Influence of visually perceived shape and brightness on perceived size, expected weight, and perceived weight of 3D objects

Michele Vicovaro; Katia Ruta; Giulio Vidotto

doi:10.1371/journal.pone.0220149

Abstract

In the size–weight illusion, when two objects of identical weight but different volume are lifted, the smaller object is typically perceived to weigh more than the larger object. A well-known explanation for this and other weight illusions is provided by the hypothesis that perceived weight results from the contrast between actual and expected weight. More recently, it has been suggested that an object’s size may exert a direct and automatic effect on its perceived weight, independently of expected weight. Here we test these two hypotheses by exploring two illusions that have been known for a long time but have remained relatively underexplored, namely the shape–weight and brightness–weight illusions. Specifically, we measured the influence of visually perceived shape and brightness on the perceived size, the expected weight, and the perceived weight of 3D plastic objects. A numerical rating task was used in Experiment 1, and a paired comparison task was used in Experiment 2. The results showed that spheres were perceived to be heavier than tetrahedrons and cubes, and cubes were perceived to be heavier than tetrahedrons. We did not find any consistent relationship between brightness and perceived weight. A systematic comparison between perceived size, expected weight, and perceived weight showed that the visual shape–weight and brightness–weight illusions are partially inconsistent with the hypothesis that perceived weight results from the contrast between actual and expected weight and with the hypothesis that perceived weight results from the contrast between actual weight and perceived size. The results appear to suggest that there may be a dissociation between the processing of variables that contribute to the conscious experience of size, such as brightness and vertical height, and the processing of variables that contribute to perceived weight, such as surface area.

Citation: Vicovaro M, Ruta K, Vidotto G (2019) Influence of visually perceived shape and brightness on perceived size, expected weight, and perceived weight of 3D objects. PLoS ONE 14(8): e0220149. https://doi.org/10.1371/journal.pone.0220149

Editor: Gavin Buckingham, University of Exeter, UNITED KINGDOM

Received: March 8, 2019; Accepted: July 9, 2019; Published: August 8, 2019

Copyright: © 2019 Vicovaro et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All the raw data from the experiments are available from the Open Science Framework database (link: https://osf.io/4m8yu/). The original physical study materials used to conduct the research (i.e., stimuli) will be made available upon request.

Funding: The study was supported by a grant form Ministero dell’Istruzione, dell’Università e della Ricerca (Dipartimenti di Eccellenza DM 11/05/2017 n.262) to the Department of General Psychology. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

The idea that the subjective impression of weight is simply a function of physical weight has intuitive appeal, as it would mean that our sensory system works similarly to a weight scale. However, research has shown that the perceived weight of objects is actually affected by a variety of factors, such as the resistance of objects to rotation in space [1] and complex visual-motor relationships between motor commands and feedback signals [2,3,4]. Furthermore, research in the field of weight illusions has shown that perceived weight depends on various non-weight properties. For instance, when two objects of the same physical weight but different sizes are lifted, the smaller object usually feels heavier than the larger object, a phenomenon known as the size–weight illusion [5,6]. Another well-known illusion is the material–weight illusion, which refers to the phenomenon whereby, when two objects of the same physical weight but different surface materials are lifted, the object with the less dense surface material (e.g., polystyrene) usually feels heavier than the object with the denser surface material (e.g., iron [7,8,9]).

A unified explanation of weight illusions is provided by the so-called expectation model of weight perception [10,11], according to which perceived weight results from the contrast between sensory information about the actual weight of an object and cognitive information about its expected weight. Because of that contrast, whenever two objects of the same physical weight are expected to differ in weight, the object that is expected to be heavier is perceived to be lighter than the other object. Support for the expectation model has been provided by studies showing that weight expectations generated prior to the lifting action can directly cause the size–weight illusion [12] (cf. [13]) and the material–weight illusion [7] and by studies showing that the size–weight illusion can be inverted after extensive experience with objects that reverse the natural size-weight correlation [10,14,15]. Interestingly, Ellis and Lederman [16] showed that golfers had different expectations about the weights of golf balls with respect to non-golfers, and as a result, golfers and non-golfers had different perceptions of the weight of golf balls. In sum, there is converging evidence that expectations exert a top-down influence on weight perception (see [17]).

The size–weight illusion may result not only from top-down but also from bottom-up perceptual processes (see [1,17,18,19,20]). For instance, Ellis and Lederman [21] showed that the size–weight illusion is larger in magnitude when participants are asked to lift the stimuli by grasping them (i.e., the haptic size–weight illusion), as compared to when the stimuli are lifted by means of a string or a pulley (i.e., the visual size–weight illusion). Since somatosensory information about the rotational inertia [1] and density [20] of the stimuli is available when objects are grasped and lifted but not when objects are lifted by means of a string or a pulley, the fact that the haptic size–weight illusion is comparatively stronger than the visual size–weight illusion suggests that participants’ sensitivity to rotational inertia or density may contribute to the illusion. Despite this, the existence of the visual size–weight illusion indicates that the illusion can be generated by expectations alone [12] (cf. [13]).

In light of the results of previous studies, the idea that expectations can affect the perceived weight of objects can hardly be questioned. However, a challenge for the expectation model is provided in studies showing that, in some cases, perceived weight does not appear to be the result of the simple contrast between actual and expected weight [22,23,24,25]. For instance, Vicovaro and Burigana [25] presented the participants with pairs of cubic boxes varying in surface material and volume. In the expected weight condition, participants had to judge which stimulus appeared to be heavier, or if the two stimuli appeared to have the same weight (touching the stimuli was not allowed). In the perceived weight condition, the participants had to lift the stimuli using a string attached to the top surface in order to compare their perceived weight. The results showed that, inconsistently with the predictions from the expectation model, large stimuli made of a low-density surface material (i.e., polystyrene) were both expected and perceived to be lighter than small stimuli made of a relatively dense surface material (i.e., wood). In a similar vein, Dijker [23] found that large toy dolls that appeared to be weak in physical strength (e.g., Barbie) were both expected and perceived to be lighter than small toy dolls associated with apparently high physical strength (e.g., Action Man). A common feature of these two studies is that some of the large stimuli were expected and perceived to be lighter than some of the small stimuli (see also [22,24]). Consistent with this finding, Saccone and Choiunard [19] recently argued that, with respect to other non-weight properties, size has a unique ability to influence weight perception, and suggested that this might be due to the fact that size is processed rapidly before conscious awareness by the magnocellular pathway. Therefore, size would strongly and automatically affect perceived weight even when it exerts a relatively small influence on expected weight.

A contribution to the current debate on the relationship between perceived weight, expected weight, and perceived size [12,17,18,19,22,23,25] may come from the study of two illusions that have been known for a long time but that have remained relatively underexplored, namely the shape–weight illusion [26,27,28] and the brightness–weight illusion [29,30]. The main aim of the current study is to test the extent to which the two illusions can be explained in terms of the contrast between actual and expected weight, and/or in terms of the contrast between actual weight and perceived size. In other words, the shape–weight illusion and the brightness–weight illusion are here conceived as a test bench for two models of weight perception. A detailed review of the two illusions is provided in Sections 1.1 and 1.2 below.

The shape–weight illusion

The first systematic study on the influence of the shape of objects on their perceived weight was conducted by Dresslar [26], who showed that shape had a systematic influence on perceived weight, and speculated that this was due to a size–weight illusion because the objects that were perceived to weigh more (e.g., the circle) were probably perceived to be smaller than the objects that appeared to weigh less (e.g., the triangle). Although Dresslar [26] did not measure the perceived size of the stimuli, if his hypothesis is correct, then the shape–weight illusion would simply be a by-product of the size–weight illusion, namely a size–weight illusion wherein differences in perceived size are due to differences in shape.

In a more recent study, Kahrimanovic, Bergmann Tiest, and Kappers [27] used 13 triplets of stimuli, each including three brass solids of identical physical mass and volume: a cube, a sphere, and a tetrahedron. The stimuli in each triplet differed from the stimuli in the other triplets both in physical mass and physical size. Blindfolded participants were asked to weigh one stimulus at a time by enclosing it with their hand and were allowed to perform up and down movement with the hand to perceive the weight. Results showed that the perceived weight of tetrahedrons was underestimated compared to that of cubes, whereas no difference in perceived weight emerged between tetrahedrons and spheres or between spheres and cubes. These results were replicated in Kahrimanovic, Bergmann Tiest, and Kappers’s [28] Experiment 1. In Experiment 2 of the latter study, in which the method of magnitude estimation was used instead of the method of paired comparisons, the results showed that tetrahedrons were perceived to be lighter than cubes and spheres. Again, no statistically significant difference emerged between cubes and spheres.

Using the same stimuli that they had used in their experiments on weight perception, Kahrimanovic, Bergmann Tiest, and Kappers [31] also measured the influence of shape on the haptically perceived size of objects. A direct comparison between the influence of shape on haptically perceived size and the influence of shape on haptically perceived weight showed that differences in perceived weight between tetrahedrons and cubes were actually larger than could be predicted from differences in perceived size [27,28]. Symmetrically, differences in perceived weight between tetrahedrons and spheres and between cubes and spheres were smaller than could be predicted from differences in perceived size. These results provide support for the hypothesis that the influence of shape on perceived weight is at least partially independent of the influence of shape on perceived size, which is inconsistent with Dresslar’s [26] hypothesis that the shape–weight illusion can be reduced to the size–weight illusion [27,28].

Because the expected weight of the stimuli was not measured in previous studies on the shape–weight illusion, it remains unclear whether the effect of shape on perceived weight could be explained in terms of the contrast between actual and expected weight [17,10,11]. The contribution of the current study to a deeper understanding of the shape–weight illusion can be summarized by two main points: 1) Whereas the studies conducted by Kahrimanovic, Bergmann Tiest, and Kappers [27,28] provide support to the existence of a haptic shape–weight illusion (i.e., participants enclosed the stimuli in the hand and could not see them), in the present study we test the existence of a visual shape–weight illusion, in that participants could see the stimuli but they could not touch them during the lifting action (i.e., participants were required to lift the stimuli by means of a string attached on the top). 2) By comparing the influence of shape on the perceived size, the expected weight, and the perceived weight of the stimuli, we could test whether the shape–weight illusion can be explained in terms of the contrast between actual and expected weight and/or in terms of the contrast between actual weight and perceived size, in line with the idea that perceived size may directly affect perceived weight independently of expected weight [19].

The brightness–weight illusion

In his pioneering study on the influence of the color of objects on their perceived weight, De Camp [29] observed a weak positive relationship between the brightness of the stimuli and their perceived weight. For instance, a white cube tended to be judged as heavier than a black cube, and an orange cube tended to be judged as heavier than a blue cube. However, De Camp [29] himself found a somewhat opposite pattern of results in an experiment similar to the one just described in which participants were presented with three cubes varying in achromatic surface colors (i.e., white, gray, and black). Based on these conflicting results, De Camp [29] concluded that the influence of brightness on perceived weight was weak and inconsistent across the sets of stimuli, the participants, and the response methods (see also [32]). In a more recent study, Walker, Fracis, and Walker [30] presented participants with a set of eight cue snooker balls of equal size and mass, differing from each other in hue and brightness. The results showed a positive relationship between brightness and perceived weight: for instance, the white ball had a probability of about 0.8 to be judged as heavier than the black ball.

Whereas evidence of a relationship between brightness and perceived weight are somewhat mixed, there appears to exist converging evidence supporting a negative relationship between brightness and expected weight. For instance, in De Camp’s [29] study, a clear negative relationship emerged between brightness and visually estimated weight, as the light-colored objects were expected to be light in weight and the dark-colored objects were expected to be heavy in weight (see also [30, 33, 34]). Recently, Walker, Scallon, and Francis [35] provided support for the hypothesis that the correspondence between brightness and weight is bidirectional; that is, not only brightness affects expected weight but perceived weight affects expected brightness. The authors asked participants to rate the expected brightness of unseen lifted objects, and the results showed that heavy-feeling objects were expected to be darker than light-feeling objects. Overall, the results of these studies suggest that the brightness–weight illusion may stem from the contrast between actual and expected weight (i.e., dark objects are expected to weigh more, and therefore they are perceived to weigh less than bright objects).

Interestingly, the brightness–weight illusion appears to be at least partially inconsistent with the hypothesis that perceived size can directly and automatically affect perceived weight independently of expected weight [19]. Indeed, previous studies show that the apparent size of objects tends to increase with their brightness (i.e., light objects are perceived to be larger than dark objects; see [36,37,38]), therefore if perceived size exerted a direct influence on perceived weight, then perceived weight should decrease–rather than increase–with brightness. In other words, the brightness–weight illusion appears to be more consistent with the predictions of the expectation model than with the hypothesis of a direct and automatic influence of perceived size on perceived weight. The contribution of the current study to a deeper understanding of the brightness–weight illusion can be summarized in two points: 1) To the best of our knowledge, only one study has so far provided convincing support for the existence of a brightness–weight illusion [30]. Here we test the replicability and generalizability of the brightness–weight illusion using a set of stimuli, a weighing mode, and a psychophysical method different from those originally employed by Walker, Francis, and Walker [30]. 2) By comparing the influence of brightness on the perceived size, the expected weight, and the perceived weight of the stimuli, we could test the extent to which the brightness–weight illusion can be explained in terms of the contrast between actual and expected weight and/or in terms of the contrast between actual weight and perceived size.

Experiment 1

Participants

Thirty-four right-handed participants (15 females, 19 males) took part in the experiment in exchange for course credit. They were undergraduate students at the University of Padua ranging in age from 21 to 43 years (M = 24.94, SD = 3.87). All had normal or corrected-to-normal vision and normal function of their upper limbs. They were naive as to the purposes of the experiment. All the participants took part in the expected weight and the perceived weight sessions (see below), and a subset of 20 participants (11 females, 9 males; mean age = 24.4, SD = 2.56) also took part in the perceived size session.

Stimuli and design

The stimuli were 18 plastic 3-D objects each weighing 220 g that were built according to a 3 (Size) × 3 (Shape) × 2 (Brightness) factorial design. All the stimuli were built using a Zmorph3D printer and could be opened and filled with a variable amount of cotton wool and plasticine, which allowed us to regulate their mass. When closed, all the stimuli had a uniform appearance (see Fig 1). The volume of the stimuli could be 216, 729, or 1728 cc. The shapes could be tetrahedron, cube, or sphere (detailed measures of the stimuli are reported in Table 1). The color of the stimuli could be white or black (in the room where the experiment took place, the average luminance measured using a Minolta LS-100 photometer was 3.82 cd/m² for the black stimuli and 49.85 cd/m² for the white stimuli). An orange 4580 cc, 440 g cylinder (height = diameter = 18 cm, luminance = 20.72 cd/m²) served as the standard stimulus (see Fig 1). A thin string (diameter = 0.3 cm) was looped through holes in the top of the experimental stimuli and of the standard stimulus (see Fig 1). Due to the technical features of the stimuli, the method of attaching the strings was different across the three shapes. For tetrahedrons, the string formed a loop passing through two holes close to the apex; for cubes, spheres, and the cylinder, the string was glued to the inside of a small hole in the top surface of the object, and the loop was formed making a knot to the external part of the string (see Fig 1). The overall height of the loop from the surface of the stimulus was about 5 cm, and all the stimuli could be lifted easily with the thumb and index fingers (see Fig 1, left panel). The lifting action did not vary with the method of attaching strings to the stimuli, therefore the latter variable should have a negligible influence on the forces with which the stimuli were lifted.

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Fig 1.

Left panel: a picture of the stimuli used in Experiment 1. Right panel: a depiction of the lifting action.

https://doi.org/10.1371/journal.pone.0220149.g001

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Table 1. Detailed measures of the stimuli.

https://doi.org/10.1371/journal.pone.0220149.t001

There were three experimental sessions, which we call the perceived size session, the expected weight session, and the perceived weight session. The perceived size and expected weight sessions lasted about 20–25 minutes, whereas the perceived weight session lasted about 25–30 minutes. The sessions took place at least two days apart. Half of the 20 participants who took part in all three sessions first completed the perceived size session, then the expected weight session, and then the perceived weight session; the order of the first two sessions was inverted for the other half of these 20 participants. All the participants who took part in only two experimental session first completed the expected weight session and then the perceived weight session.

In each session, the participants were presented with four repetitions of the 18 experimental stimuli and of the standard stimulus. The responses in the first repetition were considered practice and therefore were not analyzed. Each repetition started with the presentation of the standard stimulus followed by the randomized presentation of the 18 experimental stimuli. As detailed below, in each of the three sessions, a numerical rating response method with fixed anchors was used. Specifically, participants were asked to rate the perceived size, the expected weight, or the perceived weight of each experimental stimulus relative to the extremes of the rating scale, which were 0 (i.e., the absence of size or weight) and 100 (i.e., the size or the weight of the standard). Numerical rating tasks with fixed anchors provided measures on an interval scale of the response variable [39,40], and had been previously employed to measure the magnitude of the size–weight illusion [13,41].

Procedure

In each experimental session, the participants were seated at a square table (height 85 cm) with their elbows leaning on it. Before starting the first experimental session, they read and signed an informed consent form approved by the local ethics committee (Department of General Psychology, University of Padua). All procedures performed were in accordance with the ethical standards of the 1964 Helsinki declaration and its later amendments. The experimenter sat in front of the participant. A cardboard screen was used to hide from participant the positioning and removal of the stimuli, which were manually performed by the experimenter. The stimuli were placed on a thin layer of soft rubber to muffle the noise of the contact of the stimuli with the hard surface of the table. To warrant the equilibrium of the sphere, a small hollow was created in the rubber layer exactly in where the stimuli were placed by the experimenter. Once the stimulus was set into place by the experimenter, the occluding screen was removed and the participant was allowed to evaluate the perceived size, the expected weight, or the perceived weight of the stimulus, as detailed below. The stimuli were placed at a distance of about 30 cm from the side of the table where the participant was seated. After the participant had responded, the experimenter reset the occluding screen in its position, removed the previous stimulus from the table, and placed a new stimulus on the table.

At the beginning of the perceived size session, written instructions informed the participants that they would be presented with an orange cylinder, the size of which was 100. They were then informed that, after the presentation of the cylinder, stimuli varying in volume, shape, and color would have been presented, and that their task was to evaluate the size of each stimulus using a number between 0 and 100, where 0 would be the size of an invisible object, and 100 was the size of the cylinder. The instructions further specified that neither the cylinder nor the other stimuli could be touched at any moment during the experiment. In the expected weight session, the instructions were the same as those in the perceived size session, except that participants were informed that their task was to visually estimate the weight of each stimulus using a number between 0 and 100, where 0 was the absence of weight and 100 was the apparent weight of the cylinder. In the perceived weight session, participants were instructed to lift the stimuli (including the standard) by grasping the string on their tops using the thumb and index fingers of their right hands. The participants were further instructed to keep their elbows on the table during the lifting action, to avoid oscillations of the stimuli as much as possible, and to respond only after the lifting action had been completed. This procedure should prevent or minimize access to information about the rotational inertia of the stimuli. Participants were then informed that their task was to evaluate the weight of each object using a number between 0 and 100, where 0 was the absence of weight and 100 was the weight of the cylinder.

Results and discussion

Fig 2 shows the mean perceived size (top panels), the mean expected weight (middle panels), and the mean perceived weight (bottom panels) of the stimuli, averaged across participants and repetitions as a function of volume and shape (left panels) and as a function of volume and brightness (right panels). As detailed below, perceived size, expected weight, and perceived weight were analyzed using separate ANOVAs.

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Fig 2. The results of Experiment 1.

Top, middle, and bottom panels represent mean perceived size, mean expected weight, and mean perceived weight, respectively. In the left panels, the means of the response variable are represented as a function of the three levels of volume and as a function of the three shapes. The letters C, S, and T stand for cube, sphere, and tetrahedron, respectively. In the right panels, the means are represented as a function of the three levels of volume and as a function of the two levels of brightness. The letters W and B stand for white and black, respectively. The error bars represent the standard error of the mean. The * symbol indicates a statistically significant comparison (p < .05; Bonferroni correction method was applied).

https://doi.org/10.1371/journal.pone.0220149.g002

Perceived size.

We performed a three-way within-participants ANOVA in perceived size with the factors volume, shape, and brightness. There were main effects of all three factors (volume: F(2,38) = 416.43, p < .001, η_G² = 0.92; shape: F(2,38) = 37.99, p < .001, η_G² = 0.21; brightness: F(1,19) = 6.51, p = .019, η_G² = 0.007). The volume × shape and volume × brightness interactions were also statistically significant (F(4,76) = 14.76, p < .001, η_G² = 0.14; F(2,38) = 4.54, = .017, η_G² = 0.01 respectively). The shape × brightness interaction and the three-way interaction were not statistically significant (F(2,38) = 1.03, p > .10, η_G² = 0.001; F(4,76) = 0.67, p > .10, η_G² = 0.002 respectively).

Effects of volume. Post hoc paired comparisons tests with Bonferroni correction method showed that large stimuli were perceived to be significantly larger than medium and small stimuli (t(19) = 17.08, p < .001, d_z = 3.82; t(19) = 24.77, p < .001, d_z = 5.54 respectively), and medium stimuli were perceived to be significantly larger than small stimuli (t(19) = 14.39, p < .001, d_z = 3.22).

Effects of shape. We further explored the main effects of shape on perceived size using corrected post hoc paired comparisons tests. The results showed that tetrahedrons were perceived to be larger than spheres (t(19) = 6.26, p < .001, d_z = 1.4) and cubes (t(19) = 6.64, p < .001, d_z = 1.48) and that spheres tended to be perceived as slightly larger than cubes, although the difference was not statistically significant (t(19) = 2.49, p = .066, d_z = 0.56). Regarding the volume × shape interaction, the top left panel in Fig 2 clearly shows that the effect of shape on perceived size tended to increase with physical size. Moreover, corrected post hoc paired comparisons showed that tetrahedrons were perceived to be significantly larger than spheres for the small and the large set (small set: t(19) = 5.0, p < .001, d_z = 1.12; medium set: t(19) = 2.75, p > .10, d_z = 0.61; large set: t(19) = 5.38, p < .001, d_z = 1.20), that tetrahedrons were perceived to be significantly larger than cubes at all the levels of volume (small set: t(19) = 3.74, p = .012, d_z = 0.84; medium set: t(19) = 3.7, p = .014, d_z = 0.83; large set: t(19) = 5.89, p < .001, d_z = 1.32), and that no statistically significant differences emerged between spheres and cubes at any of the levels of volume (small set: t(19) = 0.73, p > .10, d_z = 0.16; medium set: t(19) = 2.0, p = > .10, d_z = 0.45; large set: t(19) = 1.43, p > .10, d_z = 0.32).

Effects of brightness. The ANOVA shows a main effect of the brightness factor. This is due to the fact that, on average, white stimuli were perceived to be larger than black stimuli. However, the significant volume × brightness interaction suggests that the effects of brightness on perceived size were modulated by physical volume. Corrected post-hoc paired comparisons showed that white stimuli were perceived to be significantly larger than black stimuli only for the large set (small set: t(19) = 0.81, p > .10, d_z = 0.18; medium set: t(19) = 0.28, p > .10, d_z = 0.06; large set: t(19) = 2.59, p = .049, d_z = 0.58). Overall, the results of the perceived size session confirm the existence of a positive relationship between brightness and perceived size, which had been previously reported for two-dimensional stimuli [36,37,38]. The results also suggest that the influence of brightness on perceived size tended to increase with volume.

Expected weight.

We performed a three-way within-participants ANOVA in expected weight with the factors volume, shape, and brightness. There were main effects of volume and shape (F(2,66) = 267.09, p < .001, η_G² = 0.83; F(2,66) = 64.7, p < .001, η_G² = 0.13, respectively), but not of brightness (F(1,33) = 1.75, p > .10, η_G² = 0.004). The volume × shape and shape × brightness interactions were statistically significant (F(4,132) = 17.19, p < .001, η_G² = 0.06; F(2,66) = 4.06, p = .022, η_G² = 0.003, respectively). The volume × brightness interaction and the three-way interaction were not statistically significant (F(2,66) = 1.26, p > .10, η_G² = 0.002; F(4,132) = 0.92, p > .10, η_G² = 0.001, respectively).

Effects of volume. Post hoc paired comparisons tests with Bonferroni correction method showed that large stimuli were expected to be significantly heavier than medium and small stimuli (t(33) = 14.92, p < .001, d_z = 2.56; t(33) = 17.09, p < .001, d_z = 2.93 respectively), and medium stimuli were expected to be significantly heavier than small stimuli (t(33) = 15.65, p < .001, d_z = 2.68).

Effects of shape. We further explored the main effects of shape using corrected post hoc paired comparisons tests. The results showed that tetrahedrons were expected to be heavier than spheres (t(33) = 9.48, p < .001, d_z = 1.62) and cubes (t(33) = 8.65, p < .001, d_z = 1.48), and that spheres were generally expected to be slightly heavier than cubes, although the difference was not statistically significant (t(33) = 2.49, p = .053, d_z = 0.43). Regarding the volume × shape interaction, the middle left panel in Fig 2 clearly shows that the effect of shape on expected weight tended to increase with physical volume. Moreover, corrected post hoc paired comparisons showed that, at all levels of volume, tetrahedrons were expected to be significantly heavier than spheres (small set: t(33) = 4.63, p < .001, d_z = 0.79; medium set: t(33) = 5.12, p < .001, d_z = 0.88; large set: t(33) = 7.87, p < .001, d_z = 1.35), and cubes (small set: t(33) = 4.31, p = .00014, d_z = 0.74; medium set: t(33) = 5.65, p < .001, d_z = 0.97; large set: t(33) = 7.03, p < .001, d_z = 1.21). No statistically significant differences emerged between spheres and cubes at any of the levels of volume (small set: t(33) = -0.29, p > .10, d_z = -0.05; medium set: t(33) = 1.0, p > .10, d_z = 0.17; large set: t(33) = 2.40, p > .10, d_z = 0.41).

The comparison between the top and middle left panels in Fig 2 shows that shape had very similar effects on perceived size and expected weight, which may suggest that the influence of shape on expected weight was largely or totally mediated by perceived size. To further explore this hypothesis, we performed a hierarchical linear regression analysis on expected weight using mixed-effects models (R package lme4 [42]). Specifically, we fit six nested linear mixed-effects models of increasing complexity to the data of the expected weight sessions, limited to the data of the 20 participants who also took part in the perceived size session. Each model included all the features of the models preceding it in the hierarchy of complexity plus some additional feature (see Table 2). For each model, Table 2 shows two indexes of goodness of fit, namely the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), as well as the value of the chi-square statistic for the log-likelihood ratio test comparing that model with the model preceding it in the hierarchy of complexity (together with the associated p-value). If p < .05, then the more complex model (e.g., Model 3) fits the data significantly better than the simpler model (Model 2). The results show that the model including perceived size and shape as fixed effects (i.e., Model 4) did not fit the data significantly better than the model including perceived size alone as a fixed effect (i.e., Model 3). This suggests that when the effects of perceived size on expected weight were taken into account, shape did not affect expected weight directly. In other words, the influence of shape on expected weight was fully mediated by the influence of shape on perceived size.

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Table 2. Results of the hierarchical regression analysis on expected weight (mixed models).

Bold typeface indicates the model with the best fit to the data according to the log-likelihood ratio test.

https://doi.org/10.1371/journal.pone.0220149.t002

Effects of brightness. Brightness did not appear to have a consistent effect on the expected weight of the stimuli (see the middle right panel in Fig 2). Besides the lack of main effect of the brightness factor, a Bayesian paired-sample t-test provided support for the null hypothesis of equivalence between black and white stimuli in expected weight, since the JZS BF₀₁ = 2.45 ± 0%. Although the volume × brightness interaction was not statistically significant, for the sake of comparison with the results of the perceived size session, we further explored the interaction using paired comparison tests with Bonferroni correction. These comparisons showed that, for the small set, black stimuli were expected to be significantly heavier than white stimuli (t(33) = 2.62, p = .04, d_z = 0.45). No statistically significant difference emerged between black and white stimuli for the medium and large sets (t(33) = 0.09, p > .10, d_z = 0.02; t(33) = 1.04, p > .10, d_z = 0.18, respectively). Although the shape × brightness interaction was statistically significant, post-hoc paired comparisons tests with Bonferroni correction failed to reveal any statistically significant difference between black and white stimuli at any level of shape (tetrahedrons: M_black = 34.26, M_white = 32.78, t(33) = 2.10, p > .10, d_z = 0.36; spheres: M_black = 29.19, M_white = 28.12, t(33) = 1.47, p > .10, d_z = 0.25; cubes: M_black = 27.45, M_white = 27.65, t(33) = -0.33, p > .10, d_z = -0.06).

The lack of an overall effect of brightness on expected weight is inconsistent with the results of previous studies [29,30,33,34]. A plausible explanation for this finding is that, in the current study, the negative relationship between brightness and expected weight was overridden by the positive relationship between brightness and perceived size (i.e., white stimuli were perceived to be larger, so they were expected to be slightly heavier than black stimuli). Support for this hypothesis is provided by the fact that black stimuli were expected to be significantly heavier than white stimuli only for the small set (see the middle-right panel in Fig 2), namely the level of volume at which the influence of brightness on perceived size was at its minimum (see the top-right panel in Fig 2). We can speculate that in Experiment 1, the variations in perceived volume were made particularly salient by the manipulation of volume, which might explain why the effects of shape and brightness on expected weight were strongly mediated by perceived size. An indirect test of this hypothesis will be provided in Experiment 2.

Perceived weight.

We performed a three-way within-participants ANOVA in perceived weight with factors volume, shape, and brightness. Main effects existed for the factors volume and shape (F(2,66) = 101.67, p < .001, η_G² = 0.65; F(2,66) = 19.6, p < .001, η_G² = 0.07, respectively) but not for brightness (F(1,33) = 0.5, p > .10, η_G² < 0.001). The volume × brightness and shape × brightness interactions were statistically significant (F(2,66) = 10.79, p < .001, η_G² = 0.012; F(2,66) = 8.54, p < .001, η_G² = 0.008, respectively). The volume × shape interaction was not statistically significant (F(4,132) = 1.69, p > .10, η_G² = 0.005), whereas the three-way interaction was statistically significant (F(4,132) = 5.81, p < .001, η_G² = 0.013).

Effects of volume. Consistent with the size–weight illusion, there was a strong negative relationship between volume and perceived weight. Corrected post hoc paired comparisons showed that small stimuli were perceived to be significantly heavier than medium and large stimuli (t(33) = 10.1, p < .001, d_z = 1.73; t(33) = 10.39, p < .001, d_z = 1.78 respectively), and medium stimuli were perceived to be significantly heavier than large stimuli (t(33) = 8.24, p < .001, d_z = 1.41).

Effects of shape. Corrected post hoc paired comparisons showed that spheres were perceived to be heavier than tetrahedrons (t(33) = 5.28, p < .001, d_z = 0.9) and cubes (t(33) = 3.6, p = .0031, d_z = 0.62) and that cubes were perceived to be heavier than tetrahedrons (t(33) = 3.39, p = .0054, d_z = 0.58). For each level of the volume factor, the results of the statistically significant paired comparison tests between shapes (with Bonferroni correction) are shown in the bottom-left panel in Fig 2. The observed shape–weight illusion was not fully consistent with the predictions from the expectation model. Indeed, if the shape–weight illusion resulted from the contrast between actual and expected weight, then because the spheres were generally expected to be slightly heavier than cubes, they should have been perceived to be slightly lighter–rather than heavier–than cubes. Moreover, because the differences between the three shapes in expected weight tended to increase with volume, the influence of shape on perceived weight should also have increased with volume. The lack of a statistically significant volume × shape interaction suggests that this was not the case. We fit four nested linear mixed-effects models of increasing complexity to the data from the perceived weight session for all 34 participants (see Table 3). Models with complexity higher than that of Model 4 failed to converge; thus, they were removed from the analysis. The results show that the influence of shape on perceived weight was at least partially independent of the influence of shape on expected weight, since shape had a statistically significant effect on perceived weight, even when the effects of expected weight on perceived weight were taken into account.

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Table 3. Results of the hierarchical regression analysis on perceived weight (mixed models).

Bold typeface indicates the model with the best fit to the data, according to the log-likelihood ratio test.

https://doi.org/10.1371/journal.pone.0220149.t003

The results are also somewhat inconsistent with the hypothesis that perceived size would exert a direct and automatic influence on perceived weight [19]. Indeed, if perceived weight depended on the contrast between actual weight and perceived size, then because spheres were perceived to be slightly larger than cubes, spheres should have been perceived as being slightly lighter–rather than heavier–than cubes. Additionally, the influence of shape on perceived weight should have increased with volume, but this was not the case. We fit four nested linear mixed-effects models of increasing complexity to the data from the perceived weight session, limited to the data of the 20 participants who also took part in the perceived size session (see Table 4). Models with complexity higher than that of Model 4 failed to converge; thus, they were removed from the analysis. The results show that the influence of shape on perceived weight was at least partially independent of the influence of shape on perceived size, since shape had a statistically significant effect on perceived weight, even when the effects of perceived size on perceived weight were taken into account.

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Table 4. Results of the hierarchical regression analysis on perceived weight (mixed models).

Bold typeface indicates the model with the best fit to the data, according to the log-likelihood ratio test.

https://doi.org/10.1371/journal.pone.0220149.t004

Effects of brightness. As for the effects of brightness on perceived weight, the main effect of the brightness factor was not statistically significant, and a Bayesian paired-sample t-test provided support for the null hypothesis of equivalence between black and white stimuli in perceived weight, since JZS BF₀₁ = 4.35 ± 0%. However, the bottom-right panel in Fig 2 reveals a surprising pattern of results, which can explain the statistical significance of the volume × brightness interaction. Specifically, paired-comparison tests with Bonferroni correction showed that black stimuli were perceived to be significantly heavier than white stimuli for the small set (t(33) = 3.73, p = .0021, d_z = 0.64), whereas white stimuli were perceived to be significantly heavier than black stimuli for the large set (t(33) = 2.82, p = .024, d_z = 0.48). No statistically significant difference in perceived weight emerged between white and black stimuli for the medium set (t(33) = 0.1, p p > .10, d_z = 0.02). In other words, a regular brightness-weight illusion emerged for large stimuli, whereas an inverted illusion emerged for small stimuli. Importantly, both the regular and the inverted brightness-weight illusions appear to be inconsistent with the predictions from the expectation model. Specifically, the results of the expected weight session showed that, for the small set, black stimuli were expected to be heavier than white stimuli; therefore, a regular rather than an inverted brightness–weight illusion should have appeared. Moreover, for the large set, a regular brightness–weight illusion emerged, despite the lack of difference in expected weight between black and white stimuli. The observed influence of brightness on perceived weight was also inconsistent with the idea that perceived size exerted a direct influence on perceived weight. Indeed, the results of the perceived size session showed that, for the large set, white stimuli were perceived to be larger than black stimuli; hence, an inverted–rather than a regular–brightness–weight illusion should have appeared. Moreover, no influence of brightness on perceived weight should have emerged for the small set, since there was no difference in perceived size between black and white stimuli at that volume level.

Effects of brightness: Individual differences.

The main effects of brightness on perceived size, expected weight, and perceived weight were small or null. In principle, this can be due to the fact that these effects were small or null at the individual level, or to the fact that some of the participants showed a strong effect in one direction (e.g., a strong, regular brightness–weight illusion) and others showed a strong effect in the opposite direction (e.g., a strong, inverted brightness–weight illusion). In order to clarify this point, Fig 3 shows, for each participant, the mean for the white stimuli as a function of the mean of the black stimuli, relatively to perceived size (top panel), expected weight (middle panel), and perceived weight (bottom panel). The means are averaged across shape, and different colors are used to represent volume. Each point represents a participant, and a point lying below the diagonal line indicates that the mean of the dependent variable was larger for black stimuli than for white stimuli. Visual inspection of Fig 3 shows that, with a few exceptions, the points tend to cluster along the diagonal line. This suggests that, for all the three dependent variables, the effects of brightness were small and consistent across participants.

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Fig 3. A representation of the effects of brightness on the three dependent variable at the individual level.

The top, middle, and bottom panel refer to perceived size, expected weight, and perceived weight, respectively.

https://doi.org/10.1371/journal.pone.0220149.g003

Experiment 2

To summarize, neither the expectation model nor the hypothesis that perceived size would exert a direct influence on perceived weight can fully account for the results of Experiment 1. In Experiment 2, we sought to test the generalizability of Experiment 1’s outcomes using a different method. Specifically, we used random conjoint measurement (RCM), a method based on a paired comparison task in which participants have to provide a binary response [43,44,45] (for the application of a variant of the RCM paradigm in the context of weight illusions, see [25]). In order to keep the length of the experiment within reasonable limits, we used a subset of the stimuli that we had used in Experiment 1, namely the stimuli of the medium set (this choice was arbitrary).