Discriminability in length oflines in the Mtiller-Lyer figure

Pereption & Psyhophysis 1998.60(3).511-517 Disriminability in length oflines in the Mtiller-Lyer figure MIAO-FEN WANG, R. JOHN IRWIN, and MICHAEL J. RAUTUS University ofaukland, Aukland, New Zealand Reeiver-operating harateristisfor the disriminability in the length of the lines of the Milller-Lyer figure were obtained by the rating method of detetion theory. Six observersjudged the shaft length of the lines of the figure with referene to the same standard line. Maximum-likelihood estimates of the index of disriminability, d', were a linear funtion of the differene in the length of the lines, but the funtions did not pass through the origin beause of a onstant error of judgment. Beause disriminability was determined by ROCanalysis, the onstant error ould not be attributed solely to hanges in riterion plaement; instead, it showed that the Milller-Lyer figure indued a hange in the disriminability of the lines. Despite its apparent relevane, detetion theory has seldom been used to eluidate the proesses underlying visual illusions. Here we attempt to show how detetion theory an help larify some ofthe fators that underlie the Miiller-Lyer illusion. There are many variants ofthis famous illusion (see Day & Knuth, 1981, for some history), but the prototypial example onsists oftwo lines ofequal length, presented side-by-side, in whih one of the lines has, in addition, either arrowheads or tail fins (> <) on eah end. Arrowheads make the line appear shorter than a physially equal referene line, and fins make it appear longer. A vast number ofinvestigations have attempted to unover the auses ofthis illusion. We find Coren and Girgus's (1978) lassifiation of theories into two typesstrutural theories and strategy theories-to be ofvalue. Briefly, strutural theories attribute the illusion to optial or neural proesses. For example, Ginsburg (1986) explainedthe illusion in terms ofthe limited apaity ofthe visual system to transmit the spatial frequenies that make up the Miiller-Lyer figure. Strategy theories asribe the illusion to ognitive proesses-for example, "to biases introdued by judgmental and attentional proesses" (Coren & Girgus, p. 73). Pressey and his olleagues (e.g., Pressey & Pressey, 1992) have explained the illusion as stemming from the deployment of an attentive field-a deployment that may be, at least partly, at the disposal of the observer. Nevin (1991) made an insightful observation by identifying these two kinds oftheories with the two independent proesses measured by detetion theory: disriminability and response bias. If this identifiation is valid, strutural auses ought to show up as affeting the dis- We thank Tony Nevin and two other referees for their thoughtful and hallenging reviews ofprevious versions ofour manusript. Correspondene onerning this artile should be addressed to R. 1. Irwin, Department of Psyhology, The University ofaukland, Private Bag 92019, Aukland, New Zealand (e-mail: Iji@aukland.a.nz). riminabilityofaspets ofthe stimulus, whereas ognitive auses ought to be manifested in hanges in an observer's riterion for reporting the illusory effet. Therefore, by analyzing the Miiller-Lyer illusion with the onepts of detetion theory, Nevin reasoned that it might be possible to parel out the ontributionto the illusion ofthose ognitive proesses that are refleted in response biases and the ontribution ofthose strutural proesses that are refleted in disrimination. Suh an analysis, oupled with the psyhoanatomial dissetion suggested by Coren and Girgus (1978), might identify more preisely the nature of the strutural or ognitive proesses that ontribute to the illusion. Therefore, a detetion-theoreti analysis of illusory experienes ought to offer a new insight, even if not a definitive one, into the origins ofillusions, beause ofits ability to separate hanges in disriminability from hanges in riterion plaement. There are, however, some issues that ompliate the appliation ofdetetion theory to the understanding of visual illusions. These issues an be highlighted through a desription of two previous attempts to undertake a detetion-theoreti analysis ofthe Miiller-Lyer illusion. Detetion-Theoreti Analyses ofthe MiiUer-Lyer illusion Lown (1988) may have been the first to apply detetion theory to the measurement of the Miiller-Lyer illusion. In his investigation, eah trial ontained both a standard and a omparison line. The standard line had arrowheads at eah end, and the omparison line had fins. The standard line was always 100 mm long. On "noise" trials, the omparison line was 75 mm long (it was then judged to be equal to the standard line), and on "signal"trials, it was 80,90, 100, or 110 mm long. On eah trial, "a subjet was required to say whether the line with fins out (omparison) was equal to or greater than the standard" (Lown, p. 101). In order to ompute the detetion-theory index ofdisriminability, d'; the frequeny with whih observers judged the omparison line to be longer on noise trials 511 Copyright 1998 Psyhonomi Soiety, In.

512 WANG, IRWIN, AND HAUTUS was ompared with the frequeny with whih they judged it to be longer on signal trials. The index so obtained measured the extent to whih lines of80, 90, 100, or 110 mm were disriminable from a line of75 rom when all the lines had tail fins. The stimulus-response matrix for Lown's experiment is depited in Table I, where we have labeled the noise as SI and the signal as S2. The standard line with arrowheads served as a referene line for judging the length ofthe omparison line. Inspetionofthe stimulus arrangement in Table I, however, shows that the relation between p("longer" IS I) and p("longer" IS2) does not quantify the magnitude ofthe Miiller-Lyerillusion. Rather, it quantifies the extent to whih one line that is terminated with fins is disriminable from another line that is terminated with fins. Despite its objetive, therefore, Lown's study does not provide a detetion-theoreti analysis of the illusion. (A similar study that derived nonparametri indies of disrimination has been reported by Brosvi et ai., 1994.) Nevin (1991) onduted an experiment whose design was a substantial improvement over Lown's (1988). As in Lown's design, every trial ontained both a standard and a omparison line, displayed ontiguously side-byside. There were two kinds oftrials. In one kind, the two lines had short vertial bars at eah end. Suh a figure does not give rise to an illusory effet. In the other kind of trial, the standard line had arrowheads at eah end, and the omparison line had fins. This seond kind of trial, therefore, presented one ofthe normal versions of the Miiller-Lyer illusion. After every trial, the observers judged whether the omparison line was longer or shorter than the standard. On both kinds oftrials, illusory and nonillusory, the omparison line ould take on one ofeight lengths, whereas the standard line was always the same length. The stimulus-response matrix for Nevin's experiment is shown in Table 2. Nevin (1991) did not attempt to ompute a detetiontheoreti index of disriminability. Instead, Nevin derived separate psyhometri funtions for the stimuli presented on illusory trials (S2) and for those presented on nonillusory trials (S I). These funtions showed how the ratio of"longer" to "shorter"judgments for eah kind oftrial depended on the differene in length between the omparison and the standard lines, whih also was expressed as a ratio. He found that the slopes ofthe obtained funtions were the same for both illusory and nonillusory onfigurations of the lines. From this, Nevin onluded that the Miiller-Lyer onfigurationdoes not ause a hange in the disriminability of line length. In our Table 1 The Stimulus-Response Matrix for Lown's (1988) Experiment and the Possible Outomes ofa Trial Stimulus Response Stimulus Standard Comparison "Co longer" "Co equal" S! ~ >----< p("!onger" IS!) p("equa]" ISI) S2 ~ >----< p("!onger" I52) p("equa!" IS2) Note-s-Co, omparison. Table 2 The Stimulus-Response Matrix for Nevin's (1991) Experiment Stimulus Response Stimulus Standard Comparison "Co longer" "Co shorter" S! f-----1 f-----1 p("longer" I5 I) p("shorter" I5 I ) 52 ~ >----< p("]onger" I52) p("shorter" IS2) Note-s-Co, omparison. view, however, the slope of Nevin's funtions indiates how disriminable in length one line is from another when both lines have idential terminations-that is, when no illusory effet is observable. For this reason, we suggest that in orderto assess the illusoryeffet, Nevin's psyhometri funtions ould have been onstruted from the ratios ofthe responses to SI and S2. Nevin (1991) found that the horizontal loation ofthe funtions for the illusory onfiguration was displaed relative to that for the nonillusory onfiguration. He haraterized this displaement in position as stemming from a hange in response bias. This is an important onlusion beause, taken in onjuntion with the finding that there was no hange in the disriminability ofline length, it appears to make a strong ase that the origin ofthe Muller Lyer illusion is attributable to ognitive proesses, rather than to strutural ones. Requirements ofa Detetion-Theoreti Study ofthe Miiller-Lyer Illusion With some refinements, Nevin's (1991) design an provide a detetion-theoreti analysis ofthe Miiller-Lyer illusion. One refinement is to arrange the stimuli so that the same standard line appears in both kinds oftrials, illusory and nonillusory (see Table 3). Judgments about a omparison line that gives rise to an illusory effet an then be ompared with judgments about a omparison line that does not give rise to an illusion. The omparison lines are not diretly ompared with eah other, but rather with a ommon standard line. This is an important differene from the arrangement employed by Nevin. This refinement is based on the assumption that the sensory events ofthe standard line and ofthe omparison line are Gaussian and independent. DeLuia (1993) offers some support for the assumption ofindependene. She showed that the magnitude ofthe total illusion is not signifiantlydifferent from the sum ofthe magnitudes ofits omponents parts-that is, from the sum of the illusions from lines with arrowheads and from lines with fins. A seond refinement is to obtain judgments that yield a riterion-free index ofdisrimination, suh as d', from a full reeiver-operating harateristi (ROC). A full ROC allows the goodness-of-fit ofthe detetion-theory model to be assessed-anassessment that is unavailable, for example, when the yes-no method is used to yield only a single point in the ROC square, beause any urve an pass through one point. An effiient method for this purpose is the rating method, in whih observers are asked to rate theironfidene that one line is longer than another.

DISCRIMINABILITY OF MOLLER-LYER LINES 513 By presenting several omparison lines in an experimental session, as is done in the method ofonstant stimuli, the ratings ofone omparison line with referene to a standard line an be ompared with the ratings ofanother omparison line with referene to the same standard. By this means the disriminability of any pair of lines an be found, even though the lines were not diretly ompared with eah other. The disriminability of the lines an then be presented in separate psyhometri funtions for various onfigurations ofthe Miiller-Lyer illusion as well as for nonillusory figures, and the magnitude ofthe illusion an be assessed by the horizontal separation between the appropriate riterion-free funtions. METHOD Subjets Six volunteers took part, 2 men and 4 women. They all had normal or orreted-to-normal vision. Apparatus and Stimuli All stimulus figures (white on blak) were displayed on a 14-in. VGA monitor with 640 X 480 resolution. Three kinds ofhorizontal lines were used: lines with short vertial bars at eah end, lines with arrowheads at eah end, and lines with tail fins at eah end (see Figure I). The sloping terminations subtended an angle of45 with respet to the horizontal line. All terminating elements were IS mm long. On every trial, a pair of lines was presented side-byside for 1.0 se. The left-hand line was always 70 mid long and was always terminated by two vertial bars; it is designated the standard line. The right-hand line was.erminated with vertial bars, arrowheads, or fins; it is designated the omparison line. The omparison line ould be 64, 66, 68, 70, 72, 74, or 76 mm long. The standard and omparison lines were always separated by 38 mm. During an experimental session, only two onfigurations of the pairs oflines ourred. In one onfiguration (nonillusory), the omparison line, like the standard line, had vertial bars at eah end; in the other onfiguration (illusory), the omparison line had either arrowheads or tail fins at eah end, depending on the experiment. In an experimental session, the two onfigurations, illusory or nonillusory, were presented with equal probability. Eah of the seven omparison lengths was presented in random order in eah experimental session. Proedure There were two experiments. In Experiment I, the omparison line in the illusory onfiguration had arrowheads at eah end; in Experiment 2, the omparison line in the illusory onfiguration had tail fins at eah end. The observers were seated at a viewing dis- ( ) )>-----«Figure 1. The top onfiguration was presented in both Experiments 1 and 2, the middle onfiguration in Experiment 1, and the bottom onfiguration in Experiment 2. tane of approximately 400 mm from the omputer sreen. On every trial, the standard line and a omparison line were presented together; after eah observation, the observers rated their onfidene, on a six-point sale, that the omparison line was longer or shorter than the standard. Their judgments were entered by pressing a numeri key on the omputer's keyboard; I indiated that they were very onfident that the omparison line was shorter than the standard, and 6 indiated that they were very onfident that the omparison line was longer than the standard. Intermediate numbers represented intermediate degrees ofonfidene. No feedbak was given, and the next trial began immediately after a response. An experimental session omprised 112 trials, preeded by 12 warm-up trials that were disarded from the analysis. Eah experiment entailed seven sessions, so there were 112 X 7 = 784 experimental trials per observer per experiment. All observers undertook both experiments (1,568 trials in all), but in ounterbalaned order. During an experimental session, the seven omparison lengths for both illusory and non illusory lines were eah presented eight times in random order. Analysis ROCs were onstruted from the ratings in the following way. For Experiment I, an ROC for disriminating between the length of two lines, both with vertial bars at their ends, ould be onstruted from the ratings given to separate omparison lines with referene to the standard line. The tehnique is analogous to that used by Irwin and Whitehead (1991) and Irwin, Hautus, Dawson, Welh, and Bayly (1994). Seond, an ROC ould be onstruted for disriminating between the length oftwo lines, one ofwhih was terminated by arrowheads and the other by vertial bars, from the ratings given to the omparison line with arrowheads as opposed to those given to the omparison line with vertial bars. Third, an ROC ould be onstruted for disriminating between the lengths oftwo lines, both terminated by arrowheads, from the ratings given to eah line in omparison with the standard line. These ROCs were available from the results of Experiment 1. An analogous set ofrocs for lines with and without tail fins was available from Experiment 2. Table 3 illustrates a simplified version ofthe stimulus-response matrix for Experiment I. The matrix is a simplifiation in that it supposes that only a binary response (longer or shorter) was allowed, whereas in fat a six-point rating was available. An important differene between our design and Nevin's (1991) is that, in our experiments, the same standard line appeared in both kinds oftrials, nonillusory (S 1) and illusory (S2). Beause of this, the pereived length ofany omparison line-for example, a line terminated with arrowheads-eould be ompared with the pereived length of another omparison line-for example, one terminated with finseven though they had not been judged with respet to eah other. Our experiments allowed six responses, not just two; for this reason, it was desirable to ombine the ratings of the 6 observers in order to obtain average ROCs, beause some of the ratings were used infrequently, espeially for lines that were judged as being very different in length. This made it diffiult to ompute the path ofan ROC for eah observer separately. In order to onstrut Roes for the 6 observers ombined, a jakknifing proedure (see Dorfman & Berbaum, 1986) was therefore implemented. This proedure avoids the pitfalls that an arise from the simple pooling ofratings when different riteria are adopted by different observers. RESULTS Reeiver-Operating Charateristis Figure 2 illustrates some of the ROCs obtained from eah experiment. The three left-hand ROCs are from Experiment I (where the omparison line in the illusory onfiguration was terminatedwith arrowheads), and the three

514 WANG, IRWIN, AND HAUTUS Table 3 A Simplifiation ofthe Stimulus-Response Matrix for Experiment I Stimulus Response Stimulus Standard Comparison "Co longer" "Co shorter" SI 1----\ 1----\ p("longer" lsi) p("shorter" lsi) S2 1----\ ~ p(")onger"is2) p("shorter" IS2) Note-Co, omparison. right-hand ROCs are from Experiment 2 (where the omparison line in the illusory onfiguration was terminated with tail fins). The data points are from the pooled ratings ofall observers, and the theoretial ROCs are the best fits to the ratings ofthe normal-normal equal-variane model, whih is the standard model ofdetetion theory. The fits were obtained by maximum-likelihood estimation ombined with the jakknifing tehnique. In these examples, eah ROC shows the disriminability ofa omparison line of76 mm from a omparison line of70 rom. The ROCs were obtained from the ratings given to a omparison line of76 mm and the ratings given to a omparison line of70 mm. Both these sets ofratings were made with respet to the same standard line of70 mm. To larify our analysis still further, onsider Table 4, whih shows some ofthe data from whih the ROCs for Experiment I were onstruted. The ratings ofeah row in the table were ompared with the ratings given, in the same experimental session, to a omparison line of 70 mm that was terminated with vertial bars. The frequeny ofthe six ratings to the 70-mm omparison line with vertial bars turned out to be 2, 27, 238, 302, 83, and 20. Hene, the ROC depited in the middle left-hand panel of Figure 2 was onstruted by relating these ratings to those shown in Table 4 for the 76-mm line that was terminated with arrowheads. As another example, the ROC shown in the bottom left-hand panel was onstruted from the ratings shown in Table 4 for the 76-mm line and from the ratings, also shown in Table 4, for the 70-mm line. Both these lines were terminated by arrowheads. Our proedure, however, did not provide an ROC for disriminating between two 70-mm lines when both were terminated in the same way, beause only one set of ratings was available for this ase. The two top ROCs in Figure 2 illustrate independent estimates from Experiments 1 and 2 of the disriminability of two lines, one of 76 mm and the other of 70 mm, that were terminated with vertial bars. The best-fitting values ofd ' for these two ases were 1.41 (Experiment I) and 1.52 (Experiment 2). The other ROCs illustrate the effet ofother terminationson the disriminabilityoftwo lines of these lengths, as illustrated in eah panel (the onfigurations illustrated in the legend ofeah graph are not drawn to sale). The best-fitting d' for the left-hand middle panel has a negative value of -0.18; this signifies that, although the omparison line with arrowheads was longer than the omparison line with vertial bars (a line idential to the standard), it was judged to be slightly shorter-an example of the Miiller-Lyer illusion. The right-hand middle panel illustrates the other side of the illusion, in whih tail fins on the omparison line exaggerated the judged differene between the two lines (d' = 2.57). The two bottom ROCs illustratehow arrowheads or tail fins on both lines affeted their disriminability (d' = 1.16 for the arrowheads and 0.89 for the tail fins). Psyhometri Funtions The six ROCs in Figure 2 illustrate the disriminability of a omparison line of 76 mm from a omparison line 000 rom. Altogether, seven different lengths ofthe omparison line were presented, however, so other ROCs of this type were onstruted, and the best-fitting estimate ofd ' found for eah. Figure 3 shows how, in Experiment I, these estimates depended on the differene in length between two lines when one of the lines was 70 mm long. (Only six points are shown when the two lines had idential terminations beause, in this ase, the method did not yield independent ratings for lines of equal length.) Figure 4 shows the omparable results for Experiment 2. In these psyhometri funtions, a negative d' means that the omparison line was pereived to be shorter than another line of70 mm. The lines fitted to eah set ofpoints are least-squares linear funtions with two free parameters-interept and slope. The psyhometri funtions in Figures 3 and 4 have the form d' = mx +, where x is the differene in length between the omparison line and a line of 70 mm, m is the slope ofthe psyhometri funtion, and is its interept. The onstant error is therefore equal to <elm, whih is the value ofx when d' = O. For an illusory onfiguration, the onstant error speifies the magnitude and diretion of the illusion. For the omparison lines with arrowheads (the triangles in Figure 3), the onstant error was 7.8 mm, and for the omparison lines with tail fins (inverted triangles in Figure 4), the onstant error was -9.1 mm. The size ofthe illusion therefore amounted to II % and 13% for the two onfigurations. The onstant errors for the two estimates of the nonillusory onfiguration (irles in Figures 3 and 4) were alulated to be 0.1 mm and -0.7 mm. The slope of a psyhometri funtion an provide an estimate of the just notieable differene for disriminating between the length oftwo lines. If the just notieable differene is defined as the differene between two omparison lines that yields Sd' = 1 (equivalent to 76% orret in a two-alternative fored-hoie experiment), the reiproal ofthe slope ofthe fitted funtion is the just notieable differene. The estimates from the two experiments ofthe slope ofthe psyhometri funtions when the omparison line, like the standard line, was terminated by vertial bars are 0.25 and 0.20 mm:", giving two estimates ofthe just notieable differene of4.0 mm and 5.0 mm for a standard line of 70 mm. The slope of the psyhometri funtion when the omparison line was terminated with the arrowheads (Experiment I) is 0.16, whih orresponds to a just notieable differene of

DlSCRIMINABILITY OF MULLER-LYER LINES 515 1.0 - en 0.8 "0 UO.6 'L ~0,4 0...J ~0.2 a.. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 r:--------=::=~-e 0.8 0.2 0.2 0.4 0.6 0.8 1.0 1.0 ~ en 0.8 "0 UO.6... ~0.4.9 ::...0.2 a.. :> 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 r.-----::::::t:~------,e_-e 0.8 0.2 0.2 0.4 0,6 0.8 1.0 ~ en 0.8 1.0 r;--------r===-'61 "o 00.6 -... ~0.4.9 ::...0.2 D.. O.O~_...I--_.l...--'"----'---' 0.0 0,2 0.4 0.6 0.8 1,0 P(ILonger" I Co = St) 1.0 r.---------f~~ 0,8 0.6 0.4 0.2 < 0.0 "---'"--'"----''-----'---' 0.0 0,2 0,4 0.6 0.8 1.0 P(ILonger" I Co = St) Figure 2. The urves show the best-fitting ROes (normal-normal equal-variane model) to the obtained ratings for the lines illustrated in eah panel. The lengths of the two lines were 70 mm and 76 mm. 6.3 mm. When the omparison line was terminated with tail fins (Experiment 2), the slope is 0.17, whih orresponds to ajustnotieable differene of5.9 mm All these just notieable differenes are for lines presented for 1.0 se, whih did not allow prolonged inspetion. Figures 3 and 4 eah show a third psyhometri funtion for the disrimination between two lines that were both terminated with arrowheads (Figure 3) or both terminated with tail fins (Figure 4). Inspetionof Figures 3 and 4 suggests that the slope of the funtion for disriminating between two lines that are terminated with vertial bars is steeper than are the slopes ofthe psyhometri funtions when one or both ofthe lines had sloping terminations. An analysis of variane (ANOYA) onfirms this. The slopes of the psyhometri funtions of Experiment 1 in Figure 3 are not the same, for the interation between the three types of termination and the six differenes in length was signifiant [F(10,50) = 3.78,p <.001]; the same result was found for Experiment 2 (Figure 4) [F(10,50) = 4.35, p <.001]. (In these analyses, the error term was estimated from the pseudovalues returned by the jakknife.) Hene the just notieable differ-

516 WANG, IRWfN, AND HAUTUS Table 4 Frequeny With Whih Eah of Seven Comparison Lines, Terminated With Arrowheads, Were Rated Longer Than a Standard 76-Millimeter Line Terminated by Vertial Bars Rating Comparison (mm) I 2 3 4 5 6 Sum 64 \52 354 \43 16 3 4 672 66 96 304 238 29 3 2 672 68 5\ 224 352 39 4 2 672 70 26 \42 42\ 78 5 0 672 72 7 \02 432 \21 \0 0 672 74 8 52 385 203 20 4 672 76 2 20 299 292 49 10 672 Note-The ratings are pooled from 6 observers. ene in length oftwo lines that were terminated by vertial bars was smaller than the just notieable differene in length oflines that were terminated in other ways. Another question of interest is whether the slopes of the four psyhometri funtions for lines that are terminated with sloping elements differ among themselves. An ANOVA showed that the interation between four types ofsloping elementand the six differenes in length was not signifiant [F(l5,75) = 0.41, n.s.]. The shape ofa psyhometri funtion that is linear in d' assumes a different form when disriminability is measured in units ofproportion orret [p()]. The relation between d' and p() for an unbiased observer is p() = «I>(d'/~2), where «1>( ) is the normal distribution funtion. Hene, linear psyhometri funtions like those in Figures 3 and 4 an be represented by the normal distribution funtion when disriminability is speified in units ofp(). Indeed, this is the expeted theoretial shape ofsuh a psyhometri funtion when the stimuli being disriminated are presented separately, as they are here (see Irwin, 1989; Laming, 1986). DISCUSSION The results show that the Miiller-Lyer onfiguration auses a hange in the disriminability of the length of its two lines, a hange that is manifested as a onstant error. Beause we measured disriminability by ROC analysis, this result annot be attributed to hanges in response bias, for the ROC ontains all possible biases for a given disriminability. In this respet, our onlusion is the opposite ofthat of Nevin (1991), who attributed the illusion to hanges in response bias and not to disriminability. Although our onlusion differs from Nevin's, we do not think that Nevin's results are neessarily in onflit with our own. Although Nevin's psyhometri funtions ontrol for response bias, their slopes measure the disriminability oftwo lines with the same terminations. Moreover, the displaement between two suh psyhometri funtions is akin to a lassial index ofthe illusion (see, e.g., Restle & Deker, 1977), an index whih does not allow the ontribution ofresponse bias and disriminability to be separated. An examination of the middle left-hand ROC in Figure 2 shows that a 76-mm line that is terminated by arrowheads is barely disriminable in length from a 70-mm line that is terminated by vertial bars (d' = 0.18). By ontrast, the same two lines terminated by vertial bars are more readily disriminable-as is shown, for example, by the top left-hand ROC in Figure 2, for whih d ' = 1.41. Our results, therefore, do not support strategy theories, insofar as those theories attribute the illusion, as Nevin (1991) has stated, to biasing fators stemming from riterion plaement. Our results also show that the way in whih the lines are terminated affets the slope oftheir psyhometri funtion and, therefore, the size ofthe justnotieable differene in length, irrespetive ofany illusory effet that the terminations may produe. We found that the slope ofa psyhometri funtion for two lines that were terminated by vertial bars was signifiantly greater than the slope of psyhometri funtions for two lines that were terminated by other onfigurations (see Figures 3 and 4). Two of these psyhometri funtions were for illusory onfigurations (one line terminated with vertial bars and the other line terminated with either arrowheads or tail fins) and two were for nonillusory onfigurations (both lines terminated with arrowheads or both with tail fins). The differene may stem from the fat that vertial bars provide more information on the length ofthe lines, or a learer definition oftheir ends, than do sloping terminations. Judd (1899) attributed the Miiller-Lyer illusion to 3.0 2.5 2.0 1.5 'b ~ 1.0 : 0.5 t13 'E 0.0.;:: 0-0.5 til is -1.0-1.5-2.0-2.5-8 -6 o both vertial bars 6. vertial bars and arrowheads o both arrowheads 440 2 468 Differene in length (mm) Figure 3. Psyhometri funtions from Experiment I showing the disriminability of two lines as a funtion oftheir differene in length. One line was always 70 mm long. The different symbols indiate the terminations of the lines: both lines terminated by vertial bars (irles), both terminated by arrowheads (squares), or one line terminated by vertial bars and the other by arrowheads (triangles).

DISCRIMINABILITY OF MOLLER-LYER LINES 517 3.0 2.5 2.0 1.5 ~ ~ 1.0 :0 as 0.5 :: 'E 0.0 '1:: o en -0.5 is -1.0-1.5-2.0 o both vertial bars V vertial bars and fins o both fins -2.5-8 ~ 4 ~ 0 2 4 6 8 Differene in length (mm) Figure4. Psyhometri funtions from Experiment2 showing the disriminability of two lines as a funtion of their differene in length. One line was always 70 mm long. The different symbols indiate the terminations of the lines: both lines terminated by vertial bars (irles), both terminated by fins (squares), or one line terminated by vertial bars and the other by fins (inverted triangies). a similar proess, but OUi evidene on this point is onfined to studyinghow the slope ofthe psyhometrifuntion for disriminating the length oftwo lines with idential terminations is affeted by those terminations. Again, this onlusion differs from Nevin's (1991), for he reported no effet of the angular orientation of the elements ofa terminating arrowhead, inluding an angle of 90 that formed a vertial bar, on the slope of the psyhometri funtion. A related investigation by Morgan, Hole, and Glennerster (1990) found that the just notieable differene in the length ofa line was unaffeted by its being embedded in a longer line with tail fins that ould give rise to the Miiller-Lyerillusion. From this they onluded that the illusion did not alter an observer's sensitivity to hanges in line length. We found, by ontrast, that the just notieable differene in the length ofa line was affeted by the nature ofits terminations. The task that Morgan et al. set for their observers was different from ours. Their observers had to judge the length of part of the shaft (a segment marked by vertial lines) embedded in a longerline with tail fins, whereas our observers judged the whole shaft, whih was sometimes in an illusory onfiguration and sometimes not. Morgan et al. used vertial markers to delineate the line segment to be judged, and it may be that this proedural variation gives rise to the differene between theirresultsand ours. Moreover, Morganet al. did not study the effet ofdifferent terminations on the just notieable differene for line length, and so, in this respet, they did not onsider the same question that we did. CONCLUSION We have shown how a refinement of Nevin's (1991) design an provide a detetion-theoreti analysis of the Muller-Lyer illusion. 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