.8 cmhorizontally and 27.6 cm vertically, which subtended20.85 × 15.71 deg at a distance of 100 cm. In Boston,stimuli were displayed in monochrome mode on an EIZOFlex-Scan FX·E7 color monitor with a spatial resolution of1,024 × 600 pixels, a luminance resolution of 2 15 graylevels, and a frame rate of 122 Hz. The image area spanned40 cm horizontally and 23.4 cm vertically, subtending22.62 × 13.35 deg at a distance of 100 cm. The voltage-toluminancenonlinearity was compensated for via lookuptables arising from a calibration procedure thatrendered correlations of .999961 and .999983 (in Madridand Boston, respectively) between actual and nominalluminance.The target stimulus was a Gabor patch with ahorizontal carrier of 2 c/deg and a circular Gaussianenvelope with a standard deviation of 0.35 deg (resultingin half-amplitude spatial-frequency and orientationbandwidths of 0.792 octaves and 31.06 deg, respectively).2 The target was created within a 98 × 98 pixel array(90 × 90 in Boston) and was thus truncated to approximately±2.84 SDs of the Gaussian envelope. The stimuluswas represented internally with 24.44 pixels per cycle ofthe carrier (22.48 in Boston) and was displayed with amean luminance of 36 cd/m 2 (38 cd/m 2 in Boston) thatblended in with a uniform background of the same meanluminance and covering the entire image area. A smallblack circle (2 pixels in radius) was permanently displayedat each of the four corners of the square area at which thetarget stimulus would appear, and the observers wereasked to maintain fixation around the center of the areawhose perimeter was thus marked. This fixation aid wasnot extinguished during stimulus presentation, and itsperimetric location with respect to the target helped toprevent masking effects (see Summers & Meese, 2009).The standard stimulus in discrimination experiments wasthoroughly analogous except that its contrast was fixed atthe applicable level. Standard and target were separatestimuli whose contrasts were manipulated independently.The temporal course of stimulus presentation was arectangular pulse of 181.1 msec (23 video frames) inMadrid and 180.3 msec (22 video frames) in Boston.In separate blocks in each experiment, the stimulusfield either consisted of the target (or the standard,when applicable) just described or included also twoflanking Gabor patches of the same frequency, orientation,and size, but with a fixed suprathreshold Michelsoncontrast, c = .4, which were located to the left andright of the target (and standard, when applicable),centered 2 deg (i.e., four wavelengths) away from it, anddisplayed on both intervals of each 2AFC trial. 3 Flankerswere created within pixel arrays of the same size as thoseused to create the target (and the standard) so that theflankers and target (or standard) did not overlap spatiallywhen presented simultaneously. Simultaneous presence of theflankers at a fixed contrast requires that some slots in thedisplay look-up table be reserved for them. Thus, the flankerswere displayed with a total of 126 fixed gray levels, whereasthe target (or standard) was also displayed with 126 graylevels that were drawn from a palette of 2 15 levels to renderthe contrast required for the presentation. To avoid differencesin these respects between blocks with and without2 Our software uses the conventional expression for the argument ofthe exponential in a Gaussian probability density function (whosedenominator includes a factor of two), as opposed to that adopted inmany psychophysical studies (which omit this factor). Then, the(conventional) standard deviation we are reporting for our Gaussianenvelope implies ultimately that our stimuli are identical to those inwhich the (unconventional) standard deviation has been reported toequal the wavelength of the carrier.3 We omit a figure showing our stimuli because they are identical tothose that have been shown in many previous studies (see, e.g., Polat& Sagi, 1993, 1994b; Woods et al., 2002).

834 Atten Percept Psychophys (2011) 73:829–853flankers, in the latter, the target (and the standard) wasalso displayed with 126 gray levels only.ProcedureThemonitorwasallowedtowarmupfornolessthanhalf an hour before any session started. Binocularviewing with natural accommodation and pupils wasused. Observers sat 100 cm away from the display, andtheir heads were not restrained, although they wereasked to maintain a fixed viewing distance throughoutthe experiment. The room was dark except for the lightfrom the display monitor. The background luminanceand the fixation aid were present throughout theexperimental session.All data were gathered using a temporal 2AFCparadigm. A trial consisted of two intervals; the targetwas displayed only in one of these (newly decided withequiprobability on each trial, but with the constraint thatexactly half of the trials display the target in eachinterval), whereas the other interval displayed meanluminance (in detection experiments) or a standard ofthe required contrast. In sessions with flankers, thesewere present in both intervals. The two temporalintervals were marked by beeps of different pitch andwere separated by a gap of 503.9 msec (64 frames) inMadrid or of 516.4 msec (63 frames) in Boston, withintertrial intervals of at least the same duration as theinterstimulus intervals. The observer’s task was toindicate by a keypress the interval in which the targethad been presented (in detection experiments) or theinterval in which the stimulus had higher contrast (indiscrimination experiments). If both intervals appearedto have displayed a stimulus with the same contrast (ora blank), observers were instructed to use a third, “don’tknow” key to indicate their indecision. 4 This event wasrecorded in the data file, but it also made the computergenerate a random response so that the adaptive staircasesto be described next could proceed. If the observers hadmissed a trial for whatever reason, they could use a fourthkey to ask for the trial to be discarded and repeated (notnecessarily immediately afterward). The session was selfpaced;the next trial did not start until the observer hadresponded. This is the reason that intertrial intervals mighthave variable length, but are still with the minimumduration specified above.Data were gathered using an adaptive method ofconstant stimuli governed by 1–up/1–down staircases. Full4 Strictly speaking, this three-response format with a “don’t know”option should not be properly referred to as 2AFC. Kaernbach (2001)dubbed it “two-alternative unforced-choice,” but for simplicity we willcontinue to refer to it as 2AFC.details and justification for the use of these staircases aregiven in García-Pérez and Alcalá-Quintana (2007b; see alsoGarcía-Pérez & Alcalá-Quintana, 2005), but their setup isbriefly described next. In detection experiments, steps uptripled the size of steps down (0.3 and 0.1 log units,respectively), and two separate staircases were interwoventhat differed only by 0.05 log units in their starting points(log contrast of –1 versus –0.95)—that is, half the base stepsize of 0.1 log units used in each staircase, so that the twostaircases ran on interlaced lattices. The two starting pointsrepresent contrast levels at which the target was well abovethreshold. The detection experiment was carried out firstand thus helped select the set of standard contrasts to beused in the discrimination experiment. Each of the variousstandard contrast levels thus selected was used in a separatesession of the discrimination experiment.The discrimination experiment involved 12 standardlevels placed around each observer’s detection thresholddetermined earlier. The levels used with each observer inthe conditions with and without flankers are listed inTable 1 and are labeled from s 1 to s 12 . Note that the spacingof standard levels around the detection threshold (whichdefines standard level s 3 ) is finer than the spacing ofstandard levels well above the detection threshold: 0.1 logunits between levels s 1 and s 6 , 0.15 log units between levelss 6 and s 9 , and 0.2 log units between levels above s 9 . 5 Thereason for these different spacings is that we wanted toevaluate discrimination performance when the standardlevel was within the nonasymptotic regime of the psychometricfunction for detection (standard levels s 1 to s 6 ) andalso when the standard level was within the upperasymptotic regime of the psychometric function fordetection (standard levels s 7 to s 12 ).The setup of staircases in discrimination experimentsvaried with the contrast of the standard. When the standardwas at or below 0.2 log units above the detection threshold(i.e., for standard levels between and including s 1 and s 5 ;see Table 1), one of the interwoven staircases used steps upthat were triple the size of steps down (0.45 and 0.15 logunits, respectively), whereas the other used steps up thatwere double the size of steps down (0.30 and 0.15 log units,respectively), and their respective starting points were0.450 log units above and 0.525 log units below thecontrast of the standard. At higher standard levels (s 6 tos 12 ), the interwoven staircases also had starting points thatwere 0.450 log units above and 0.525 log units below thecontrast of the standard, but they applied inverse rules so5 This statement is true for standard levels in the condition withoutflankers. Since we also wanted standard levels s 7 to s 12 to be the samewith and without flankers, the gap between levels s 6 and s 7 in thecondition with flankers is sometimes different from the 0.15 log unitsjust mentioned.

836 Atten Percept Psychophys (2011) 73:829–853the stimulus level for which performance is at the midpointof the range of the psychometric function. The sameEquations 4 and 5 were used with discrimination data,where PSEs were conventionally obtained relative to π T =.5 and discrimination thresholds were obtained relative toπ T = .75, whereas measures of support were obtainedmaking d ¼ ð1 l gÞ=100. Figure 3 illustrates all ofthese measures using sample psychometric functions fordetection, near-threshold discrimination, and abovethresholddiscrimination.Detection data were also subjected to a second analysisby which the psychometric function was fitted using thedenoising approach described in García-Pérez (2010). Inpractice, denoising merely implies (a) replacing computergeneratedresponses arising from the observer’s use of the“don’t know” key with wrong responses, and (b) fitting ageneral form for the psychometric function in which thelower asymptote is a free parameter ξ. For additionaldetails, see García-Pérez (2010). Discrimination data werenot denoised in any way.Some of our analyses require tests of equality (e.g.,whether discrimination thresholds with and withoutflankers are identical). In all of these cases, the Bradley–Blackwood test was used, which is a robust simultaneoustest of equality of means and variances with paired data(Bradley & Blackwood, 1989).ObserversFive experienced psychophysical observers with normal orcorrected-to-normal vision participated in the study.Probability correct10.0.5support (lower curve)support0.0–3 –2 –1 0Log contrastsupportsupport0.75Fig. 3 Graphical illustration of the estimates of detection thresholdand support obtained from psychometric functions for detection (curveon the left), and the estimates of PSE, discrimination threshold, andsupport obtained from psychometric functions for discrimination nearthe detection threshold (second curve from the left) and at twodifferent levels above the detection threshold (two curves on theright). For simplicity, this illustration assumes 1 = 0. Discriminationthresholds relative to a 75%-correct performance level on thecorresponding curve are given by the distance between the referencedashed and solid vertical lines for each curve; PSEs (the 50%-correctpoint in discrimination curves) are indicated by solid vertical lines0.5Observers M1, M2, and M3 participated in Madrid;Observers B1, B2, and B3 participated in Boston. ObserversM1 and B1 are the same person (one of the authors);Observers M2 (an author) and B2 were also aware of thedesign and goals of the study; the remaining observers (M3and B3) were naive in all these respects. Prior to theirparticipation, all observers read and signed an informedconsent form that had been approved by the InstitutionalReview Board in accordance with NIH regulations.ResultsEstimated psychometric functions for detectionFigure 4a shows raw data and fitted psychometric functionsfor detection for each observer in each condition. Severalaspects of these data are worth pointing out. First, B3 is theonly observer showing threshold elevation in the presenceof flankers, in an amount of 0.05 log units; the remainingobservers did show some facilitation in an amount thatvaried between 0.02 log units (Observer B1) and 0.09 logunits (Observer M1). Second, Observer M1 showedfacilitation in the typical amount (a threshold reduction of0.09 log units in the presence of flankers), whereasObserver B1 (who is indeed the same person, as indicatedabove) showed a smaller effect (0.02 units), although it isunclear whether this is simply a question of sampling error.At the same time, Observer B1 showed lower absolutesensitivity than did Observer M1: Detection thresholds forB1 were 0.15 log units higher without flankers and 0.22log units higher with flankers. These differences inabsolute sensitivity cannot be attributed to the differenthues of the phosphors in each monitor (a yellowish-greenphosphor for Observer M1 compared to an RGB whitelookingphosphor for Observer B1), because contrastsensitivity in green and white monochromatic light isvirtually identical, provided the mean luminance ismatched (Nelson & Halberg, 1979; Watanabe, Mori,Nagata, & Hiwatashi, 1968; Zulauf, Flammer, & Signer,1988). A more subtle difference in hardware seems to haveproduced these variations in absolute thresholds, but adiscussion of this side issue (which has no impact on thepaired comparisons involved in this study) is deferred tothe Appendix.Figure 4b shows analogous psychometric functionsfitted to denoised data, which confirm the aforementionedpoints. In particular, estimates of threshold andsupport are virtually identical for each observer andcondition whether they are obtained from raw data(Fig. 4a) or denoised data (Fig. 4b). Both approachesdiffer only as to how “don’t know” responses are treated:They are regarded either as half right and half wrong(Fig. 4a) or as wrong responses. As discussed in García-

Atten Percept Psychophys (2011) 73:829–853 837(a) Raw dataProbability correctProbability correct1.00.80.6M10.4λˆ 0.000 0.0070.2θˆ –1.99 –2.08σˆ 0.829 0.9540.0–2.5 –2.0 –1.5 –1.0 –0.5(b) Denoised dataProbability correctProbability correct1.00.80.61.00.8B10.4λˆ 0.000 0.0000.2θˆ –1.84 –1.86σˆ 0.689 0.7920.0–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, xM10.60.40.2ξˆθˆ0.103–1.980.121–2.06λˆσˆ0.0000.7480.0050.9050.0–2.5 –2.0 –1.5 –1.0 –0.51.00.8B10.60.40.2ξˆθˆ0.071–1.840.079–1.87λˆσˆ0.0000.6960.0000.8600.0–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, x1.00.80.6M20.4λˆ 0.000 0.0000.2θˆ –2.08 –2.15σˆ 0.637 0.8530.0–2.5 –2.0 –1.5 –1.0 –0.51.00.80.60.40.20.01.00.80.6B2M2λˆθˆσˆ–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, x0.40.2ξˆθˆ0.050–2.080.086–2.14λˆσˆ0.0000.6490.0000.8300.0–2.5 –2.0 –1.5 –1.0 –0.51.00.80.60.40.20.0B2ξˆλˆθˆσˆ0.000–1.890.8960.0280.000–1.900.9140.000–1.970.9300.1220.000–1.960.851–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, x1.00.80.6M30.4λˆ 0.009 0.0000.2θˆ –1.94 –1.99σˆ 0.577 0.8330.0–2.5 –2.0 –1.5 –1.0 –0.51.00.80.60.41.00.80.6B3M3λˆθˆσˆ0.013 0.0000.2–1.83 –1.780.710 0.6950.0–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, x0.40.2ξˆθˆ0.046–1.940.038–1.98λˆσˆ0.0090.6000.0000.8200.0–2.5 –2.0 –1.5 –1.0 –0.51.00.80.6B3ξˆλˆθˆσˆ0.40.000 0.0140.013 0.0000.2–1.83 –1.790.611 0.6770.0–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, xFig. 4 Data and fitted logistic psychometric functions for detection ofthe target without flankers (open symbols and dashed curves) and withthem (solid symbols and solid curves). To avoid clutter, data points areshown only if at least 10 trials had been administered at the applicablecontrast level, but all data were used for parameter estimation. Part (a)Pérez (2010), one of the advantages of denoising is thatthe free lower asymptote in the fitted psychometricfunction indicates compliance of the observer with theinstructions to use the “don’t know” response key,something that is impossible to check out with raw data.A second advantage is that estimates of support fromdenoised data have smaller standard errors.A summary picture of the relations between thresholdand support with and without flankers is given in Fig. 5inthesameformatusedinFig.1, but for our raw (Fig. 5a)and denoised (Fig. 5b) data. In either case, our resultsshow mild facilitatory effects of flankers (left panels inshows results for raw data; part (b) shows results for denoised data.Each panel in each part shows results for a different observer, asindicated in the top left corner. Estimated parameters are given in theinsets; parameter ξ in part (b) is an estimate of the lower asymptotefor denoised data (see García-Pérez, 2010)Fig. 5a, b), stronger effects on the support of thepsychometric functions in the presence of flankers (centerpanels in Fig. 5a, b), and a lack of relation betweenthreshold and support (right panels in Figs. 5a, b): Therelation was negative but not significantly different fromzero in the presence of flankers whether for raw data [r =–.74; 95% confidence interval (–.97, .18) in Fig. 5a] orfordenoised data [r = –.62; 95% confidence interval (–.95,.38) in Fig. 5b], and a weaker sample correlation was alsonot significantly different from zero for the target alonewhether for raw data (r = .20 in Fig. 5a) or for denoiseddata (r = .09 in Fig. 5b).

838 Atten Percept Psychophys (2011) 73:829–853Fig. 5 Relation between detectionthresholds with and withoutflankers (left panels), betweensupport with and withoutflankers (center panels), andbetween support and detectionthreshold (right panels; solidsymbols for the condition withflankers and open symbols forthe condition without flankers)in the results shown in Fig. 4 forraw data (a) and denoised data(b). When larger than symbolsize, error bars (vertical forestimated values along thevertical axes and horizontal forestimated values along thehorizontal axes) are standarderrors taken from García-Pérez(2010; see his Fig. 3) for rawor denoised estimates arisingfrom sets of 1,000 trials(a) Raw dataThreshold with flankers–1.7–1.8–1.9–2.0–2.1–2.2–2.2 –2.1 –2.0 –1.9 –1.8 –1.7Threshold without flankers(b) Denoised dataThreshold with flankers–1.7–1.8–1.9–2.0–2.1–2.2–2.2 –2.1 –2.0 –1.9 –1.8 –1.7Threshold without flankersSupport with flankersSupport with flankers1.510.500 0.5 1 1.5Support without flankers1.510.500 0.5 1 1.5Support without flankersSupportSupport1.510.50–2.2 –2.1 –2.0 –1.9 –1.8 –1.7Threshold10.50–2.2 –2.1 –2.0 –1.9 –1.8 –1.7ThresholdEstimated psychometric functions for discriminationFigure 6 shows data and fitted psychometric functions fordiscrimination for each observer (rows) in each condition(distributed across two panels to reduce clutter). At lowstandard levels (between s 1 and s 6 ), for which thepsychometric function for detection has not yet reachedits upper asymptotic regime, the psychometric functionsfor discrimination have the high lower asymptoteexpected from the formal analysis in García-Pérez andAlcalá-Quintana (2007b); at higher standard levels (at andabove s 7 ), where detection performance is at its ceilinglevel, the lower asymptote of the psychometric functionfor discrimination is at or near zero. These characteristicshold with and without flankers.Figure 7 compares the location and support of thepsychometric functions for discrimination in the presence(solid symbols and curves) and in the absence (opensymbols and dashed curves) of flankers at each standardlevel separately for the case of standard levels around thedetection threshold (s 1 to s 6 ; Fig. 7a) and for the case ofstandard levels well above the detection threshold (s 7 to s 12 ;Fig. 7b). In the former case, the specific standard levelsused in the presence and in the absence of flankers aregenerally different for any given observer (see Table 1)because they are defined relative to the detection thresholdin each condition. For this reason, parameter estimatesobtained with and without flankers are first plotted as afunction of standard contrast in the top row of Fig. 7a. Theleft panel in the top row of Fig. 7a plots the PSE for eachobserver and standard level against the actual standardlevel, with open symbols when target and standard werepresented alone and solid symbols when target and standardwere presented with flankers. In principle, the PSE shouldmatch the standard level within sampling error in all cases,and the data bear this expectation with no apparentdifferences between conditions with and without flankers.The correlation between standard level and PSE in theabsence of flankers was .995 (it was .991 with flankers),with the 95% confidence interval ranging from .990 to .998(or from .981 to .995 with flankers); on the other hand,least-squares regression lines fitted to the data yieldedintercepts of –0.112 without flankers and –0.068 with themthat were not significantly different from zero [95%confidence intervals (–0.172, 0.052) without flankers and(–0.147, 0.010) with them] and slopes of 0.932 withoutflankers and 0.957 with flankers that were not significantlydifferent from unity [95% confidence intervals (–0.202,2.066) without flankers and (–0.505, 2.418) with them].The center panel in the top row of Fig. 7a plotsdiscrimination thresholds as a function of standard levelfor the same conditions for which PSEs were plotted in theleft panel. As is clear from the sketch in Fig. 3, the distancebetween the PSE and the point that serves as a reference todefine the discrimination threshold increases as thepsychometric function flattens. Then, variations in thediscrimination threshold (i.e., the distance between thosetwo points) as a function of standard level are indirectindications of concomitant changes in the support of theunderlying psychometric functions. A quick look at thecenter panel in the top row of Fig. 7a suffices to note thatdiscrimination thresholds decrease as standard level

Atten Percept Psychophys (2011) 73:829–853 839increases and that the sets of conditions with and withoutflankers (solid and open symbols, respectively) do not seem toyield different outcomes. Quantitatively, the significantcorrelation between standard level and discriminationthreshold in the absence of flankers was –.774 (it was–.722 with flankers), with the 95% confidence intervalranging from –.879 to –.598 (or from –.849 to –.516with flankers).Finally, the right panel in the top row of Fig. 7a showshow support varies with standard level, something forwhich the center panel provided only indirect evidence. Thecorrelation between support and standard level was notsignificantly different from zero whether without flankers[open symbols; r = .219; 95% confidence interval (–.118,.511)] or with flankers [symbols; r = –.157; 95%confidence interval (–.462, .181)]. The different outlookgiven by the center and right panels in the top row ofFig. 7a clearly reveals that discrimination thresholds areonly indirect and are somewhat inaccurate indices of howthe support of the psychometric function varies withstandard level. This is because at low standard levels, thepsychometric functions for discrimination have asymmetriclower and upper asymptotes (see Fig. 6).As mentioned previously, near-threshold standard levelswere generally different with and without flankers for eachobserver, something that precludes a direct comparison ofthe characteristics of the psychometric functions with andwithout flankers. Nevertheless, the differences weregenerally small (see Table 1): 0.1 log units (2 dB) forObserver M1, and ±0.05 log units (1 dB) for ObserversM2, M3, B2, and B3, whereas standard levels with andwithout flankers for Observer B1 were actually identical.With the necessary precautions in the interpretation ofthese results, the bottom row of Fig. 7a plots PSEs,discrimination thresholds, and support of psychometricfunctions with and without flankers against one another,with solid symbols for Observer M1 (for whom standardlevels with and without flankers differed by 0.1 log units),open symbols for Observer B1 (for whom standard levelswith and without flankers did not differ), and graysymbols for the remaining observers (for whom standardlevels with and without flankers differed by ±0.05 logunits). The left panel in the bottom row of Fig. 7a showsdifferences between PSEs with and without flankers thatmostly reflect the fact that the standard levels were notalways the same in either case and, thus, that solidsymbols lie relatively far below the diagonal, graysymbols lie closer to the diagonal, and open symbols lievirtually on the diagonal. The center panel in the bottomrow of Fig. 7a shows that discrimination thresholds areslightly higher with flankers, but this seems only thenatural consequence of the lower standard levels used withflankers, which are indeed associated with higher discriminationthresholds (see the center panel in the top row ofFig. 7a). Finally, the right panel in the bottom row ofFig. 7a shows that support with flankers is again generallylarger than without them, something that cannot beattributed to any relation between support and standardlevel (see the right panel in the top row of Fig. 7a). In thiscase, the average support with flankers was 0.742 with astandard deviation of 0.145, whereas the average supportwithout flankers was 0.634 with a standard deviation of0.103. The difference is significant by the Bradley–Blackwood test, F(2, 34) = 16.485, p < .0001.In sum, at standard levels near the detection threshold,the support of psychometric functions for discriminationseems slightly larger in the presence of flankers. There arealso clear and significant traces that discrimination thresholdsdecrease as the level of the standard stimulus increases(center panel in the top row of Fig. 7a), whether with orwithout flankers, although no such relation appears to existin either case in terms of the support of the underlyingpsychometric functions (right panel in the top row ofFig. 7a).Figure 7b shows the relationship between psychometricfunctions for discrimination with and without flankers atstandard levels well above the detection threshold. Sincethe set of levels used with and without flankers was thesame for each observer (see Table 1), data plotted in thepanels of Fig. 7b can be subjected to thorough statisticalanalyses. Figure 7b shows that, whether with or withoutflankers, the PSE is approximately at the same location (leftpanel), the discrimination threshold is also similar (centerpanel), and the support of the underlying psychometricfunctions is also similar (right panel). Furthermore, thecenter and right panels in Fig. 7b look about the same, aconsequence of the fact that discrimination thresholds aremore accurate indices of the support of the underlyingfunctions in the absence of large differences between theupper and lower asymptotes of the psychometric function.Across the board, Bradley–Blackwood tests revealednonsignificant differences only in the case of the PSEs inthe left panel of Fig. 7b, F(2, 34) = 0.007, p = .9925.Concerning the center panel of Fig. 7b, the averagediscrimination threshold with flankers was 0.098 with astandard deviation of 0.027, whereas the average discriminationthreshold without flankers was 0.089 with astandard deviation of 0.018— differences that werenevertheless significant, mostly as a result of the quitedivergent variances, F(2, 34) = 7.016, p = .0028. Similarly,in the right panel of Fig. 7b, the average support withflankers was 0.802 with a standard deviation of 0.223,whereas the average support without flankers was 0.734with a standard deviation of 0.143, differences that werealso significant for the same reason, F(2. 34) = 7.290, p =.0023. Thus, the larger support of the psychometric

Probability target reported higherProbability target reported higher840 Atten Percept Psychophys (2011) 73:829–853Probability target reported higher1.00.80.60.40.2M1s1s3s5s7s9s110.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.2M20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.8M31.00.80.60.40.2s2s4s6s8s10s120.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.2B10.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.00.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.8B21.00.80.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.2B30.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.0Target contrast, x0.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.01.00.80.60.40.20.0–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.0Target contrast, xFig. 6 Data (symbols) and fitted logistic psychometric functions(curves) for discrimination. Results for different standard contrasts areshown with different symbols and are separated into two columns ofpanels to reduce clutter (see the legend at the top of each column).Open symbols and dashed curves represent results without flankers;solid symbols and solid curves represent results with flankers. Toavoid clutter, data points are shown only if at least 10 trials had beenadministered at the applicable contrast level, but all data were used forparameter estimation. Each row shows results for a different observer,as indicated in the top left corner in the panels on the left column

Atten Percept Psychophys (2011) 73:829–853 841Fig. 7 a Variations in PSE,discrimination threshold, andsupport of the psychometricfunctions for discrimination as afunction of standard contrast atlevels near detection threshold(top row) and relation betweenPSE, discrimination threshold,and support with and withoutflankers at the same standardlevels (bottom row). Data fromdifferent observers arerepresented with differentsymbols. In the top row,open symbols denote resultswithout flankers and solidsymbols denote results withthem; in the bottom row, opensymbols denote results in whichthe standard levels with andwithout flankers were identical,gray symbols denote results inwhich the standard levels withand without flankers differed by±0.05 log units, and solidsymbols denote results in whichthe standard levels with andwithout flankers differed by0.10 log units. b Relationbetween PSE, discriminationthreshold, and support with andwithout flankers at standardlevels above detection threshold.Data from different observersare represented with differentsymbols(a) Near detection thresholdPSEPSE with flankers–1.4–1.6–1.8–2.0–2.2–2.4–2.4 –2.2 –2.0 –1.8 –1.6 –1.4–1.4–1.6–1.8–2.0–2.2Standard level, x s–2.4–2.4 –2.2 –2.0 –1.8 –1.6 –1.4PSE without flankersDiscrimination thresholdDisc. threshold with flankers(b) Above detection thresholdPSE with flankers–0.3–0.6–0.9–1.2–1.5–1.8–1.8 –1.5 –1.2 –0.9 –0.6 –0.3PSE without flankersDisc. threshold with flankers0.40.30.20.10.0–2.4 –2.2 –2.0 –1.8 –1.6 –1.40.40.30.20.1Standard level, x s0.00.0 0.1 0.2 0.3 0.4Disc. threshold without flankers0.20.10.00.0 0.1 0.2Disc. threshold without flankersSupportSupport with flankersSupport with flankers1.51.00.50–2.4 –2.2 –2.0 –1.8 –1.6 –1.41.51.00.5Standard level, x s00 0.5 1 1.51.51.00.5Support without flankers00 0.5 1 1.5Support without flankersfunction for detection in the presence of flankers (with adifference of about 0.13 units; see the center panels inFig. 5a, b) persisted to a slightly lesser extent (about 0.11 units)at near-threshold contrast levels and was further reduced toabout 0.07 units (although the difference was still significant) atcontrast levels well above the detection threshold.Efficacy of the manipulation to eliminate order effects dueto response biasWe argued in the introduction that the provision of a “don’tknow” response option and application of Fechner’s (1860/1966) half right and half wrong rule eliminates order effectscaused by response bias and renders psychometric functionsthat are not significantly displaced in oppositedirections when fitted to the subset of trials in which thetest was presented first versus to the subset of trials inwhich the test was presented second. Analyses presented inthe preceding section were carried out on aggregated datafrom both trial types, but it is worth seeking traces of ordereffects. For this purpose, sets of psychometric functionswere fitted as described above, but separately to datacoming from the subsets of trials in which the test waspresented first and to data coming from the subset of trialsin which the test was presented second. The magnitude oforder effects was then defined as in Alcalá-Quintana andGarcía-Pérez (in press), namely, as the difference betweenthe estimated PSE and the level of the standard stimulus—that is, as PSE–x s . This difference can also be computedwhen the PSE is estimated from aggregated data acrosspresentation orders, even though the name “order effect”does not make sense in such case and presumably reflectsonly sampling error.The left panel of Fig. 8 plots PSE–x s when the test ispresented second, against PSE–x s when the test is presentedfirst. In a plot such as this, order effects show in that datapoints are concentrated along the negative diagonal andsignificantly away from the origin of coordinates. Theobvious negative correlation in the left panel of Fig. 8 issignificant whether for data with flankers [r = –.66; 95%

842 Atten Percept Psychophys (2011) 73:829–853PSE–St when test second0.20.10.0–0.1–0.2–0.2 –0.1 0.0 0.1 0.2PSE–St when test firstPSE–St for aggregated data0.20.10.0–0.1–0.2–0.2 –0.1 0.0 0.1 0.2PSE–St when test firstPSE–St for aggregated data0.20.10.0–0.1–0.2–0.2 –0.1 0.0 0.1 0.2PSE–St when test secondFig. 8 Scatterplot of the difference between the estimated PSE andthe standard stimulus level (St) across psychometric functions fitted todata from trials in which the test stimulus was presented first, to datafrom trials in which it was presented second, or to aggregated datafrom both presentation orders. Open symbols denote conditionswithout flankers; solid symbols denote conditions with them. Datafrom different observers are plotted with different types of symbols.Only data at the higher standard levels (at and above s 6 ) are includedin this analysis, because the lower asymptote of the remaining datasets is much too high (see Fig. 6)confidence interval (–.80 , –.44)] or for data without them[r = –.78; 95% confidence interval (–.88 , –.63)], but thevast majority of data points actually lie around the origin ofcoordinates, indicative of a lack of order effects, except in afew cases. It should be noted that order effects that areexclusively caused by response bias further result in that thedifference PSE–x s from aggregated data is unrelated to thedifference PSE–x s computed either when the test ispresented first or when the test is presented second, andthis is a defining property of what Ulrich and Vorberg(2009) called “Type-A order effects.” Contrary to thisexpectation, the center panel of Fig. 8 reveals a significantpositive correlation whether with flankers [r = .48; 95%confidence interval (.20 , .68)] or without them [r = .56;95% confidence interval (.31 , .74)]. And, in the right panel,the correlation is significantly different from zero withflankers [r = .32; 95% confidence interval (.02 , .57)], butnot without them [r = .06; 95% confidence interval (–.25 ,.36)]. The presence of these correlations is suggestive ofwhat Ulrich and Vorberg called “Type-B order effects,”which cannot be attributed to response bias and, hence,cannot be eliminated with our strategy. Then, thus far,Fig. 8 suggests that order effects arising from response biashave actually been eliminated, although some otherunknown source of order effects still exists that hascontaminated the data from a few observers in a fewconditions. 7 Interestingly, these residual Type-B ordereffects do not seem to have contaminated our primaryestimates (namely, the support of psychometric functions),as will be discussed next.7 It should be stressed in this respect that data points departing mostfrom the diagonal identity line in the left panel of Fig. 8 generallycame from the same two observers at the highest standard levels (at orabove s 10 ). This may be indicative of another type of order effects thatresults from temporary desensitization caused by interstimulus orintertrial intervals that were too short for these particular observers(see Alcalá-Quintana & García-Pérez, in press), a source of ordereffects that cannot be eliminated with our method.The left panel of Fig. 9 shows that, with flankers,estimates of support when the test is presented second(average of 0.699 with a standard deviation of 0.222) aresomewhat smaller than when the test is presented first(average of 0.830 with a standard deviation of 0.280), andthe difference is significant, F(2, 40) = 9.263, p = .0005.The same was true and in the same direction withoutflankers [averages of 0.654 and 0.730, with standarddeviations of 0.156 and 0.162; F(2, 40) = 6.373; p =.0040]. This is the defining property of Type-B order effectswhich, in principle, have the drawback that the support ofthe psychometric function (and, in turn, the discriminationthreshold) varies with presentation order and, in consequence,may potentially inflate estimates of support (ordiscrimination threshold) obtained from aggregated data(García-Pérez & Alcalá-Quintana, 2010a). Interestingly,this inflation has not occurred here, as the center and rightpanels of Fig. 8 reveal: Estimates of support fromaggregated data are similar to estimates obtained from thesubset of trials in which the test was presented first (centerpanel in Fig. 9), although they are necessarily larger thanestimates obtained from the subset of trials in which the testwas presented second (right panel of Fig. 9). In thepresence of Type-B order effects—and the ensuing uncertaintyas to whether the actual support (or discriminationthreshold) is that estimated when the test is presented firstor second—one can only hope that support estimates fromaggregated data are not further contaminated by thischaracteristic and that they are not still larger than estimatesobtained from the least favorable presentation order. This isactually the case here: From the center panel in Fig. 9,average support from aggregated data is 0.806 (SD: 0.210)with flankers and 0.731 (SD: 0.141) without them, figuresthat compare well with those of estimates coming fromtrials in which the test was presented first (averages of0.830 and 0.730, respectively, already reported in thediscussion of the left panel of Fig. 9). The Bradley–Blackwood test came out significant only with flankers and

Atten Percept Psychophys (2011) 73:829–853 843Support when test second1.61.20.80.40.00.0 0.4 0.8 1.2 1.6Support when test firstSupport for aggregated data1.61.20.80.40.00.0 0.4 0.8 1.2 1.6Support when test firstSupport for aggregated data1.61.20.80.40.00.0 0.4 0.8 1.2 1.6Support when test secondFig. 9 Scatterplot of estimated support across psychometric functionsfitted to data from trials in which the test stimulus was presented first,to data from trials in which it was presented second, or to aggregateddata from both presentation orders. Open symbols denote conditionswithout flankers; solid symbols denote conditions with them. Datafrom different observers are plotted with different types of symbols.Only data at the higher standard levels (at and above s 6 ) are includedin this analysis, because the lower asymptote of the remaining datasets is much too high (see Fig. 6)only because the standard deviations differed meaningfully;a more relevant (for our present purpose) paired-samples ttest revealed that the average support is not significantlydifferent between the two conditions whether with flankersor without them.In sum, Type-A order effects caused by response biasappeared to be eliminated by application of the half rightand half wrong method, but traces of Type-B order effectsare visible in our data whose cause is unknown. Nevertheless,estimates of support obtained from aggregated data donot seem to be inflated by these Type-B order effects.Estimated TvC curvesFigure 7 suggests that psychometric functions for discriminationdo not differ meaningfully with and withoutflankers. No other study that we know of has measuredpsychometric functions in these conditions, but discriminationthresholds obtained with adaptive procedures havebeen used to determine TvC curves (Adini & Sagi, 2001;Chen & Tyler, 2001, 2002, 2008; Zenger-Landolt & Koch,2001). We thus plotted our results also in this form, asshown by the symbols in Fig. 10. In the aforementionedstudies, empirical TvC data were used to fit modelsinvolving a particular transducer function. Here, we fittedthe typical transducer functionðmðcÞ ¼cS EðcS I Þ q þ ZÞ Pð6Þto our raw data (i.e., to the entire set of data displayed inour Figs. 4a and 6, not just to the summaries plotted inFig. 10) separately for each observer and condition.Equation 6 is the well-known transducer function proposedby Foley (1994) and further described by Foley andSchwarz (1998), which assumes that responses are basedon excitatory (represented in the numerator) and inhibitory(represented in the denominator) influences; for additionaldetails regarding the parameters of Equation 6, see alsoGarcía-Pérez and Alcalá-Quintana (2007b). We fittedseparate transducer functions for the conditions with andwithout flankers because our only aim was to obtain anaccurate closed-form summary of empirical data in eachcase, rather than to test particular hypotheses or fitparticular models. The functions were fitted as describedin García-Pérez and Alcalá-Quintana (2007b) and, as usual,parameter S E was set equal to 100 to fix the scale for theremaining (free) parameters. The resultant parameter estimatesare given in Table 2.The transducer functions thus fitted actually generatepredicted psychometric functions for detection and fordiscrimination at any standard level (see García-Pérez &Alcalá-Quintana, 2007b). Then, the 75% and the 50%points x 50 and x 75 on each of these predicted functions weredetermined, and the increment threshold Δc ¼ 10 x 7510 x 50was plotted as a function of standard contrast x s , yieldingthe dashed (for the condition without flankers) and solid (forthe condition with flankers) curves in the panels of Fig. 10. Ingeneral, TvC curves with and without flankers only differedeither at the low-contrast region (Observers M1 and B2) oras to the depth of the dipper (Observers M2, M3, and B1);they did not differ in any meaningful respect for ObserverB3, and only for Observer M2 was a difference observed inthe high-contrast region. Also, no consistent pattern ofvariation across observers could be appreciated.DiscussionWe have measured psychometric functions for detection anddiscrimination with and without flankers, and our resultsconfirm again that psychometric functions for detection areflatter in the presence of flankers. Our results also show that aflattening of the psychometric functions for discrimination ismildly present near the detection threshold, but that it virtuallydisappears well above the detection threshold. At the same

844 Atten Percept Psychophys (2011) 73:829–853Fig. 10 TvC function for eachobserver. Open symbols anddashed curves reflect resultswithout flankers; solid symbolsand curves reflect results withflankers. Data points indicate thelog increment contrastlogðΔcÞ ¼ logð10 x75 10 x50 Þ,where x 75 is the target contrastrequired to attain a discriminationperformance level π T = .75,and x 50 is the PSE, plotted as afunction of the standard contrastx s . Points plotted outside theframe on the left of each panelrepresent detection thresholds atπ T = .75. Dashed and solid lines(for conditions without and withflankers, respectively) are predictedTvC curves from Equation6 with maximum-likelihoodestimates of its parametersIncrement threshold, log(Δc)Increment threshold, log(Δc)–1–2alonewith flankersM1M2–3–3–3 –2 –1 0 –3 –2 –1 0–1–2–3B1–3 –2 –1 0Standard level, x s–1–2–1–2–3B2–3 –2 –1 0Standard level, x s–1–2M3–3–3 –2 –1 0–1–2B3–3–3 –2 –1 0Standard level, x stime, detection thresholds are generally lower in the presenceof flankers, but (with the exception of Observer M2)discrimination thresholds generally remain the same whetherflankers are or are not present. Finally, our results corroboratea feature that was also reported in earlier studies, namely, theheterogeneity of flanker effects across observers.Comparison with previous resultsThe TvC data and curves in our Fig. 10 reveal a new typeof pattern as compared with two other patterns that havebeen described in the literature (see Adini & Sagi, 2001;Table 2 Maximum-likelihood estimates of the parameters of thetransducer function in Equation 6 for each observer and conditionp q S E S I zM1 alone 3.112 2.707 100 57.209 1.067with flankers 2.526 2.117 100 48.380 0.581M2 alone 3.135 2.723 100 61.305 0.587with flankers 2.534 2.134 100 59.114 0.512M3 alone 3.019 2.619 100 60.674 1.284with flankers 2.464 1.990 100 64.773 0.789B1 alone 3.119 2.719 100 56.213 2.888with flankers 3.012 2.552 100 62.581 2.319B2 alone 3.059 2.607 100 59.748 1.614with flankers 2.990 2.576 100 56.543 0.826B3 alone 3.692 3.230 100 73.456 4.118with flankers 3.152 2.752 100 63.423 4.161Chen & Tyler, 2001, 2002, 2008; Zenger-Landolt & Koch,2001) and that were also mentioned in the introduction. Incomparison, our data show instead little differencesbetween TvC curves with and without flankers. This lackof major differences in TvC curves was accompanied by alack of major differences in the support of the psychometricfunctions for discrimination with and without flankers athigh standard levels—a characteristic that cannot becontrasted against previous results because those studiesmeasured only discrimination thresholds.How, then, can these three patterns of results bereconciled? It could certainly be that the discriminationthresholds measured in other studies were contaminated byorder effects (one of whose sources was eliminated by ourmethod) and that the unknown source of these order effectsdifferentially affects conditions with flankers and withoutthem. But, at the same time, the three patterns have eachbeen obtained with a different psychophysical procedure:adaptive threshold estimation using up–down staircases(Adini & Sagi, 2001; Zenger-Landolt & Koch, 2001),adaptive threshold estimation using Bayesian methods(Chen & Tyler, 2001, 2002, 2008), and estimation ofpsychometric functions with elimination of order effectscaused by response bias (the present study).Zenger-Landolt and Koch (2001) and Adini and Sagi(2001) both obtained similar patterns by using up–downstaircases with equal sizes for the steps up and down, whichthey claimed to converge on the 79.3%-correct (or 79%-correct) point on the psychometric function. It has beenshown, however, that these staircases do not converge on

Atten Percept Psychophys (2011) 73:829–853 845their presumed points and that use of steps up and down ofthe same size make their convergence point highlydependent on both the starting point of the staircase andthe relative size of the steps with respect to the support ofthe psychometric function (Faes et al., 2007; García-Pérez,1998, 2000). Furthermore, these staircases were designedfor use in 2AFC detection tasks in which the psychometricfunction has a lower asymptote at .5; 2AFC discriminationtasks with high-contrast standards render instead psychometricfunctions with a lower asymptote near zero (seeFig. 3), thus similar to those that apply in yes–no tasks andfor which staircases set up as those in the studies of Zenger-Landolt and Koch and Adini and Sagi have an even moreerratic behavior (García-Pérez, 2001). Thus, each of thedata points estimated by Zenger-Landolt and Koch or Adiniand Sagi for their TvC curves actually reflects a distinctlydifferent (and largely unknown) percentage-correct level,and variations in this particular level across the board mayactually be very large owing to variations in the support andlower asymptote of the underlying psychometric functions.Neither Zenger-Landolt and Koch nor Adini and Sagimeasured psychometric functions and, therefore, it isuncertain how large the effect may be, but for reference,we have plotted in Fig. 11 the estimated support of thepsychometric function of each of our observers in eachcondition (with and without flankers) as a function ofstandard level. This plot shows that support may vary by asmuch as a factor of three. At the same time, our Fig. 6showed large variations in the lower asymptote of psychometricfunctions for discrimination at near-threshold standardlevels along with lower asymptotes at or near zero inpsychometric functions for discrimination at above-thresholdcontrast levels. It is therefore uncertain what it is thatdiscrimination thresholds measured by Zenger-Landolt andKoch or Adini and Sagi actually represent. And, it should beremembered that they were also obtained under the assumptionthat the PSE lies exactly at the standard level (i.e., nosampling error) and under the potential contamination of ordereffects.We should also stress that the discrepant results canhardly be attributed to other differences between our studyand that of Zenger-Landolt and Koch (2001). For instance,according to results presented in García-Pérez et al. (2005),our temporal 2AFC paradigm should have increased (ratherthan decreased) the magnitude of facilitation found byZenger-Landolt and Koch using a spatial 2AFC paradigm.Also, according to results presented in Giorgi et al. (2004),our foveal viewing should have increased (rather thandecreased) the magnitude of facilitation found by Zenger-Landolt and Koch using eccentric viewing.Chen and Tyler (2001), on the other hand, used QUEST(Watson & Pelli, 1983) to measure detection and discriminationthresholds defined as the 91.5%-correct point on theunderlying psychometric functions (which they did notestimate independently either), also on the assumption thatthe PSE lies exactly at the standard level and withoutmeasures to prevent order effects. With small numbers oftrials, this method has been shown to provide estimates thatare biased toward starting point and that are also affected bylarge systematic errors if the actual psychometric functiondiffersfromthatassumedby QUEST, regardless of whetherthe discrepancy bears upon the mathematical form of thefunctions, their support, or their lower and upper asymptotes(Alcalá-Quintana & García-Pérez, 2004). 8 It has also beenshown that the dependability of estimates obtained withQUEST and its variants is compromised when used to estimatethresholds away from the 80%-correct point on the psychometricfunction for detection or from the 50% point on thepsychometric function for discrimination (García-Pérez &Alcalá-Quintana, 2007a). Subsequently, Chen and Tyler(2002, 2008) used a variant of QUEST that was designed forconcurrent estimation of threshold and slope (Kontsevich &Tyler, 1999). This procedure involves a larger number ofassumptions than QUEST, and its configuration also requiresdecisions that were not described by Chen and Tyler (2002,2008), except for the fact that the procedure was set up toestimate the 75%-correct point using 40 trials. In any case,this variant of QUEST was also designed (and partially tested)for use in cases in which the psychometric function has alower asymptote at .5 (i.e., for 2AFC detection tasks) and itsproperties, when used with discrimination tasks in which thelower asymptote of the psychometric function lies at a lowerand undetermined level, are actually unknown. At the sametime, results reported by Kontsevich and Tyler (1999; seetheir Fig. 1b) indicate that threshold estimates obtained withtheir method are actually positively or negatively biased toan extent that varies with the actual slope of the psychometricfunction (an issue that is most relevant in the presentcontext), and that this bias is not eliminated unless more than100 trials are administered.In contrast, we have estimated psychometric functionsfor detection and discrimination, so that their support canbe measured instead of inferred from discrimination thresholds.Our estimates of psychometric functions wereobtained using a sampling plan whose performance hasbeen extensively studied (García-Pérez & Alcalá-Quintana,2005), and its setup was tailored to the characteristics of thepsychometric functions at each standard level (including thecase of detection, which implies a null standard) so as tomaximize estimation accuracy. In addition, we allowed a“don’t know” option and we applied Fechner’s (1860/1966)8 It should be stressed here that the actual lower asymptote of thepsychometric function for discrimination at near threshold contrastlevels (see Fig. 6) is largely unpredictable in advance, and thresholdestimates obtained with QUEST will thus be seriously affected by amismatch with the assumed lower asymptote.

846 Atten Percept Psychophys (2011) 73:829–853Fig. 11 Support of the psychometricfunctions for each observer,as a function of standardcontrast. The support of thepsychometric function for detectionis plotted at an abscissaof -∞. Open symbols pertain tothe condition without flankers;solid symbols to the conditionwith flankersSupport of Ψ1.41.21.00.80.60.40.2M10.0–∞ –2.0 –1.0 0.01.41.21.00.80.60.40.2M20.0–∞ –2.0 –1.0 0.01.41.21.00.80.60.40.2M30.0–∞ –2.0 –1.0 0.01.41.41.41.21.21.2Support of Ψ1.00.80.60.41.00.80.60.41.00.80.60.40.2B10.0–∞ –2.0 –1.0 0.00.2B20.0–∞ –2.0 –1.0 0.00.2B30.0–∞ –2.0 –1.0 0.0Standard level, x sStandard level, x sStandard level, x shalf right and half wrong rule in order to eliminate spuriousinflation of support due to Type-A order effects. Our resultsthus showed that the effect of flankers is different atcontrast levels around the detection threshold (where theyflatten psychometric functions) and at contrast levels wellabove the detection threshold (where they only seem tohave a minimal effect on the support of the psychometricfunction and, hence, on discrimination thresholds).Models of flanker facilitation effectsThe way in which psychometric functions with and withoutflankers vary along the contrast continuum must constrainmodels of the flanker facilitation effect. In this section, wediscuss whether or not the most prevalent models for theflanker facilitation effect can accommodate variations ofone or another type in the support of psychometricfunctions.Petrov et al. (2006) claimed that flanker facilitation iscaused by uncertainty reduction. Their modeling consistedonly of fitting psychometric functions for detection withand without flankers with an approach that estimates thenumber M of detectors that are presumably monitored bythe observer during the task. The larger the number ofdetectors that have to be monitored, the larger theuncertainty. It has also been shown that, under this model,if the number of detectors decreases, then the psychometricfunction flattens (Pelli, 1985), although the reverse is notnecessarily true. Given that this is an inherent property ofthe model, a mere visual comparison of the support of twopsychometric functions would suffice to assert which caseputatively involved more detectors. This model can thenaccommodate any observed pattern of changes in thesupport of psychometric functions with and withoutflankers, and will attribute these changes to variations inthe number of receptors that are involved, with no moresupporting evidence than the estimated number of receptorsprovided by the fit. Petrov et al. argued that this uncertaintyreduction occurs because flankers indicate the location ofthe target. If this were the actual cause of flankerfacilitation, a similar reduction should be observed ifuncertainty about the location of the target were eliminatedin some other way. For instance, in our foveal conditionswith fixation markers, there is no more uncertainty aboutthe location of the target without flankers than with them,and yet, our data indicate that detection thresholds in theseconditions are still generally lower in the presence offlankers. The same results were observed in other studiesthat used foveal presentations with fixation aids. It seems,then, that uncertainty reduction is not a tenable explanation(besides not being supported by any other evidence than anintrinsic property of a model). A similar argument againstuncertainty reduction has been elaborated by Chen andTyler (2008). Morgan and Dresp (1995), Williams and Hess(1998), Huang and Hess (2007), Huang, Mullen, and Hess(2007), Summers and Meese (2009), Meese and Baker(2009), and Wu and Chen (2010) also presented data thatprompted them to rule out uncertainty reduction as thecause of flanker facilitation.Conventional psychophysical modeling of the flankerfacilitation effect has estimated contrast response (transducer)functions from TvC data with and without flankers.This has been done under the assumption that flankers havea multiplicative effect on contrast (Chen & Tyler, 2001,2002, 2008), under the assumption that flankers haveadditive and multiplicative effects at different contrastlevels (Zenger-Landolt & Koch, 2001), or avoidingassumptionsofthistypebyindependently fitting separate

Atten Percept Psychophys (2011) 73:829–853 847transducer functions for the conditions with and withoutflankers (as was done in the present study). The effectsof flankers are thus expressed directly as an alteration ofthe contrast response function inlinewiththeresultsofphysiological studies (Polat, 1999).Atthesametime,thevariance of sensory effects with flankers or without themhas been assumed to be independent of stimulus levels(i.e., the additive noise assumption). Under this approachto modeling flanker effects, the data and fitted psychometricfunctions in our Figs. 4a and 6 or the derived TvCdata and fitted TvC curves in our Fig. 10 couldbeusedtoestimate transducer functions with and without flankersand to interpret the results along these lines, but thisapproach suffers from a problem that we will describe andillustrate next.The validity of any conclusion regarding flankercontingentchanges in the transducer function rests in turnon the validity of the additive noise assumption underwhich the transducer functions are estimated. But it mightinstead be that the presence of flankers affects the varianceof sensory effects (i.e., the multiplicative noise assumption,not to be confused with multiplicative effects on contrastsuch as those discussed in the preceding paragraph, whichaffect the contrast response function without altering thetype of noise), whether altering also the transducer functionor leaving it intact. Unfortunately, current psychophysicalparadigms do not allow the determination of whether noiseis additive or multiplicative: García-Pérez and Alcalá-Quintana (2009; see also Katkov, Tsodyks, & Sagi,2006a, 2006b) have shown that empirical TvC curves canbe accounted for equally accurately by a convenientlychosen transducer function coupled with additive noise orby an alternative transducer function coupled with multiplicativenoise. Next, we will demonstrate this interchangeability,since this is useful in a discussion of the flankerfacilitation effect.Given that with our data the estimated TvC functionswith and without flankers do not differ much (see Fig. 10),we will consider instead the data reported by Chen andTyler (2001), which show larger differences and thus allowa clearer illustration of our point. But this illustrationrequires a brief description of Chen and Tyler’s (2001)method. The transducer functions fitted by Chen and Tyler(2001) had the formmðcÞ ¼K eðcS eK i ðcS i Þ q þ s ; ð7ÞÞ pand note that parameter σ in Equation 7 is not the same asthe parameter that was defined in Equation 2 as a measureof the support of a psychometric function. Under theirparameter-estimation approach, the transducer function forthe condition without flankers was constrained to have K e =K i = 1, whereas that for the condition with flankers wasunconstrained in this respect. In addition, the two functionswere jointly fitted to the two data sets (with and withoutflankers) so that the remaining parameters (S e , S i , p, q, andσ) had the same estimated values in both functions. In otherwords, and given the role of parameters K e and K i inEquation 7, the strategy of Chen and Tyler (2001) fits amodel that embeds the assumption that flankers have amultiplicative effect on contrast, whereas noise is additive.In any case, these response functions express the averagesensory effect of (or the internal response to) a target as afunction of contrast and, under the standard differencemodel with normally distributed sensory effects and fixedvariance, the probability of a correct response in a 2AFCtask is given by 9Z1ΨðxÞ ¼ 8 d; m 10xð ð Þ mð10 xsÞ; nÞd d ¼ Φ m ð 10x Þ mð10 xs Þpffiffi0nð8Þ0:7287550:728755(see García-Pérez & Alcalá-Quintana, 2007b), where8(d; m, v) is the probability density of a normallydistributed random variable D with mean m and variancev, andΦ is the unit-normal distribution function. Chen andTyler (2001) estimated thresholds as the 91.5% point onpsychometric functions and further assumed that thresholdis reached when mð10 x Þ m 10 x sð Þ ¼ 1. Taken together,pffiffithese two assumptions require v ¼ 0:728755 in Equation8 to satisfy the definition that, at threshold,ΨðxÞ ¼Φð1=0:728755Þ ¼ :915. Chen and Tyler (2001)thus sought functions μ 1 (for the case without flankers)and μ 2 (for the case with flankers) such that the families ofpsychometric functions generated by Equation 8, that is, Φ m 1ð10 x Þ m 1 ð10 xs Þand Φ m 2ð10 x Þ m 2 ð10 xs Þ, give a satisfactoryaccount of their empirical TvC data. This strategyassumes that flankers alter only the transducer function.Chen and Tyler (2001) and others (Chen & Tyler, 2002,2008; Yu, Klein, & Levi, 2002, 2003; Zenger-Landolt &Koch, 2001) actually succeeded in accounting for empiricaldata in this way, but the question arises as to whether thedata could also be accounted for by a model in whichflankers leave the transducer function intact and alter thevariance of the noise. If this is the case, the success atfitting one or the other type of model would not revealanything about the mechanism of flanker facilitation.Testing the feasibility of the alternative model impliesdetermining whether a transducer model with multiplicativenoise (i.e., one in which the variance of sensory effects9 As will become clear in Equation 10, the Gaussian assumptionembedded in Equation 8 is inconsequential, and the argument holdsalso for alternative distributions.

848 Atten Percept Psychophys (2011) 73:829–853changes with stimulus level according to some variancefunction v) can be found for data with flankers such thatΦ!m 1ð10 x Þ m 1 10 x sð Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ Φm 2ð10 x Þ m 2 10 x sð Þ; ð9ÞnðxÞþnðx s Þ0:728755where the right-hand side is the family of psychometricfunctions generated by the additive noise model with theresponse function μ 2 originally fitted to data with flankersby Chen and Tyler (2001), and the left-hand side is thefamily of functions generated by an alternative version ofthe transducer model in which the response function is stillμ 1 (i.e., the same one that was fitted to data withoutflankers) but variance is no longer constant. Clearly, such amodel must exist, and its variance function v can beobtained given the necessary equality of the arguments of Φon either side of Equation 9, first yieldingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðÞþvx ð s Þ ¼ 0:728755 m 1ð10 x Þ m 1 10 x sð Þm 2 ð10 x Þ m 2 ð10 x sÞ : ð10ÞFinding a closed-form expression for v in Equation 10takes some doing and may not always be possible, but thisis immaterial: The TvC data of Chen and Tyler (2001), orany other data, can be accounted for on the assumption thatflankers affect the variance of noise while leaving theresponse function unchanged. For illustration purposes, anapproximation to the form of the variance function v caneasily be obtained by noting that Equation 10 must alsohold when x s is the null stimulus, so that 10 x s¼ 0,m 1 10 x sð Þ ¼ m 2 10 x sð Þ ¼ 0, and v(x s )=v 0 , yieldingvðxÞ ¼ 0:728755 m 1ð10 x Þ 2m 2 ð10 x v 0 ; ð11ÞÞwhere v 0 is a free parameter. Then, the variance function inEquation 11 coupled with the transducer function μ 1 willproduce the same psychometric function for detection in thepresence of flankers as the additive noise model of Chenand Tyler (2001) with the transducer function μ 2 , and it willproduce slightly different psychometric functions fordiscrimination owing to the fact that Equation 11 is onlyan approximation to the variance function that will satisfyEquation 10.Figure 12 shows the outcomes of the multiplicative noisemodel for data with flankers in comparison with the originaladditive noise model considered by Chen and Tyler (2001).The first column shows the fitted transducer functions μ 1 andμ 2 for each observer, redrawn from Chen and Tyler’s (2001)Fig. 5. The second column shows the (constant) variancefunction that was implied in Chen and Tyler’s (2001)additive noise model and also shows the nonconstantvariance functions that arise from Equation 11 with v 0 =0.15 for Observer C.-C.C., and v 0 = 0.06 for Observer M.D.L.The third column shows the fit of the additive noise model ofChen and Tyler, redrawn from their Fig. 4, but reverses theassignment of symbol and line styles in accordance withconventions in the present article. The fourth column showsthe fit of the alternative model for data with flankers inwhich the transducer function is the same that holds withoutflankers, but the variance function implies multiplicativenoise (continuous curves; the dashed curves are unchangedwith respect to those in the third column). Despite the grossapproximation that Equation 11 represents, the fit of thismultiplicative noise model for flanker effects is no worsethan that of the additive noise model considered by Chen andTyler (2001), which shows that the question as to what is itthat flankers alter is very slippery. Success at fitting somemodel to data does not actually disclose the cause of thephenomenon. We have chosen to leave the data withoutflankers accounted for by the additive noise model of Chenand Tyler (2001) and to fit the data with flankers with amultiplicative noise model, but Equations 9–11 make clearthat we would also have succeeded at ascribing additivenoise and multiplicative noise models in reverse.The bottom line of the preceding demonstration is thatdifferences in TvC data (or psychometric functions) in theabsence or the presence of flankers do not by themselvesreveal anything about whether response functions or noisevariance (or both) are affected by the presence of flankers: Thedata with flankers can be equally accurately accounted for bya model in which the transducer function varies, whereas thevariance function remains the same (the choice of Chen &Tyler, 2001 as well as that of all others who have addressedthis issue; see Zenger-Landolt & Koch, 2001; Yu, Klein, &Levi, 2002, 2003) and by a model in which the transducerfunction is the same in the conditions with and withoutflankers but the variance function changes. This essentialindeterminacy also shows that the leftward shift of psychometricfunctions with flankers that was reported by Shaniand Sagi (2006; see also Petrov et al., 2006) can be causedby a reduction of noise variance rather by an additive effectof flankers on the contrast of the target.Given the functional equivalence of additive noise andmultiplicative noise models, experimental methods thatcould allow determining whether noise is additive ormultiplicative would be needed to clarify whether flankersalter the contrast response function, the variance of thenoise, or both. In any case, flankers seem to have a largereffect on the support of the psychometric function than theyhave on its location, particularly for detection and fordiscrimination near the detection threshold. The termflanker facilitation thus appears as a misnomer that placesthe emphasis in the small and unstable effect of flankers onthresholds while leaving out of the picture their actuallylarger and more consistent effect on the support of thepsychometric function.

Atten Percept Psychophys (2011) 73:829–853 849Response (arbitrary units)Response (arbitrary units)101(a) Observer C.-C.C.0.1–2 –1 0Target contrast, x101(b) Observer M.D.L.0.1–2 –1 0Target contrast, xVariance (arbitrary units)Variance (arbitrary units)1010.1–2 –1 0Target contrast, x1010.1–2 –1 0Target contrast, xIncrement thresholdIncrement threshold–0.6–0.8–1.0–1.2–1.4–1.6Additive noise–1.8–3 –2 –1 0Standard level, x s–0.6–0.8–1.0–1.2–1.4–1.6Additive noise–1.8–3 –2 –1 0Standard level, x sIncrement thresholdIncrement thresholdMultiplicative noise–0.6–0.8–1.0–1.2–1.4–1.6–1.8–3 –2 –1 0Standard level, x sMultiplicative noise–0.6–0.8–1.0–1.2–1.4–1.6–1.8–3 –2 –1 0Standard level, x sFig. 12 First column: Response function estimated for each of thetwo observers (rows) in the study of Chen and Tyler (2001) in theconditions with flankers (solid curve) and without them (dashedcurve). These curves are redrawn from Fig. 5 in Chen and Tyler butare done so using line styles consistent with the conventions in thepresent article. Second column: Alternative variance functionsrepresenting additive noise (horizontal line) and multiplicative noise(asymptotically increasing curve); the latter was obtained throughEquation 11 with v 0 = 0.15 in the top row (Observer C.-C.C.) and v 0 =0.06 in the bottom row (Observer M.D.L.). Third column: EmpiricalTvC data from Chen and Tyler and the theoretical curves that theyfitted to those data, involving different transducer functions and theassumption of equal additive noise (replotted from their Fig. 4).Fourth column: The same data as before, but now both theoreticalcurves arise from the same transducer function (given by the dashedcurve in the left panel) in combination either with the additive noiseassumption for the condition without flankers (dashed curve, which isthus identical to that plotted in the third column) or with themultiplicative noise assumption for the condition with flankers (solidcurve). Note that open and solid symbols and dashed and solid curvesin the two columns on the right are assigned to conditions in theopposite way to Fig. 4 in Chen and TylerConclusionWe measured psychometric functions for detection anddiscrimination with and without flankers using a robustpsychophysical method. Our results confirm that psychometricfunctions for detection are flatter in the presence offlankers, that this flattening is mildly present in psychometricfunctions for discrimination near the detectionthreshold, and that it virtually disappears well above thedetection threshold. When plotted in TvC form, ourdiscrimination data describe a pattern that is distinctlydifferent from two other patterns that have been reported inthe literature, although the differences are reasonablyattributed to the different psychophysical methods usedacross the studies that reported these three patterns.Our results did not replicate the most common finding ofearlier studies, namely that at high-contrast levels, discriminationthresholds with flankers are higher than thosewithout flankers. Because our method eliminated Type-Aorder effects that spuriously broaden psychometric functions,one might speculate that, by comparison, whatprevious studies have actually shown is that flankersincrease the magnitude of order effects, and thus producespuriously higher discrimination thresholds. The origin ofType-B order effects still found in our data is unclear,although they have been reported to have different formsand magnitudes in different conditions (Ulrich & Vorberg,2009). Although only a speculation at this point, flankercontingentType-B order effects do not seem untenable.Hopefully, further research designed also to eliminate Type-A order effects will clarify whether Type-B order effects in2AFC discrimination tasks are actually larger with flankersthan without them and, ideally, will also identify theircauses and devise means for the elimination of theircontaminating influence.Our discussion of current models of flanker facilitationeffects has questioned the validity of the hypothesis thatflankers reduce uncertainty about the location of the target.Also, the widespread claim that flankers alter the contrastresponse function has been shown to reflect only the naturaloutcome of the modeler’s decision to attribute this particularrole to the flankers by the arbitrary choice of fitting additive

850 Atten Percept Psychophys (2011) 73:829–853Fig. 13 Relation between actualand nominal luminance (leftpanel) and between actual andnominal contrast (right panel) inthe monitor used in Madrid. Thedashed diagonal is the identityline. Symbols denotemeasurementsActual luminance (cd/m 2 )7560453015Actual contrast (log units)0–0.5–1.0–1.500 15 30 45 60 75Nominal luminance (cd/m 2 )–2.0–2.0 –1.5 –1.0 –0.5 0Nominal contrast (log units)noise models to the data (and succeeding at that). We haveshown that the alternative choice of fitting a multiplicativenoise model also succeeds at accounting for the data equallyaccurately, and in this type of model, the contrast responsefunction is the same with and without flankers, whereas thevariance function differs in either case. The functionalequivalence of these alternative explanations reveals that thecause of flanker effects cannot be determined until experimentalprocedures are devised that allow separate estimationof the contrast response and variance functions.Author Note The present work was supported by Grants EY05957and EY12890 from the National Institutes of Health, by GrantSEJ2005-00485 from Ministerio de Educación y Ciencia, and byGrant PSI2009-08800 from Ministerio de Ciencia e Innovación. Wethank Rob Giorgi for his help in data collection.AppendixThis appendix discusses a generally widespread hardwareproblem that must have affected a number of studies that haveused our same calibration and experimentation system(VisionWorks; Swift et al., 1997). We should neverthelessstress that this problem affects absolute measures ofthreshold and support of psychometric functions; therefore,they may be relevant in clinical studies in which observers’absolute thresholds are compared to normative data fordiagnosis. But, this problem has no consequence in studiessuch as ours, where the issue under consideration is therelative magnitudes of threshold and support across conditions(i.e., with and without flankers). Indeed, similar oreven larger calibration problems have been described underthe Modelfest project (see Ahumada & Scharff, 2007), whichin that case affect comparisons of absolute measures acrosslaboratories in which different equipment was used.Monitor calibration protocols ensure proper gamma correctiononly under the same exact conditions that were usedduring calibration. Our calibration system uses a protocol thatmeasures the luminance of a uniform patch centered on theimage area and whose size is half the width and half the heightof the image area, whereas the outer region of the image areaon the monitor remains unlit. This was the only calibrationoption in the system until recently, and it is still the defaultoption. Our experiments, on the other hand, are carried outusing an average luminance field that covers the entire imagearea. García-Pérez and Peli (2001) showed that the RGBmonitor used in Boston in the present experiments and amonochrome monitor very similar to the one used in Madridrespond differently to these changes in image size when itcomes to ensuring gamma correction. In particular, forProbability correctProbability correct1.00.80.60.4M10.000 0.0090.2–1.81 –1.870.524 0.5930.0–2.5 –2.0 –1.5 –1.0 –0.51.00.80.60.4B10.000 0.0000.2–1.84 –1.860.689 0.7920.0–2.5 –2.0 –1.5 –1.0 –0.5Target contrast, xFig. 14 Top panel: Data and fitted psychometric functions forObserver M1 in Fig. 4a, but using actual rather than nominal contrast,transformed using the relation in the right panel of Fig. 13. Bottompanel: Data and fitted psychometric functions for Observer B1 (thesame person) merely copied without change from Fig. 4a forcomparisonλˆθˆσˆλˆθˆσˆ

Atten Percept Psychophys (2011) 73:829–853 851stimuli displayed at the center of the RGB monitor, actualluminance levels in the vicinity of the midrange (where ourstimuli were placed) are about 95% of the nominal levels,which leaves contrast unaltered (see the strand of opencircles in the bottom left panel of Fig. 5 in García-Pérez &Peli, 2001). However, on the monochrome monitor, actualluminance in this region is higher than nominal and is in aproportion that increases slightly nonlinearly with nominalluminance, although local variations within the narrowsubranges involved in our experiments are approximatelylinear (see the strand of open circles in the bottom right panelof Fig. 5 in García-Pérez & Peli, 2001). To check whetherthe monitor used in Madrid was affected by the sameproblem, we carried out measurements such as those leadingto Fig. 5 in García-Pérez & Peli (2001).Figure 13 shows the relation between actual and nominalluminance (left panel) and the relation between actual andnominal contrast (right panel) that arose from thesemeasurements. Quite apparently, the relation between actualand nominal luminance is still very approximately linear,although the slope is lower than unity. As for contrast,actual contrast is slightly higher than nominal contrastacross the entire range, but the discrepancy is slightly largerat very low contrasts (i.e., at contrast near the detectionthresholds reported in the upper panels of our Fig. 4a). Inparticular, when nominal contrast is 1% (–2 log units),actual contrast is about 0.16 log units higher. Thischaracteristic might explain why the same person appearsto have lower thresholds on the monochrome than on theRGB monitor (Observer M1 vs. Observer B1; see Fig. 4).To confirm this point, we plotted the data for Observer M1in Fig. 4a (and fitted the psychometric functions) using theactual contrasts determined by the relation shown in theright panel of Fig. 13, instead of the nominal contrasts usedin Fig. 4a. The results are shown in Fig. 14, which alsoreplots the data for Observer B1 (the same person) fromFig. 4a for comparison. Clearly, thresholds and psychometricfunctions (including measures of support) for ObserversM1 and B1 are much more coincidental here.To corroborate that the differences in absolute thresholdcould be attributed to the different behavior of the monitorsand were not caused by some idiosyncratic characteristic ofObserver M1/B1, psychometric functions for detection withand without flankers were also measured for Observer M2on the RGB monitor in Boston a few months later, and theresults showed differences of similar magnitude and sign:Detection thresholds without and with flankers were,respectively, 0.16 and 0.31 log units higher on the RGBmonitor (Boston) than they were on the monochromemonitor (Madrid; data reported in Fig. 4a for Observer M2).Note, finally, that the calibration problem just described(and corrected) affects only comparisons of absolutethresholds and supports across monitors, and not comparisonsbetween stimuli presented on the same monitor(which is the actual topic of the research reported in thepresent article). 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