Sanford’s L dissected: A partial replication and extension of Cai et al. (2017)

Abstract

Partial replications of experiments reported by Cai et al. (Attention, Perception, & Psychophysics, 79(4), 1217–1226, 2017) on the so-called Horizontal-vertical illusion confirmed that dissecting L-figures into two separate lines yields greater overestimation of (near-)verticals than do intact Ls. However, contrary to Cai et al.’s findings, which had been obtained with a staircase procedure, with the method of constant stimuli, the amount of illusion was much smaller. This divergence is explained by the self-reinforcing nature of adjustment procedures. Another finding, already reported by Cormack and Cormack (Perception & Psychophysics, 16(2), 208–212, 1974), that obtuse angles between an L’s lines yield greater bias than acute angles, was also replicated in one experiment but tended to be reversed in another. Mixing dissected, upright and top-down inverted Ls and laterally oriented Ts, both with tilted lines, within one experiment confirmed that the bias for Ts is opposite to the one for Ls: For Ts, the effect of (virtual) bisection dominates, yielding an overestimation of the length of the undivided line, whereas for Ls, the horizontal-vertical anisotropy dominates, yielding an overestimation of the length of the vertical line. The differential gap effects can possibly be explained by interactions within the neural substrate between orientation-sensitive and end-inhibited neurons, and the method effects by perceptual learning.

Introduction

Originally, the horizontal-vertical illusion (HVI) refers to the overestimation of vertical extents relative to horizontal ones, when both are visible at the same time (Fick, 1851; Verrillo & Irvin, 1979). According to Finger and Spelt (1947), it was Sanford (1898) who introduced the L-figure as the putatively most proper illustration of this illusion. Other illustrations, including the rectangles and squares used by Fick (1851), either proved to be less effective (Sleight & Austin, 1952), or they introduce additional features (most notably: bisection in the ⊥; Finger & Spelt, 1947; Künnapas, 1955a). Amounts of illusion seen with the L differed widely, depending, among other things, on the orientation of the figure (Avery & Day, 1969) and the direction of the comparison (Gardner & Long, 1960; see Avery, 1970, and Begelman & Steinfeld, 1967, for a critique of this latter work, and Hamburger & Hansen, 2010, and Zhu & Ma, 2017, for mini-reviews).

History

Prior to the bulk of research with the L and the ⊥, Shipley et al. (1949) and Pollock and Chapanis (1952) had studied individual lines in a paired-comparison paradigm and an adjustment-to-standard procedure, respectively. In both experiments, the maximum apparent length was found at a 60° tilt off horizontal, calling Fick’s (1851) original observations into doubt. However, the effect did not unanimously replicate (Cormack & Cormack, 1974; Landwehr, 2017; Zhu & Ma, 2017), and both Shipley et al. (1949) and Pollock and Chapanis (1952) reported extreme interindividual differences between observers. Another observation, reported early on by Finger and Spelt (1947), and even earlier by Valentine (1912a), that did not replicate concerned a greater HVI with an inverted, or “hanging” L as compared to an upright one (Künnapas, 1955a) – but again, as suggested by Finger and Spelt (1947), the finding may have been compromised by individual differences (in their experiment, it was only in 63% of the cases that observers responded as expected).

The issue of differences between observers was taken up by Hamburger and Hansen (2010), but no explanation for where these differences might come from was offered. From an experimentalist perspective, the following factors should be noted: homogeneity of the sample of participants, simplicity of the task and the instruction, and number of trials. Some of the studies cited above used inhomogeneous samples and rather few trials (which may inflate variable errors). Although this does not seem to have been studied systematically yet (but see Macmillan & Creelman, 2005), a decision task (e.g., two-alternative forced choice (2AFC)) seems easier than an adjustment task. For the latter, an instruction to set lengths so that they appear to be equal may confuse observers since, obviously, there is a solution to the task where lengths are equal when measured by a ruler.

With the notable exception of Avery and Day (1969), few researchers have looked at the fate of the HVI when illusion figures were rotated off their canonical orientations (see Landwehr, 2009, for rotations of the ⊥). Even more critical perhaps, few have dared to modify the classical HVI figures – an approach that is quite common with other illusion figures (e.g., Brentano, 1892; Coren, 1970; Fisher, 1968). A few years ago, Cai et al. (2017), by introducing gaps between the L’s lines, and by tilting them relative to one another, reported that the HVI was significantly enhanced when the L was dissected into two separate lines; then, a 15° tilt of the whole figure led to a slight reduction of the illusion, which did not occur with the intact figure. In a third experiment, Cai et al. found that an obtuse angle of 105° between the L’s lines yielded a stronger illusion than did an acute angle of 75°, but only when the lines were sufficiently long (4.9° or 6.5° visual angle; although effect sizes were small: η_p² ranging between .05 and .10). While the last-mentioned result concurs with what others found (Cormack & Cormack, 1974), the first-mentioned result is exactly opposite to what I found for the ⊥ (Landwehr, 2015).

Here, I set out to partially replicate and extend Cai et al.’s (2017) study in an attempt to reconcile findings in a coherent account of illusions that occur with two-line stimuli. In view of the abundance of research with intact upright Ls (see the above-mentioned mini-reviews), I mainly looked at tilted Ls, dissected tilted Ls, Ls with tilted lines, and dissected Ls with tilted lines. Going beyond replications, I combined dissected Ls and ⊥s with tilted lines in one experiment to see whether dissection and the vertical or oblique orientation of one of the figures’ lines would act in similar or different ways with modified Ls and ⊥s.

Theory

When introducing newly discovered illusion figures, authors typically hasten to offer an explanation. Mostly, these explanations have been mini-theories, tailored to the illusion figures described (e.g., Ponzo, 1912). For the HVI, explanatory accounts can be classified into optical-ophthalmological, psychological, mathematical, and neurophysiological.¹ Fick (1851) calculated that a difference of 0.31 mm between the radii of the vertical versus horizontal curvature of the cornea could explain the amount of overestimation of the vertical extents of squares and rectangles, which he had observed with his experimental subjects. Valentine (1912b) tested a fair number of people with or without different forms of astigmatism, and found the HVI to persist in subjects entirely free from astigmatism (although he still observed different amounts of illusion between the two eyes). Pearce and Matin (1969) looked at effects of retinal eccentricity on the magnitude of the HVI, and found it to reverse at great vertical eccentricities. Horizontal eccentricities did not show much of an effect, though, yielding mixed evidence for an influence of astigmatism. This line of theorizing does not seem to have been continued more recently.

The most popular psychological theory of the HVI refers to the oval shape of humans’ field of view. According to Valentine (1912a), the theory can be traced back to Külpe (1893), but its modern popularization was due to Künnapas (1957a, b). The basic idea of the theory is that, because our field of view is wider than high, horizontals have a greater distance to the field’s boundary than equally long verticals, producing a “contrast with the shape of the visual field“ (Valentine, 1912a, p. 9). Valentine tested the theory by comparing monocular versus binocular viewing conditions – an idea picked up by Prinzmetal and Gettleman (1993) – and obtained falsificatory evidence with both the method of right and wrong cases (= method of constant stimuli) and the method of mean error (= method of production). Künnapas’ (1955b et passim) work has been reviewed by Robinson (1972), who pointed out the difference between a frame drawn on paper and the indefinite “boundary” of the field of view – which difference casts some doubt on the adequacy of Künnapas’ (1957a, b) tests of the theory. Another popular psychological theory of the HVI, which goes back to Thiéry (1895), posits that verticals are seen as receding into depth; thus, being a frontoparallel projection of something more distant, they ought to appear longer. Ward et al. (1977) did not find a significant amount of depth associations with the ⊥, though, and Wolfe et al. (2005) definitely falsified the theory – Williams and Enns’ (1996) revitalization of both the framing and the perspective theories notwithstanding.

Mamassian and de Montalembert (2010) modeled response behavior to L-, ⊥-, and +-figures in terms of the known contributing factors of anisotropy and bisection as well as observers’ hypothetical sensitivity (the slope of a fitted psychometric function). Inasmuch as the authors alternatively relate differential sensitivity to uncertainties (“noise”) occurring during “image measurements” versus “the decision stage,” the modeling may be regarded as an implicit psychological theory. The modeling has been elaborated upon by Mikellidou and Thompson (2013), who split up the bisection component of the ⊥ and the + into separate “crossing” versus “abutting” components. Although both sets of models successfully predicted most of the responses of newly drawn samples of observers, the models do not – as expressly admitted by both groups of authors – provide explanations. Utilizing another variant of the HVI, the Helmholtz square (which looks wider than high with vertical stripes and higher than wide with horizontal ones), and functional magnetic resonance imaging as well as transcranial magnetic stimulation, Mikellidou et al. (2016) identified the extrastriate area LO1 to be involved in the illusion. It is uncertain, though, whether findings with a textured stimulus generalize to the simpler L-, ⊥-, and +-figures.

Caelli (1977) used zig-zag lines and sine waves with different extreme bends in a paired-comparison task and, analyzing data by means of Thurstonian scaling, found that “the lines were underestimated in length as their horizontal components increased” (p. 838). Given the close correspondence between psychophysical data on the HVI (most notably, the data of Shipley et al., 1949; Pollock & Chapanis, 1952) and a theoretical distribution of orientation judgments as a function of adaptation and test stimuli, derived from Carpenter and Blakemore (1973), Caelli (1977) suggested that perceived length may be determined by interactions between neural orientation detectors. In various publications, I applied Caelli’s idea to explain psychophysical findings with intact ⊥s, dissected ⊥s, and ⊥s surrounded by flanks (e.g., Landwehr, 2009, 2015, 2017, 2018), and in the General discussion of the present paper, I elaborate on these accounts.

A second idea that can be derived from neurophysiological research is the notion of figural schemata. Shevelev et al. (2001) obtained single-cell recordings from area 17 of the cat visual cortex in response to +-, ⊥-, and Y-like stimuli,² and identified neurons that were sensitive to a figure’s orientation, its shape, or both. Only 9.3% of the 75 sensitive neurons (out of a total of 101 tested) responded to all stimuli alike. Relating Shevelev et al.’s findings to Neisser’s (1976) concept of schematic processing, Landwehr (2016) proposed to explain the persistence of the HVI and bisection illusion in patterns of several ⊥s and in triangular and beehive figures, for which the ⊥ constituted an invisible skeleton, in terms of a “⊥-schema.” No neurophysiological data appear to exist for the L, but Changizi et al. (2006) gathered suggestive observations with regard to an L-schema.

Another possible explanation of the HVI at the neurophysiological level refers to the elongated shape of the receptive fields of retinal and cortical neurons (McGraw & Whitaker, 1999). At a horizontal orientation of these fields, a vertical bar will excite more units than a horizontal bar, presumptively yielding an overestimation of the vertical bar’s length. Psychophysical evidence for this account was obtained by presenting pairs of luminous dots in various places around fixation. Spatial misperceptions displayed a radial structure, with the usual HVI when dots were placed on the horizontal meridian, and a reversal when they were placed on the vertical one. A possible limitation of the account may pertain to the very short viewing times used (200 ms).

General method

The chief difference between my present experiments and those of Cai et al. (2017), which served as models, was that I used the method of constant stimuli, whereas Cai et al. had used a staircase procedure. The reason for this deviation was to look at possible effects of the psychophysical methods used. As suggested by Cormack and Cormack (1974), different findings obtained in studies of the ⊥ may have resulted from the use of different procedures – and I confirmed their suspicion in my partial replication of their experiment (Landwehr, 2017).

Another difference between my experiments and Cai et al.’s (2017) was that I used L-figures of negative contrast (black on white), whereas Cai et al. had used white Ls on a black background. This difference, which was motivated by comparability with my previous work on the ⊥ (Landwehr, 2015 et passim), was not considered critical because achromatic polarity has been shown to be immaterial for the HVI (Verrillo & Irvin, 1979, Experiment 2).

Participants

Thirty psychology undergraduates, drawn from the same population (Johannes Gutenberg-Universität Mainz), took part in the experiments, ten per experiment (independent samples, as in Cai et al., 2017). The number of participants had been decided upon a priori in order to achieve statistical power of 1 – β ≥ .95 for repeated-measures analyses of variance (rmANOVAs) with α = β ≤ .05, and f ≥ 1 (Faul et al., 2007). Written, informed consent was obtained from all participants, and they were treated in accordance with the Declaration of Helsinki (World Medical Association, 1964/2013). All participants had normal or corrected-to-normal vision, and all served in partial fulfillment of a class requirement. My department’s Ethics Committee had declared my experiments exempt.

Apparatus

The essential part of the apparatus was a touch-sensitive computer screen (size: 59.6 × 33.5 cm; resolution: 2,560 × 1,440 pixels; response time: 3 ms), which was used for both stimulus presentation and response registration. The screen was oriented frontoparallel at a distance of 44 cm from the observer. Stimuli, drawn from thin black lines (width: 1.25 mm; visual angle: 9.5 arcmin), were presented within a circular, light grey window (diameter: 28.5 cm; plane visual angle: 35.9°; luminance: 228 cd m^-2; CIE-coordinates: x = 0.312; y = 0.332; Weber contrast between stimulus and background: C_W = − 0.998); the rest of the screen was dark (0.355 cd m^-2), and there was only faint, indirect illumination of the room.

Stimuli and responses

The specific stimuli used will be described for each experiment separately. For Experiments 1 and 2, four different lengths were used for the two lines of the L-figures – 6, 6.1, 6.2, and 6.3 cm (7°48’, 7°56’, 8°3’, and 8°11’ visual angle) – and factorially crossed. However, the cases in which the Ls’ lines were equally long were deselected, and participants were informed about this. This was done to motivate observers to try hard to decide on the correct answer. Stimuli were presented for 1.5 s. Then, in order to nonverbally indicate the target line, one of the L’s lines changed its color from black to red for 200 ms, and then, for another 200 ms, back to black. The observer’s response started the next trial after a delay of 200 ms.

The response format was two-alternative forced choice (2AFC). Observers had to compare the two lines of the stimulus and say whether the target line had been longer or shorter than the comparison line. Both lines of the L served as target and as comparison line. Responses were delivered by gently touching response buttons that appeared on the computer screen after the stimulus had been turned off. Effective response registration was signaled to participants by the buttons shining light blue.

Data analysis

Data were analyzed by fitting psychometric functions with cumulative Gaussians. Parameters were computed via the binary logistic regression routine of SPSS™. Points of subjective equality (PSEs) were found at the 50% points of the necessarily symmetric functions for the longer and shorter judgments that had been plotted against an abscissa defined by the difference in length between the two lines of the L (points were found numerically by transferring the data to customized software in Mathematica™). Difference thresholds (just noticeable differences (JNDs)) were read off the psychometric functions from half the difference between the 25% and 75% points of the functions. Both sets of data were further analyzed by means of repeated-measures analyses of variance (rmANOVAs).

Experiment 1

Utilizing a fixed angular deviation from vertical (here, following Cai et al., 2017, 15° was used), the L may be rotated to eight qualitatively different oblique orientations, which –depending on the length calibration – are also perfect or approximate mirror images of one another (Fig. 1). While Cai et al., in their Experiment 2, had used only the clockwise tilts of the L, I used the counterclockwise tilts, too. As in Cai et al.’s experiment, there were both intact and dissected Ls, but, in contrast to their experiment, no upright or top-down inverted, nontilted Ls. Cai et al. had created disconnected Ls by a simultaneous, horizontal and vertical offset of the L’s two lines by a randomly determined, variable amount of 1.2–5.1 cm. I also used a simultaneous offset, but by an equal amount (two times 1 cm), because only this case retains the figure’s bilateral symmetry.³ The hypothesis of my first experiment was that the more vertically oriented lines would be judged longer than the more horizontally oriented lines. Following Cai et al., it was further assumed that this kind of overestimation would be much greater for the dissected Ls than for the intact ones.

Fig. 1

The eight different orientations of the L-figure used in Experiment 1. Note. In text, the lines tilted 15° off verticality are referred to as “more vertical,” the others, tilted by the same amount with regard to horizontality, as “more horizontal.” In order to illustrate the symmetry relations between the differently oriented Ls, all lines in the figure have the same length, which they never had in the experiment

Design

There were 2 types of stimuli (connected vs. dissected Ls) × 8 orientations × 12 length combinations × 2 directions of comparison, making for a total of 384 unique trials, which were analyzed for the two types of stimuli and the two directions of comparison. Hence, there were 96 trials to estimate psychometric functions.

Results and discussion

Responses did not differ between the different orientations of the stimuli; therefore, data were aggregated across this variable. According to the design, this left four psychometric functions to be fitted per observer, making for a total of 40 functions. Functions could not be fitted for two observers, leaving 32 functions, and in another three cases of this remaining number of functions. Only the gap between the two lines of the L-figures showed a significant effect on the PSEs, F(1, 5) = 23.512, p < .005, η_p² = .825, yielding a greater illusion for the gap stimuli than the intact Ls. There was no effect of the target line, F < 1, but a marginally significant interaction effect between gap and target, F(1, 5) = 5.560, p < .065, η_p² = .527, consisting of a greater response bias with the vertical line as target in intact Ls.

Although the gap effect replicated Cai et al.’s (2017) findings, the amount of illusion seen in my experiment was much smaller than the one calculated by Cai et al. for their observers: 4.7% versus 10.5%. A possible explanation for this difference is that it is due to the different methods used: The staircase procedure used by Cai et al. seems to have favored the HVI, and the method of constant stimuli, which I used, seems to have disfavored the illusion. An explanation for this divergence, as well as one for the gap effect as such, is deferred to the General discussion.

There were no effects of the experimental variables on the JNDs (all Fs < 1), the grand mean of which was 0.19 cm, SD = .07 cm. PSEs and JNDs were uncorrelated, r(29) = .153, p < .428. Finally, it should be mentioned that there were considerable interindividual differences, both for the PSEs, F(1, 5) = 22.095, p < .005, η_p² = .815, and for the JNDs, F(1, 5) = 374.669, p < .000, η_p² = .987.

Experiment 2

Experiment 2 was a partial replication of Cai et al.’s (2017) Experiment 3, which, in turn, had been a partial replication of Cormack and Cormack (1974). Therefore, only upright Ls were used. There were both intact and dissected Ls, and the vertical line was tilted to 75° or 105° relative to the horizontal one, creating an acute versus an obtuse angle between the two lines (Fig. 2). Line length was varied as in Experiment 1, and the apparatus and procedure also remained the same.

Fig. 2

Selected specimens of the L-figures used in Experiment 2. Note. The experimental stimuli consisted of the two bold lines of the configurations only

Design

There were 2 types of stimuli (connected vs. dissected Ls) × 2 orientations (right-left reversals) × 2 angles of internal tilt × 12 length combinations × 2 directions of comparison × 2 repetitions = 384 trials, which were analyzed separately for the two types of stimuli and the two angles of tilt, making for 96 trials per psychometric function. Since the same set of data was analyzed twice, for the rmANOVAs, α was corrected to 0.025.

Results and discussion

Responses did not differ between the mirror images of the stimuli; therefore, data were aggregated across this variable. According to the design, this left two times four psychometric functions to be fitted per observer, making for a total of two times 40 functions. Fits worked well in 31 cases for the gap variable and in 32 cases for the variable angle. There were significant effects of the variables gap and angle on the PSEs, F(1, 4)_Gap = 17.120, p < .014, η_p² = .811; F(1, 4)_Angle = 25.066, p < .007, η_p² = .862, but no effects of the target line (both Fs < 1) and no significant interaction effects, F(1, 4)_{Gap x Target} = 2.320, p < .202, η_p² = .367; F(1, 4)_{Angle × Target} = 1.843, p < .246, η_p² = .315.⁴ In terms of means and percentages (overestimation of the vertical line relative to the horizontal one, referenced to the mean line length of 6.15 cm, and calculated from the data of all observers), effects were M_Gap0cm = 0.31 cm (5%), SD_Gap0cm = 0.17 cm, M_Gap2cm = 0.40 cm (6.5%), SD_Gap2cm = 0.10 cm, M_AngleObtuse = 0.42 cm (6.8%), SD_AngleObtuse = 0.14 cm, M_AngleAcute = 0.29 cm (4.7%), SD_AngleAcute = 0.12 cm. There were no effects of the experimental variables on the JNDs, the means of which were M_Gap0cm = 0.18 cm, SD_Gap0cm = 0.06 cm, M_Gap2cm = 0.14 cm, SD_Gap2cm = 0.05 cm, M_AngleObtuse = 0.17 cm, SD_AngleObtuse = 0.07 cm, M_AngleAcute = 0.16 cm, SD_AngleAcute = 0.07 cm. PSEs and JNDs were uncorrelated, r(31)_Gap = .208, p < .262, r(32)_Angle = .115, p < .532, but, as in Experiment 1, there were great interindividual differences between observers (all Fs > 30, all ps < .005, and all η_p²s ≈ .900).

The observations of Cormack and Cormack (1974) and the findings of Cai et al. (2017) were replicated: There was a larger response bias for obtuse angles as compared to acute ones, and there was a larger bias for dissected as compared to intact Ls. Cai et al. did not offer an explanation for the effect of the relative angle of tilt of the Ls’ lines. Cormack and Cormack (1974), with reference to Morinaga et al. (1962), as well as to comments of their own participants, suggested that the nearness of the tips of the lines with acute angles made judgments easy, because “lengths could be equated by fairly simple alignment” (Cormack & Cormack, 1974, p. 211). This reasoning intermingles sensitivity and bias, though, and is only superficially supported by my present data. A factor that may have supported the larger response bias for obtuse angles may be the trend towards collinearity the greater the angles become. However, support for this idea so far only came from upright ⊥s, but not from Ls or laterally oriented ⊥s (ibid.; Landwehr, 2017).

Experiment 3

Experiment 3 was conceived to reconcile the contradictory findings that dissecting an L increased response bias (Cai et al., 2017, and the present Experiments 1 and 2) whereas dissecting a ⊥ decreased it (Landwehr, 2015, 2017). The stimuli shown in Fig. 3 have been derived from L- and ⊥-figures. Always, one vertical line was paired with one oblique line which hovered at a distance of 1 cm to the left or right of one of the vertical line’s endpoints – forming dissected Ls – or to the left or right above its midpoint – forming dissected ⊥s. The oblique lines either pointed 45° upwards or -45° downwards. For comparability’s sake, line length was varied as in Landwehr (2015, 2017). Three lengths – 6.5, 7, and 7.5 cm (8.45°, 9.1°, and 9.7° visual angle) – were factorially crossed, and the pairs with identical lengths were kept. The apparatus and procedure remained the same as in Experiments 1 and 2. The prediction was that the length of the oblique line would be overestimated with the ⊥, but that it would be underestimated with the L. Since, for the L-figures, the obliques created either acute or obtuse enclosed angles, angles were expected to modulate the predicted effects.

Fig. 3

Selected specimens of the stimuli used in Experiment 3. Note. The experimental stimuli consisted of the two bold lines of each configuration only

Design

There were 3 types of stimuli (two types of Ls and one type of ⊥) × 4 configurations (right-left positions and up-down orientations of the oblique lines) × 9 length combinations × 2 directions of comparison = 216 unique trials, which were repeated once to obtain a sufficient number of trials to estimate psychometric functions. For the analysis of the data, it is important to note that the design of the experiment contained an irremovable confound: Although there were two types of figures and two up-down orientations of the oblique lines, it was only for the L-figures that the enclosed angle between verticals and obliques could assume one definite value (acute versus obtuse), whereas for the ⊥, both types of angles always occurred simultaneously. For the statistical analysis, the focus was on angle instead of pattern because of the equal number of trials for type of angular relationship.

Results and discussion

A first impression of the empirical findings may be gained from the plots of the raw data (Figs. 4 and 5): The oblique lines of L-figures were more often judged shorter at acute angles, and the vertical lines more often judged longer at these angles; the effects were attenuated at obtuse angles, and for ⊥-figures, effects were reversed. The probabilities of the respective judgments apparently followed linear trends.

Fig. 4

The aggregated raw data of Experiment 3 for the oblique lines as targets

Fig. 5

The aggregated raw data of Experiment 3 for the vertical lines as targets

According to the design, six psychometric functions had to be fitted per observer, making for a total of 60 functions. Fits worked well in 57 cases. A rmANOVA on the PSEs revealed a main effect of angle, F(1.564, 9.384) = 27.064, p < .000, η_p² = .819, but no significant effect of target line, F(1, 6) = 5.722, p < .054, η_p² = .488, and no significant interaction, F(2, 12) = 4.414, p < .060, η_p² = .424. Figure 6 displays the results. Obviously, the (multiple) linear trend seen in the raw data (Figs. 4 and 5) was preserved. Repeated contrasts, however, revealed that the difference between acute versus obtuse angle failed to reach significance, F(1, 6) = 4.633, p < .075, η_p² = .436, possibly due to the large variance in the data for the acute-angle condition. The other comparison (obtuse vs. acute + obtuse) was highly significant, F(1, 6) = 32.130, p < .001, η_p² = .843, but this effect also mirrors the difference between the L and the ⊥.

Fig. 6

Points of subjective equality and 95% confidence intervals for the angular relationships used in Experiment 3

As is evident from Fig. 6, response biases went in opposite directions for L- versus ⊥-figures. The reason why the bias is positive for the ⊥, but negative for the L, is simply a consequence of how the difference in line length that was used as covariate for the logistic regressions had been calculated (vertical minus oblique). Consistent with the raw data plots (Figs. 4 and 5), the length of the vertical line was overestimated with L-figures (4.9% and 1.4%, for acute vs. obtuse angles, respectively), and the length of the oblique line was overestimated with ⊥-figures (1.9%).

What is more surprising is that, contrary to the results of Experiment 2, the bias with acute angles in L-figures tended to be more extreme than the one with obtuse angles. This potential difference could not have been noted in previous research because oblique lines in L-configurations had only been compared with horizontal, not vertical lines (Cai et al., 2017; Cormack & Cormack, 1974). If it stabilizes in future replications, the inversion of the effect of type of angle might be understood as a meta-effect of the HVI itself.

As in the preceding experiments, PSEs and JNDs were uncorrelated, r(57) = .086, p < .524, but this time, there was an effect of the major experimental variable on the JNDs, F(1.111, 6.668)_Angle = 7.239, p < .031, η_p² = .547 (the other Fs < 1). Repeated contrasts showed that the effect was restricted to the comparison between acute versus obtuse angles in L-figures, F(1, 6) = 32.344, p < .001, η_p² = .844 (the other F < 1), yielding smaller JNDs for acute angles, M_Acute = 0.21 cm, SD_Acute = 0.10 cm, M_Obtuse = 0.28 cm, SD_Obtuse = 0.12 cm, M_⊥ = 0.29 cm, SD_⊥ = 0.10 cm, thus supporting Cormack and Cormack’s (1974) earlier cited claim that these judgments were easier.

For the PSEs, there were no significant individual differences, F(1, 6) = 2.354, p < .176, η_p² = .282, but for the JNDs, there were, F(1, 6) = 49.086, p < .000, η_p² = .891. A simple histogram (not shown) revealed that the effect was mainly due to two participants, one of whom showed mediocre performance (all JNDs > 0.37), whereas the other one showed superb performance (all JNDs < 0.20).

General discussion

Gap effects

As admitted by Cai et al. (2017), their finding of an increased HVI with dissected Ls was not novel. As mentioned in the Introduction, several authors have studied the HVI with individual lines, and typically, the illusion seen was much greater than with configurational stimuli (also: Armstrong & Marks, 1997) – a finding that Cai et al. (2017) replicated with the length perception task in their Experiment 1. In all of these studies, though, as well as in most others (including mine), the stimuli were not presented devoid of some kind of context. Most often, a circular presentation window was used (e.g., Zhu & Ma, 2017). As argued by Künnapas (1957a, b), such kind of “surrounding field” may affect length estimates, and empirically, he demonstrated such effects for the (intact) L by comparing circles and squares with “standing” versus “lying” ellipses and rectangles as contexts (a square white screen had actually been used by Armstrong & Marks, 1997, and by Pollock & Chapanis, 1952; and in Cai et al.’s (2017), experiments, the rectangular shape of the computer screen seems to have been available as a reference).

As far as I am aware, there have been only three studies that were conducted in a darkroom.⁵ Avery and Day (1969) found no differences for an intact L when presented in the dark or under dim or bright light conditions, concluding that “elimination and obscuring of visual field contours neither eradicate the HV illusion nor significantly reduce it” (p. 377). Verrillo and Irvin (1979), on the other hand, found no differences between individual horizontal versus vertical lines, when these were presented in complete darkness – concluding that “When care is taken to eliminate cues …, the Horizontal-Vertical Illusion does not exist” (p. 270). On a positive note, Avery and Day’s (1969) results suggest that, for configurational stimuli, context effects may often be negligible (in their study, the context had been a square box, the internal illumination of which produced the stimulus, plus furniture and laboratory equipment). The effects observed by Künnapas (1957a, b) may have been due to the small size of the gap between stimulus and surrounding frame (1° 25ʹ; cf. Landwehr, 2015, pp. 2149-2152). Still, Verrillo and Irvin’s (1979) finding leaves open the possibility that the differences between connected versus disconnected lines seen in Cai et al.’s (2017) as well as in my present experiments were a joint effect of the feature of connectedness and the free-floating lines’ greater susceptibility to context effects.

Perhaps, the notion of levels of context may help to render results intelligible: The second line in an L- or ⊥-figure can be said to constitute a first level, and other lines, borders, etc., a second level. For both levels of context alike, the connectivity or relatedness of the context elements to the target elements seems to be the prime moderating variable. However, effects need not go in the same direction, and they may also depend on task demands. In Avery and Day’s (1969) experiments, an unconnected, global context did not exert any influence, but in Cai et al.’s (2017) experiments, a local, unconnected context did (similarly: Armstrong & Marks, 1997). In Landwehr’s (2016) Experiment 2, the context of abutting or intersecting ⊥s, added to a target ⊥, eliminated the ⊥-illusion, albeit for a haptic response measure only.

Since null effects may eventually turn out to be minuscule effects, what needs to be explained first is why and how connectivity versus dissection in two-line figures modulates response bias. With reference to Behrmann et al. (1998) and Feldman (2007), Cai et al. (2017) tried to explain the differential effect of the lines’ connectedness in L- and ⊥-figures in terms of perceived objectness: An intact L would be perceived as a single object, but an intact ⊥ as two objects. This kind of explanation encounters difficulties when applied to a dissected ⊥. Here, the illusion remained strong at the canonical, upright and top-down inverted orientations of the ⊥, but it was much attenuated at other orientations (Landwehr, 2015). Even more serious perhaps, there is also psychophysical and neurophysiological evidence that an intact ⊥, like an intact L, is processed as a unit (Landwehr, 2016; Shevelev et al., 2001). On the other hand, there have also been suggestions to the effect that a ⊥ would be apprehended as three objects (Schumann, 1900; cf. Landwehr, 2018).

Either way, the foregoing line of reasoning does not appear promising. For one, the studies of Behrmann et al. (1998) and Feldman (2007) were concerned with reporting features from intersecting and occluding bars, and with reaction times to features superimposed on line figures, respectively, and not with comparative judgments of length; and for another, for plane line figures, I do not see what is gained by replacing the concept of line arrangement by the notion of object. At the level of neurophysiology, Hubel and Wiesel (1962, 1965; see Priebe & Ferster, 2012, for an informed update) have established the existence of two classes of neurons that, respectively, respond to the orientation of a line or its orientation and its endpoints. Although the specification of the details of the workings of such mechanisms requires high-resolution neuroscientific methods, the application of the basic idea to the psychophysical data obtained seems straightforward: Interactions between orientation-sensitive and end-stopped neurons seem likely to differ for different line arrangements, and for connected versus dissected lines in particular (Landwehr, 2018).

The psychophysical data may now be regarded to be beyond doubt: Dissecting an L increases response bias, and dissecting a ⊥ decreases it. Three things may contribute to this difference: local processes in the neural substrate, top-down processes related to figural schemata, geometrically defined configurational singularities. The mark of the ⊥, that distinguishes it from an L, even after dissection, is the manifest or implicit bisectioning of one line by another one. As demonstrated by Künnapas (1955a) by moving the ⊥’s undivided line along the divided line, the position of the undivided line determines the amount of bias – going from a maximum for the ⊥-configuration to a minimum for L-configurations – suggesting that the divided line’s midpoint is a critical reference point. As mentioned in the Introduction and above, the observations of Shevelev et al. (2001) provide neurophysiological evidence for the processing of the ⊥ as a unit. Importantly, this does not imply the absence of local, elementaristic processing. Shevelev et al. considered two mechanisms that may generate a ⊥-schema in the first place: “excitatory convergence of two neurons with different preferred orientations …, inhibitory and facilitatory (or disinhibitory) interactions between central and peripheral RF [receptive field] zones, as well as combination[s] of these factors” (p. 547).

Psychophysics can only suggest neural mechanisms that may generate observed effects. Yet it is the behavioral data that constrain which mechanisms are feasible and which are not. Three things in particular may be inferred from psychophysical findings: the independence or additivity of mechanisms, their linear or nonlinear functioning, and quantitative relations. On the basis of his data, Künnapas (1955a) has argued that the two factors of the ⊥-illusion, bisection and HVI, combine additively and hence may be considered independent (similarly: Mamassian & de Montalembert, 2010). Part of the data of Cai et al.’s (2017) Experiment 1, as displayed in their Fig. 3, when compared to data obtained with similar methods (cf. Hamburger & Hansen’s, 2010, mini-review), suggest that verticality and (dis-)connectedness also combine additively in the L. Linear relationships are evident in another data plot of Cai et al.’s (2017) Experiment 1 (their Fig. 2), since the mean PSE increased linearly with the “labeled size” of a standard. Figures 4 and 5 of the present paper show linearities related to angular relations between the lines of Ls and ⊥s. Quantitative relations are more difficult to establish, but Landwehr (2017) predicted and confirmed such relations for dissected ⊥s with tilted lines.

Tilt effects

Absolute and relative tilt of and/or within the L-figure was Cai et al.’s (2017) second theme. Here effects were less transparent and more variegated: A 15° tilt of the whole figure affected response bias only for dissected, not intact Ls, and effects of relative tilt depended on the absolute size of the figures. In the series of experiments reported in the present paper, the effect of relative tilt between the L’s lines depended on the figure’s overall orientation and shape (i.e., the specific angles of tilt). It is here that the hypothesis of interactions between otherwise independent neural mechanisms may have its virtue. While Caelli (1977) focused exclusively on orientation-sensitive mechanisms, Landwehr (2018) assumed additional interactions with end-stopped mechanisms. For the ⊥, the original idea had been that mechanisms that respond to the ⊥’s undivided line would inhibit mechanisms that respond to its divided line (Landwehr, 2009). In view of effects of the number of gaps in three-line stimuli (⊥s with free-floating or abutting flanks), Landwehr (2018) suggested a more prominent role of end-stopped mechanisms. For the L, the same logic may be invoked. For the differences seen between the L and the ⊥, the relative orientation of neurons’ receptive fields may also be important (McGraw & Whitaker, 1999).

Two other factors may intervene. Junctions create distinctly different diffraction patterns on the retina than continuous lines or the ends of lines do. This in itself may cause a perceptual shortening of connected versus disconntected lines. If, as suggested in the preceding section, the second line of a two-line stimulus constitutes the main reference for the length and orientation of the first line, the exact position of the lines’ endpoints may be differently difficult to determine, depending on the lines’ connectedness. Reconsidering the finding of Landwehr (2016), that the ⊥-illusion vanished in patterns of abutting or intersecting ⊥s, when a haptic response measure was used, we may suppose that the free endpoints of lines are more difficult to localize than junction points. This localization uncertainty may further contribute to the larger response bias seen with dissected Ls. For dissected ⊥s, the midpoint of the originally divided line may still act as a useful reference point.

Method effects

It has long been known that different psychophysical methods may yield different findings. Concerning the ⊥-illusion, Finger and Spelt (1947) and Cormack and Cormack (1974), using the method of adjustment, found a dominance of the horizontal-vertical component of the illusion, whereas Künnapas (1955a) and Tedford and Tudor (1969), using the method of constant stimuli, found a dominance of the bisection component. For a dissected, top-down inverted L, Begelman and Steinfeld (1967), using an adjustment procedure, found complex interactions between line length, lateral separation of the lines, and which line served as standard or comparison line.⁶ Avery (1970, Experiment 1), using a double-staircase procedure, did not find any effects of the last-mentioned variable. Utilizing the method of constant stimuli, the present Experiment 1 replicated Cai et al.’s (2017) finding, obtained with a staircase procedure, of an enhancing effect of a gap between an L’s lines on the HVI, but effect sizes differed considerably.

The nagging question, of course, is why and how different methods produce different results? From an experimental-methodology point of view, of the three psychophysical methods as systematized by Fechner (1860), only the method of constant stimuli qualifies as experimental, because only this method implements the control technique of randomization. In the method of limits, experimenters provide feedback in response to observers’ responses, and in the method of adjustment, observers expose themselves to different conditions of stimulation in unpredictable and uncontrolled ways. The problem of the method of limits is exacerbated in simple staircase procedures because now observers may soon become aware of the feedback schedule (Cornsweet, 1962). Double or multiple staircases are often believed to circumvent potential effects of this awareness, but observers may still be influenced by the objective schedules of reinforcement even if they do not look through them (Premack, 1959; Spielberger, 1962). Randomly intermixing staircases has been proposed as the ultimate solution by Cornsweet (1962), but the data he presented clearly show a drift off horizontality (i.e., a trend for measured thresholds to increase with the number of trials). Another problem associated with all “adaptive” procedures is that they tend to overestimate observers’ sensitivity (Leek, 2001).

Conclusion

As exemplified by the current research (as well as its models), only radical modifications of traditional illusion figures promise to provide new insights into the causes of response biases. The effects seen with L-figures were not confined to a horizontal-vertical anisotropy, but also involved the tilt of the whole figure (Cai et al., 2017, Experiment 2) and the relative tilt of the L’s two lines (ibid., Experiment 3, and my present Experiment 2). The suggestion, thus, is to move away from traditional illusion research and instead consider line arrangements more generally (see Feldman, 2007, and Landwehr, 2018, 2022, for psychologically inspired attempts at a systematization of two-line configurations).

Of equal importance, or so others have argued decades ago (Morgan et al., 1990), seems to be to take advantage of both parameters that can be read off psychometric functions. As demonstrated in the presently reported experiments, the existence of an illusion did not imply absence of sensitivity. Importantly, on average (in fact, in the vast majority of individual cases, too), bias (PSE) and sensitivity (JND) were uncorrelated. This means that effects can be interpreted independently, but also, that the causes of high versus low bias or sensitivity may differ, and so require different explanatory accounts.