How we compare areas: The underlying mechanism of the elongation bias

Abstract

Across four experiments, we investigate the mechanism that underlies the elongation bias. We find individuals tasked with assessing the area of two objects do so by comparing the objects’ dimensions, and thus subtle changes in the objects’ dimensions can impact area assessments. Because a typical elongation bias experiment places two objects side-by-side horizontally and varies the elongation ratio while maintaining the same area, height is generally easier to compare than width. Thus, there will exist a region where the change in height noticeably crosses a perceptual just noticeable difference boundary, but the corresponding change in width does not, and individuals will tend to perceive that the taller object has a greater area or volume. Consistent with this proposed process, we suggest that, although the elongation bias occurs under a comparative judgment, it does not do so under a single judgment situation. This research contributes to our wider understanding of the visual processes underlying area comparisons.

Introduction

Visual illusions and biases have long been of theoretical interest because these failures reveal the machinery of visual information processing. For instance, researchers have investigated how stable individual differences and other factors influence several visual illusions (Cretenoud, Francis, & Herzog, 2020; Cretenoud, Grzeczkowski, Kunchulia, & Herzog, 2021). Visual biases are also of practical interest to marketers, who may use them to make their products more appealing to customers. For example, by changing the shape of a juice bottle, a marketer might increase consumers’ estimates of the amount of juice in the bottle without actually increasing the volume. More generally, consumer researchers have investigated how visual biases change consumer decision-making (Adams, Anthony Di Benedetto, & Chandvan, 1991; Britt, 1975; Raghubir & Krishna, 1999). 

This article investigates why a specific visual illusion—the elongation bias—occurs. The elongation bias usually emerges when one places two quadrilaterals (usually rectangles) next to each other. The elongation ratios of these are subtly different such that one of them is a little taller and thinner than the other. However, both have the same area. A recurrent theme is that the taller, thinner rectangle is often judged to have greater area resulting in the aptly named elongation bias. 

Prior research on the elongation bias

Piaget (1968) first described the elongation bias as a centration bias among young children. When researchers moved water from a tall, thin cylinder to a short, wide cylinder, the children perceived less water in the short cylinder than in the tall cylinder, despite both containers having the same volume. Piaget and others suggested that this perceptual bias came from observers focusing on the most salient dimension, or the “height” of the containers. Several researchers have continued to investigate the effects of what is known now as the elongation bias on the estimated sizes of different objects (Anderson & Cuneo, 1978; Raghubir & Krishna, 1999; Verge & Bogartz, 1978; Yang & Raghubir, 2005), including the perceived volume of a room (Saulton, Mohler, Bulthoff, & Dodds, 2016). Theoretical explanations for this effect included inadequate information coordination (Teghtsoonian, 1965; Verge & Bogartz, 1978) or integration (Anderson & Cuneo, 1978). Krider, Raghubir, and Krishna (2001) refined existing explanations with a psychophysical model of how consumers compare areas, postulating that when individuals are tasked with comparing areas, they over-rely on the more salient of two dimensions in two-dimensional judgments. Although they may adjust their initial judgement based on a second comparison using a secondary dimension, this adjustment is insufficient to reach the normatively correct size judgment. 

Current research

We suggest that, when observers attempt to estimate the area of a target object, errors can arise at two stages: during perception as one tries to assess the cardinal dimensions of target objects as well as any context effects that occur as one compares the target object with other presented referents. 

Perceptual processes involved in area estimation

A number of recent studies provide compelling evidence for the theory that perceptual area estimation in human subjects relies on the addition of an object's two dimensions along cardinal axes (e.g., the height and width of an object), which is often referred to as the additive area heuristic (Yousif, Aslin, & Keil, 2020; Yousif & Keil, 2019). For instance, when individuals are presented with both squares and diamonds (the diamond being identical to the square but rotated by 45°), the diamond is perceived as having a larger area because the observers use the diagonals to assess the diamond's height and width, while the observers use the sides of the square to assess the square's height and width (Yousif et al., 2020). This additive heuristic has been demonstrated among children (Yousif, Alexandrov, Bennette, Aslin, & Keil, 2022) and adult individuals tasked with assessing the volume of three-dimensional objects (Bennette, Keil, & Yousif, 2021). 

An alternative hypothesis is that individuals may rely on contour tracing or the perimeter of the object to assess area. For instance, Clearfield and Mix (1999) found that, when it comes to distinguishing small visual sets, infants tend to rely on the contours of objects instead of the number of objects present. In addition, work by Hubel and Wiesel (1962, 1968) discovered that receptive fields of neurons in the primary visual cortex of mammals are tuned to detect oriented structure and suggested that the systematic orientation had evolved to support contour analysis. Given the importance of contour tracing in visual information processing, we developed the perimeter hypothesis to suggest that individuals may estimate size by assessing how long it takes to mentally trace the contour (or outline) of an object (i.e., the perimeter). For example, a square has the minimum perimeter of a right-angled quadrilateral with a given area (see the Supplemental Material for proof). As the perimeter of a rectangle with the same area is longer than that of the square, it will require more time to mentally trace the contours. For this reason, individuals may perceive a more elongated object to be larger than a less elongated object simply because it takes longer to trace its contours. 

It should be noted that the perimeter hypothesis generates precisely the same predictions as the additive area heuristic, albeit by exploring an alternative process account (namely, that individuals do not assess the salient dimensions but mentally trace the contour of the object). Because the same area is enclosed by shapes that move from squareness toward rectangularity, the additive area (which is essentially one-half of the perimeter in the context of the elongation bias, which relies on nonrotated quadrilaterals) will inexorably increase. For instance, a 100 × 1 square has an additive area of 101, whereas a 10 × 10 square has an additive area of 20. Thus, observers will always assess a more rectangular shape as enclosing a larger area than a more square shape with constant orientation and no rotation involved. 

It is worth noting, as well, that neither the additive area heuristic nor the perimeter hypothesis involves a direct comparison between two objects, focusing instead on the process of perceiving each shape individually. Because the elongation bias seems to necessitate a comparison, we propose that context effects be considered concurrently with perceptual processes. 

Context effects in area estimation

The act of comparing two objects always invites the possibility of context effects. Whether an object serves as a context for assessing another object, however, depends on a variety of features, the most salient of which is the degree of similarity between the two objects. For instance, objects in the same category are more likely to invite comparisons than objects in different categories (Brown, 1953; Coren & Enns, 1993). Thus, if provided with the areas of a standard square and a standard circle, the standard square is more likely to be used to assess the area of a target square and the standard circle is more likely to be used to assess the area of a target circle. A second plausible criterion is the ease of comparison, where the proximity of objects makes them easier to compare, increasing the likelihood that they will serve as contextual objects. For instance, when mentally rotating and comparing objects, the accuracy of comparisons depends on the alignment of objects and their environmental frame (Pani & Dupree, 1994; Parsons, 1995), with individuals capable of rotating just one part of the object (Xu & Franconeri, 2015). Therefore, if there are no mental manipulations to perform (e.g., the rotation of a diamond to the figure of a square), then they are easier to compare and more likely to serve as contextual objects. 

In sum, we suggest that the process generating the elongation bias may be more complex than suggested before. We argue that there are two processes at work. At the perceptual level, the perimeter hypothesis and the additive area hypothesis both generate the prediction that elongated rectangles will create the perception of greater area. In addition, the presence of a comparison square has the potential to generate context effects. We expand on these ideas in greater detail in this article. 

The elongation bias

Typical examples of experiments exploring the additive area hypothesis usually make objects dissimilar. For example, researchers may increase dissimilarity by asking respondents to compare squares with diamonds. Other manipulations include separating the objects by some degree of space or by enclosing the objects in separate figural boundaries to create a perception of different categories (e.g., enclosing a set of circles within a box that is separated from another box that itself contains another set of circles) (Yousif et al., 2020; Yousif & Keil, 2019). 

In contrast, the typical elongation bias study places two objects that look extremely similar next to each other, thus allowing the objects to serve as referents for each other (Folkes & Matta, 2004; Krishna, 2006; Raghubir & Krishna, 1999). This procedure, which facilitates comparing the two objects, may influence the viewers’ estimates of the length of the two dimensions that are used to produce the additive area estimate. In more specific terms, we hypothesize that the placement of the two stimuli notably affects the just noticeable difference (JND) of cardinal dimensions, which in turn influences the individuals’ area assessment. For example, when rectangles are placed next to each other, the height dimension is generally easier to compare than the width dimension (the latter of which requires the mental manipulation of either rotating one of the widths by 180° or moving one of the widths over by some significant amount). For this reason, we predict that individuals tasked with comparing the area of two quadrilaterals placed side by side will find it easier to perceive minor changes in height as opposed to width. This comparison should reveal itself in the form of different JNDs across the two dimensions of height and width, such that the height has a lower JND threshold than the width. 

We, therefore, suggest that the format of a typical elongation bias stimulus results in a comparative process being overlaid on the principles of additive area; the cardinal dimensions suggested by the additive area hypothesis are still the cues that are perceptually sought—however, while assessing these cardinal dimensions, the comparative process results in a biasing effect on the assessments such that small differences in height (easy to compare) are noticed to a greater degree than small differences in width, resulting in the illusion of one rectangle appearing to have the same width but greater height. This factor creates the elongation bias of a slightly taller rectangle that contains more area, because the height (which is easily comparable and has a smaller JND) is easily perceived as larger, whereas the width (which is more difficult to compare) appears to be the same. 

Note that this perspective generates a different prediction from the salient dimension (Krider et al., 2001), additive area (Yousif et al., 2020; Yousif & Keil, 2019) and perimeter hypotheses. These processes predict a linear increase in the size of the elongation bias as the rectangles get taller and thinner because the perimeter (the additive area and the most salient, or longest, dimension) of a rectangle continues to increase as it moves further away from a square of equivalent area. However, the presence of context effects suggests that the elongation bias occurs when individuals perceive the illusion of a rectangle of the same width as the square but taller in height; this perception is strongest when the height dimension has crossed a JND, but the width has not yet done so. Once the width crosses a JND point, however, the illusion shatters and the individual becomes aware of the fact that while the height has increased, the width has decreased. Assessments are thus likely to become more regressive, with even the potential for overcorrection in extremely tall and skinny rectangles; at the extreme, the rectangles can look more like thick lines rather than objects enclosing space. In other words, this perspective would generate a prediction of a quadratic function to the elongation bias such that it will increase initially, but then decrease as both cardinal dimensions cross their respective JND thresholds. 

The contrast effects outlined here seem to have the potential to inform other visual phenomena as well. For example, consider the folded paper illusion (Carbon, 2016), which occurs when individuals compare the area of an A4 sheet of paper and a sheet of paper folded in half—specifically, individuals underestimate the area of a full-sized (A4) paper when attempting to center a folded sheet of A4 paper upon it. The degree of underestimation depends on how the folded sheet aligns with the full-sized paper. For instance, when both the folded sheet and the full-sized sheet align on three sides, the accuracy of the area estimation increases, with the individual perceiving that the area of the full-sized sheet of paper is double the length of the folded paper. 

When three sides align, only the difference on one cardinal dimension needs to be evaluated. If only two sides align, both cardinal dimensions have to be compared and assessed. If the difference on any dimension is split across two sides (e.g., by aligning one edge of the smaller paper in the middle of the larger one), there are separate assessments that have to be combined. Any assessment of differences in the cardinal dimensions is likely to involve a comparison of the dimension of one object with that of the other, which has the potential to create context effects. In other words, the assessment of the extra length of the larger piece of paper is compared with the larger length on the folded piece of paper and creates a contrast effect resulting in its looking smaller than it actually is in a manner similar to the Ebbinghaus illusion. 

The data

Experiment 1 provides evidence showing that an individual's area estimations cannot be fully explained by a judgment process that relies solely on additive area or perimeter assessments. Specifically, we see a quadratic relationship that is at odds with the monotonic predictions of these processes. Experiment 2 assesses the idea of JNDs that differ across the height and width dimensions, the data of which allow us to calculate a region where the elongation bias is likely to be the greatest. Experiments 3 and 4 provide evidence for the proposition that because these automatic processes occur beyond conscious scrutiny (and because an unconscious influence, by definition, is difficult to correct consciously), their ability to bias judgments can be great. 

Experiment 1: Perimeter hypothesis test

By elongating a square by various degrees to a rectangle while maintaining the same total surface area, we aim to observe how individuals’ perception of size changes as a function of the elongation (i.e., aspect ratio). 

Methods

Participants and design

The study had a 1-way design with 10 levels of elongation as a within-participant factor, with each condition manipulating the elongation difference between the target and reference stimuli. The typical elongation bias experiment has sample sizes that result in 30 to 40 participants in each cell (Folkes & Matta, 2004; Krishna, 2006). Thus, to ensure our experiment had sufficient power, we set out to recruit 150 participants and succeeded in recruiting 142 undergraduate students (35.2% female; M_age = 20.97 years) for partial course credit. Because 1 participant was excluded for failing to complete all the questions, the total number of participants was 141. There were 10 observations per participant. 

Among the 1,410 observations, 23 were excluded because their area estimates were 3 standard deviations (SD) above or below the mean. The mean of the area estimate was 142.86 dks (deckas squared, a hypothetical unit of measure created for this work), and the SD was 13.69 dks. Therefore, the total number of observations for our analysis was 1,387. The resulting sample afforded us a power of 1 (f² = 0.04; p = 0.05). 

Stimuli

We used the stimuli exhibited in Figure 1. Table 1 shows the elongation ratio (ratio between the elongation of the target and the reference object) for each set of objects. We calculated the aspect ratio (height divided by width, or the longer cardinal axis divided by the shorter cardinal axis) of each shape (A and B) and then calculated the ratio of those two ratios (elongation ratio of B/elongation ratio of A). The participants were given four foil questions as a practice trial before they were given the 10 study questions. The stimuli for the foil questions were either smaller or larger than the reference stimuli, though the stimuli for the study questions had the same area (144 × 144 pixels) across each of the pairs (reference [A] vs. target [B]). Although all the pairs were presented side by side, the elongation ratios of the two objects differed across the 10 sets of stimuli. 

Figure 1.

 

Ten stimuli with the same area (144 × 144 pixels) in Experiment 1. The number below each set indicates the elongation order that is provided in Table 1.

Table 1.

 

The ratio of the elongation ratios of A (reference) and B (target), and area perception results at each ratio of the elongation ratios of A and B (elongation ratio of B/elongation ratio of A).

 

Procedures

All procedures in this article were approved by our university's institutional review board and all participants provided us with informed consent. All 10 stimuli (Figure 1) were separately presented to each participant in a random order with a questionnaire instructing them to estimate the surface area of the target stimulus (B, on the right) by comparing it to the reference stimulus (A, on the left) by directly recording their numbers. We report the results of the direct recording experiment here; please refer to the Supplemental Material for the slider scale experiment in which the participants answered their area estimation by using a slider scale, the results of which were similar to those of the experiment described here. In the current study, we asked the participants to record their estimate of object B's (the target) area in a text box in a hypothetical unit of measurement deckas squared (dks). Object A (the referent) was labelled with one of the five possible areas in dks to ensure that the participants did not become suspicious throughout their repeated trials. If the participants overestimated the area of B (target) compared with the area of A (reference), even though both had the same areas, we consider that the elongation bias occurred. When setting these paired comparisons, we followed previous studies (Folkes & Matta, 2004; Krishna, 2006). Furthermore, we used the hypothetical unit of measurement deckas (dks) to prevent the participants from calculating or estimating actual areas using a typical metric. Familiar units may have constrained the participants’ willingness to report their actual perception of nuanced differences in size (Perfecto, Donnelly, & Critcher, 2019). We concluded the survey with demographic questions. 

Results

Error as a function of the ratio of the elongation ratios of A and B

We used a repeated measure generalized estimated equation to analyze the data. We regressed the area estimate error of B, %: (estimated area of B—actual area of B)/actual area of B × 100) on the ratio of the elongation ratios of A and B, ER_b/ER_a: elongation ratio of B/elongation ratio of A (Table 1). Note that, as a shape becomes more elongated (i.e., more rectangular), the perimeter increases monotonically and would require greater time to mentally trace the contour. As a result, the perimeter hypothesis predicts that the overestimation of the area of elongated shapes (elongation bias) should increase monotonically as a function of the extent of elongation. The area estimate errors of B, in contrast with the prediction of the perimeter hypothesis, actually significantly decreased as the ratio of the elongation ratios of A and B increased, β = −1.44, 95% confidence interval [CI] −1.81 to −1.06, p < 0.001. Moreover, when looking at the area-estimate error of B (%) at each elongation ratio (Table 1), the elongation bias occurred up to the point that the ratio of the elongation ratios of the target and reference reached 1.56. However, after this ratio, the participants underestimated the area of the target compared with its reference. 

To investigate the U-shaped patten of area estimate errors further, we ran a quadratic model by regressing the area-estimate error of B on ER_b/ER_a and (ER_b/ER_a)². The main effect of (ER_b/ER_a)² was significant, β² = 1.43, 95% CI 0.94–1.91, p < 0.001, and the main effect of ER_b/ER_a was significant, β = –8.73, 95% CI –11.25 to – 6.22, p < 0.001. The value of model fit (corrected quasi likelihood under independence model criterion [QICC]) improved when the model included (ER_b/ER_a)², QICC^b = 52661.22, compared with only including ER_b/ER_a, QICC^b = 53913.14. Figure 2 shows the plot of the quadratic model. As the elongation of B increased relative to the elongation ratio of A, the participants were more likely to underestimate the area of B, which is not consistent with the perimeter hypothesis. 

Figure 2.

 

Area-estimate error of B as a function of the ratio of elongation ratio of A and B.

Error as a function of the additive area difference between A and B

The additive area heuristic makes the same prediction as the perimeter hypothesis, namely that, as an object elongates while maintaining the same area, the participants perceive a greater area. To test this hypothesis, we regressed the area-estimate error of B on the additive area difference between A and B (AA_B–A). The additive area was calculated by adding the height and width of an object. There was a significant main effect of AA_B–A, β = –0.04, 95% CI –0.05 to – 0.03, p < 0.001. In contrast with the prediction of the additive area, however, as AA_B–A increased, the area estimate error (elongation bias) decreased. 

Discussion

The results of Experiment 1 suggest additional processes beyond models utilizing purely perceptual processes which predict a monotonic increase in the elongation bias as either the most salient (or longest) dimension increases (Krider et al., 2001) or the perimeter or additive area increases (Yousif et al., 2020; Yousif & Keil, 2019). In contrast, our results indicate that increases in the most salient dimension do not always result in increased area perception. 

It is worth noting, as well, that the significant influence of the ratio of elongation ratios implicates a comparative process in which some aspect of A is compared with that of B. That said, because the task invites the comparison of two adjacent objects, individuals may not solely perceive the height and width of each object independently; rather, the individuals may compare the cardinal dimensions of the two objects (Height_target vs. Height_reference; Width_target vs. Width_reference), as suggested by the additive area hypothesis, but these assessments are further biased by local contrast effects. 

When area assessment invites such comparison, our JND hypothesis could provide a complementary explanation. For instance, in Experiment 1, as the ratio of elongation ratios of the target and object increased, there was a decrease in the elongation bias. Our JND hypothesis clarifies that, when the changes are drastic enough to make noticeable changes to both height and width dimensions, the elongation bias should be decreased or eliminated, particularly as individuals realize that the object is both taller and thinner, thus complicating the task of assessing which object is larger. This overcorrecting process (as well as the distinguishably different heights and widths) may cause area underestimation. To further investigate this area underestimation, we ran the additional analysis to examine whether the height and width (the components of additive area), influence the area estimate error differently. Specifically, we regressed the area estimate error of B on the height and width differences between A and B (Height_B–A, Width_B–A). The interaction effect of Height_B–A and Width_B–A on the area estimate error was significant. β = –0.001, 95% CI –0.002 to –0.001, p < 0.001, indicating that when Width_B–A is small, Height_B–A would predict large area estimate errors. In contrast, as Width_B–A increases, the effect of Height_B–A decreases. 

With this in mind, we proceed to specify the additive area heuristic in more detail by explicating the JND hypothesis in the context of area comparison. Based on this evidence, we suggest that the threshold of JND of cardinal dimensions may differ based on how easy it is to compare two objects. The following sections report experimental results aimed at testing the JND hypothesis. 

Experiment 2: JND hypothesis test

To test our JND hypothesis, we measured the accuracy of participants’ area judgments while manipulating the width or the height of a target stimulus compared with a referent. 

Methods

Participants and design

We determined the sample size according to the largest number of participants that the researcher could recruit before the end of an academic semester. A total of 211 undergraduate students (55% male; M_age = 20.76 years) at a large midwestern university participated in the experiment for partial course credit. There were 10 observations per participant yielding a total of 2,110 observations. One observation was removed because the response time was 68 ms, which was both unusually fast compared with the average speed, (M = 5,627.81 ms, and well below subliminal levels (the minimum response time required for stimulus perception and motor responses for physiological processes is at least 100 ms) (Luce, 1986, as cited in Whelan, 2008), suggesting that it is likely this observation was created by an accidental key press). After excluding this 1 observation, 38 more responses were excluded for being 3 SDs greater than the mean, M = 5,630.44 ms, SD = 4,640.42 ms. In total, the number of observations for our analysis was 2,071. The design of the experiment was an elongation × orientation of an elongated target mixed-factorial design, with the 10 types of elongation conditions as a within-subject factor and the 2 types of orientation of an elongated target (vertical and horizontal) as a between-subject factor. The power achieved was 1, p = 0.05, Probability (Y = true | X = vertical) = 0.89, and Probability (Y = true | X = horizontal) = 0.76. 

Procedures

We used E-prime software to conduct this experiment. The program randomly assigned the participants to either the horizontal or vertical condition and assigned them to see a randomly generated subset (n = 10) out of 21 stimuli sets. Specifically, we had 21 stimuli sets for each vertical and horizontal condition. Among them, we had one stimulus in which both the reference (A, on the left) and the target (B, on the right) were the same height and width. For the rest of 20 stimuli, although the height and width of the reference (A, on the left) was constant across all conditions, the height or width of the target (B, on the right) was greater in comparison (Figure 3). Please note that, in this study, the elongation ratio refers to the longer cardinal axis divided by the shorter cardinal axis. Accordingly, the elongation ratio on the vertical condition was calculated by dividing the height (increasing dimension) by the width (constant dimension), whereas the elongation ratio in the horizontal condition was calculated by dividing the width (increasing dimension) by the height (constant dimension). To be specific, in the vertical condition, we made 20 stimuli by elongating the heights of the target by ratios of the starting height (the same height as the reference) from 0.01 to 0.20. Similarly, the horizonal condition used the same elongation ratios to adjust the widths rather than heights. 

Figure 3.

 

A sample screen of the questionnaire used in Experiment 2. The height of B increased per each elongation condition as a within-subject factor in the vertical condition, while the width of B increased per each elongation condition as a within-subject factor in the horizontal condition.

The participants learned that they would see several consecutive sets of figures followed by a fixation cross. They also learned that they should press appropriate keys (e.g., “P” for True, “Q” for False) to indicate whether the statement, “B has a greater area than A,” was true or false. The participants’ responses served as dependent variables. The participants were given 4 practice questions and then 10 real questions. The stimuli for the practice questions were clearly distinguishable as either smaller or larger than the reference stimulus. For real questions, each participant had 10 randomly ordered sets of either horizontally or vertically oriented stimuli. We also measured the participants’ response times. In addition, we measured two items to assess the fluency experienced in the task and one item to assess the participants’ confidence about each judgment. The participants saw these three additional items after every even-numbered or odd-numbered question. We invite interested readers to consult the Supplemental Material available online for an analysis of these additional measures (fluency and confidence). 

Results

Error analysis

Note that as the object B moves from a square to increasingly rectangular shapes, the area is not adjusted to remain the same—so, the correct answer would be for B to be judged as larger. However, if the change in shape is not discernable (i.e., at small elongation ratios), individuals might make mistakes. Thus, in general, we should expect performance to improve as elongation ratios increase. However, recall also that our hypothesis suggests that, as the shape of an object moves from a square to a rectangle of the same area, there will be a greater sensitivity on the height dimension (which is easy to compare because the two quadrilaterals are placed side by side). This suggests that participants’ response accuracy should improve more rapidly with increases in the height, in comparison with increases in the width, dimension. This should reveal itself as an interaction between elongation ratio and orientation. Also note that accuracy has a ceiling—in other words, once a JND is exceeded, accuracy should reach its maximum and level off, suggesting that we should test for quadratic terms to capture these curvilinear effects. Consistent with our prediction, the interaction between orientation and the square of the elongation ratio (ER²) was significant, β² = 326.43, Z(1, 205) = 5.26, p < 0.0001, for the probability of correct area judgments. To illustrate the interaction, we sorted the data by orientation and analyzed each orientation using a repeated measures approach with a logistic link regression function to accommodate the repeated-measures design and categorical nature of the dependent variable (error: 0 = incorrect, 1 = correct). We ran both linear and quadratic models to investigate which type of variable—ER or ER²—explains the error rates more parsimoniously. 

Vertical condition

When the ER was included as an explanatory variable, we found a significant main effect of ER, β² = 21.35, Z(1, 99) = 4.13, p < 0.0001. Also, including ER² as an explanatory variable yielded a significant main effect of ER² β² = −197.09, Z(1, 98) = –6.78, p < 0.0001. The value of each model's fit criteria indicated that the model fit was improved when ER² was included, QIC = 568.80, compared with only including ER, QIC = 600.16. Figure 4 shows the plot of the quadratic model including ER². 

Figure 4.

 

The results of correct answers in Experiment 2. Vertical: P(y = true) = \({e^{ - 197.089*{x_1}^2 + 447.8329*{x_1}-250.839}}\)/(1 + \({e^{ - 197.089*{x_1}^2 + 447.8329*{x_1}-250.839}}\)). The elongation ratio² (ER²) had a significant main effect, β² = −197.09, Z(1, 98) = –6.78, p < 0.0001, on the area comparisons’ accuracy (probability of correct answers). Horizontal: P(y = true) = \({e^{129.3425*{x_1}^2 - 257.969*{x_1} + 128.5577}}\)/(1 + \({e^{129.3425*{x_1}^2 - 257.969*{x_1} + 128.5577}}\)). The elongation ratio² (ER²) had a significant main effect, β² = 129.34, Z(1, 107) = 2.36, p = 0.0182, on the area comparisons’ accuracy (probability of correct answers).

Horizontal condition

When ER was included as an explanatory variable, we found a significant main effect of ER, β = 20.74, Z(1, 108) = 9.86, p < 0.0001. Also, including ER² as an explanatory variable yielded a significant main effect of ER², β² = 129.34, Z(1, 107) = 2.36, p = 0.0182. However, the model with ER², QIC = 973.43, was a better fit to the data than when only ER was included, QIC = 982.93. Figure 4 shows the plot of the quadratic model including ER². 

Vertical versus horizontal condition

The results of both conditions showed that, as the elongation ratio increased, participants made fewer errors. In other words, the accuracy of their judgments increased. However, the pattern of error rates was different across the two orientations (Figure 4). 

Participants in the horizontal condition chose more incorrect answers than correct answers at the early stages of ER than those in the vertical condition. At low levels of elongation ratio, not surprisingly (because a JND had not been crossed), there were relatively large error rates. As the ER increased and crossed the JND, individuals (correctly) perceived that one of the rectangles was larger and errors decreased (accuracy increased). However, the rate at which this happened differed as a function of orientation. If the elongation happened in the vertical orientation, it was noticed much earlier. This finding suggests that the JND in the vertical dimension was indeed smaller than in the horizontal dimension—that is, an equivalent small change was easier and more likely to be noticed on the vertical dimension than on the horizontal dimension. 

Discussion

Experiment 2 supports our JND hypothesis and suggests that the elongation bias is probably the result of the elongated rectangle being perceived as having greater height (owing to the smaller JND, the difference in height is perceived for smaller changes) but not having smaller width (owing to the greater JND, the difference in width is not perceived). 

Moreover, Experiment 2 provides an approximate range of elongation ratios that can be used to determine when the probability of detecting the elongation bias is highest. For example, according to Figure 4, the greatest perceptible difference in being able to perceive an accurate area between the vertical and horizontal conditions occurred when the elongation ratio was approximately 1.04 to 1.06. Around this range, although the vertical dimension had crossed its JND for most of the participants, the horizontal dimension had not yet crossed a JND. Therefore, the highest probability of detecting elongation bias is when the ratio of the height of the target and reference (Height_target/Height_reference) and the ratio of the width of the target and reference (Width_target/Width_reference) are around this range (1.04–1.06). 

Experiment 2 helps to delineate the region where an elongation bias is most likely to manifest. Specifically, the results suggest that when the ratio on the vertical dimension reaches 1.05, it seems to cross a JND and thus becomes noticeable and affects judgments. In contrast, the change on the horizontal dimension needs to reach a ratio of 1.15 before it is noticed. Thus, the elongation bias is most likely to be evidenced when the ratio on the vertical (easy to compare) dimension has exceeded 1.05 and the ratio on the horizontal (difficult to compare) dimension has not yet crossed 1.15. However, once both vertical and horizontal dimensions cross a JND, the scope for an elongation bias decreases. With this in mind, we re-examined the data from Experiment 1. Indeed, according to our post-hoc analysis, both height and width crossed JNDs in 8 out of 10 stimuli in Experiment 1, showing why we observed a decreased elongation bias in Experiment 1. 

Experiment 3: Area judgment condition (single vs. comparative) test

Because the JND hypothesis explicitly invokes a comparative process, it also suggests that the elongation bias should not occur in a between-subject design, particularly because a noticeable difference in an object's dimension cannot occur without comparing it with a reference object. For this reason, Experiment 3 investigates whether the elongation bias will only occur under a comparative judgment situation (Experiment 3A) and disappear under a single judgment situation (Experiment 3B). In so doing, Experiment 3 further demonstrates that the elongation bias automatically occurs when the necessary condition (in this case, the context of comparison) is provided. 

Experiment 3A

Methods

Participants and design

Participants were 80 MTurk workers (41.3% female; M_age = 34.83 years). We used data from 75 participants after excluding fake MTurk workers, participants who failed to pass an attention check, and participants from duplicated GPS and IP addresses. Specifically, we used Shinyapps (itaysisso.shinyapps.io/Bots) (Prims, Sisso, & Bai, 2018) to filter fake participants. For the attention check, we asked the participants what product picture they saw during the survey. The design was a one-way within-subject analysis of variance (ANOVA) with two conditions (elongated package, less-elongated package). 

Stimuli

We used a picture of two cracker packages (elongated: 200 (width) px × 400 (height) px; less elongated: 210.00 (width) px × 380.95 (height) px) that had the same surface area, although one was taller than the other. Also, we included a picture of an actual cracker between the two packages as a reference point for package size estimations. Note that in Experiment 2, we found that at around a ratio of 1.05, there was the greatest difference between the vertical and horizontal orientation conditions in terms of correct area judgment, suggesting that that the vertical dimension had definitely crossed a JND for most participants whereas the horizontal orientation had not yet crossed a JND for most participants. For this reason, we adopted a 1.05 ratio for the widths and heights of the elongated and less elongated packages (e.g., the width of the less elongated package/the width of the elongated package; the height of the elongated package/the height of the less elongated package) when creating the stimuli to maximize the elongation bias phenomenon. The elongated package was on the right side and the less elongated package was on the left. We also blurred the front of the packages to prevent any confounding effects from the product information that appeared there. 

Procedures

We asked the participants to estimate how many crackers they believed were in each package. This question served as a more distal indicator of area assessments—in other words, it serves to provide some external validity by demonstrating how the processes generating area assessments can have consequences on human judgment and behavior. For each package, we instructed the participants to use a slider scale that ranged from 10 to 100 crackers. The survey concluded with demographic questions. 

Results

Perceived relative number of crackers

The analysis was a repeated measures ANOVA that treated the two cracker packages (elongated, less elongated) as a within-subject factor. Elongation was a significant within-subject factor, F(1, 74) = 13.47, p < 0.001, η_p²= 0.15, on cracker estimates. The participants estimated that there were more crackers in the elongated package, M = 56.87, SD = 22.59, than in the less elongated package, M = 51.93, SD = 20.05. 

Experiment 3B

Methods

Participants

Participants were 101 MTurk workers (44% female; M_age = 36.38 years, SD = 11.18 years). Data from 91 participants were used after 6 participants were excluded based on the same criteria used in Experiment 3A. 

Design and procedures

The design was a one-way between-subjects ANOVA with two conditions (elongated package, less elongated package). In Experiment 3B, we used the same stimuli and the same decision task as in Experiment 3A. However, each participant saw only one package (either elongated or less elongated) and evaluated its size by using the same slider scale that ranged from 10 to 100 crackers. 

Results

Perceived relative number of crackers

The analysis was a one-way ANOVA treating the two cracker packages (elongated, less elongated) as a between-subjects factor. Elongation was not a significant between-subjects factor on cracker estimates, F(1, 89) = 0.23, p = 0.63, η_p² = 0.003. In other words, there were no differences in terms of estimated number of crackers in the elongated, M = 49.72, SD = 20.85, and less elongated, M = 52, SD = 23.95, packages. 

Discussion

Experiments 3A and 3B support our prediction that elongation bias only occurs when comparative judgment is involved. Given that Experiment 3A was within-subject and Experiment 3B was between-subject, we cannot formally run an interaction effect (e.g., package shape [elongated vs. less elongated] × condition [within vs. between]) by combining the data. However, it might be instructive to show the means of the two studies in one graph. The observed pattern of means supports our comparative judgment idea in area comparisons (Figure 5). 

Figure 5.

 

The cracker estimates as a function of package shape (elongated vs. less elongated) and condition (within subject vs. between subject) in Experiments 3A and 3B.

Although Experiment 3 provides an example of real-world implications, one aspect might still be cause for concern. A significant difference between Experiment 3 and the approach used in Experiments 1 and 2 is that Experiment 3 asks respondents to engage in mental processes regarding the number of components that could fit inside a specific area, whereas Experiments 1 and 2 focused on estimates of the area itself. We have argued that the results in Experiment 3 were the result of the perceived larger area. However, alternative accounts could be devised using changes in perceptions of the component parts. Thus, it seems prudent to study this effect without the issue of components and use just area perception. 

Experiment 4: Area judgment condition (single vs. comparative) test

Experiment 4 was devised to eliminate the issue of component parts within an area. Specifically, we wanted to replicate the results of Experiment 3, but to eliminate the issue of component parts inside a whole. 

Experiment 4A

Methods

Participants and design

Participants were 86 MTurk workers (45.3% female; M_age = 39.07 years). We used data from 58 participants after excluding participants based on the same criteria employed in Experiment 3A. Additionally, four observations were excluded owing to area estimates being either 3 SDs above the mean, M = 69.78, SD = 153.31, or zero. In total, 112 observations from 57 participants were used for analysis. The design was a one-way within-subject ANOVA with two conditions (elongated shape, less elongated shape). 

Stimuli

Participants received a cover story about a facility that manufactures printed circuit boards that need to be coated with a copper solution that is then etched away to provide the connections for components. Thus, a critical management task is to estimate the area of the substrate on which the copper solution is applied. We used a picture of two substrates, elongated: 200 (width) px × 400 (height) px; less elongated: 210.00 (width) px × 380.95 (height) px, that had the same surface area, although one was taller than the other. The elongated substrate was on the right side and the less elongated substrate was on the left (Figure 6). 

Figure 6.

 

Two substrates used in Experiment 4.

Procedure

We first showed the participants four different substrates at a 1:1 size in random order and told them how much copper was required to cover the substrate. For instance, we showed a substrate, 288 (width) px × 288 (height) px, and told the participants that it took 14.67 mL of copper to cover the substrate. Subsequently, we showed participants the main stimulus, the two side-by-side substrates. We asked participants to estimate how many milliliters it would take to cover each substrate. The survey concluded with a request for demographic information. 

Results

Perceived relative amount of copper

We used a repeated measures generalized estimated equation to analyze the data that treated the two substrates (elongated, less elongated) as a within-subject factor. Elongation was a significant within-subject factor on the milliliters of copper estimates, β = −6.65, 95% CI –11.76 to −1.54, p = 0.011. The participants estimated that more copper was needed to cover the elongated substrate, M = 50.81, SD = 37.10, than the less elongated substrate, M = 44.16, SD = 34.35. 

Experiment 4B

Methods

Participants

Participants were 172 MTurk workers (50% female; M_age = 38.10). Data from 125 participants were used after excluding participants based on the same criteria employed in Experiment 3A. Additionally, 3 observations were excluded owing to area estimates being 3 SDs above the mean, M = 34.04, SD = 33.44. In total, we used 122 observations from 122 participants for our analysis. 

Design and procedures

The design was a one-way between-subject ANOVA with two conditions (elongated shape, less-elongated shape). In Experiment 4B, we used the same stimuli and the same decision task as in Experiment 4A. However, each participant saw only one substrate (either elongated or less-elongated) and estimated how many milliliters of copper it would take to cover the substrate using the same scale as in Experiment 4A. 

Results

Perceived relative amount of copper

The analysis was a one-way ANOVA treating the two substrates (elongated, less-elongated) as a between-subjects factor. Elongation was not a significant between-subjects factor on copper estimates (F(1, 120) = 0.90, p = 0.344, η_p² = 0.007). In other words, there were no differences in terms of copper estimations in the elongated, M = 31.98, SD = 24.10, and less-elongated, M = 28.39, SD = 16.82, substrates. 

Discussion

Experiments 4A and 4B replicated the findings of Experiment 3 while addressing the Experiment 3’s problem of component parts within a whole. The results imply that elongation bias occurs exclusively in comparative judgments. 

General discussion

Over the course of four studies, we investigated the underlying process that generates the elongation bias in area judgments. We first outlined the additive area hypothesis and proposed the perimeter hypothesis (which describes the time it takes individuals to optically follow the contour of an object) as heuristics for assessing an area. However, Experiment 1 produced results that did not support these hypotheses as sole mechanisms. In general, the participants did not consistently overestimate the area of an elongated object, as the target object's elongation ratio increases relative to the reference. We, thus, proposed the notion of contrast effects in addition to the additive area heuristic as a way to explain the phenomenon. In Experiment 2, we tested this idea of context effects by assessing the JND on the cardinal axes to explain why we may not observe monotonically increasing area perceptions when a comparison is involved. According to this notion, individuals try to assess the cardinal dimensions to assess areas. However, in the comparative framework of the elongation bias, there are perceptual comparisons that are made that lead to context effects and bias the assessments of the cardinal dimensions. The extent to which individuals notice subtle length changes in these dimensions depends on how difficult it is to compare the target's dimension with the reference's dimension. With this in mind, we demonstrate that individuals are more likely to distinguish size changes easily in an elongated stimulus when the elongated axis is adjacent to the reference axis. In Experiments 3 and 4, we find that the elongation bias only occurs when a reference object can be compared with the size of a target object. 

This article contributes to the area perception literature by extending the additive area theory, which proposes that individuals perceive areas by adding two cardinal dimensions (Yousif et al., 2020; Yousif & Keil, 2019). Although the additive area heuristic can explain area assessment when individuals assess the area of a single object individually, the JND account provides evidence for comparative processes and their impact on area judgments. We suggest that the difference threshold may differ on the cardinal dimensions that make up the additive area. The threshold of JND depends on such contextual factors as the size of the objects, the physical distance between the objects, and the alignment of the objects, among others. All factors influence the level of difficulty that is involved when comparing the composition of the additive area (vertical and horizontal axes) and, by extension, the accuracy of area assessment. Specifically, when two rectangles are adjacent, the JND of height is smaller than the JND of width because heights are easier to compare than widths in this orientation; thus, a small change in height is more likely to be noticed than the small change in width that is made to keep the area the same, resulting in the taller rectangle being perceived as taller than, but the same width as, the referent. Accordingly, in this scenario, if we take two rectangles and change one of the cardinal dimensions to make it larger while maintaining the same area, it is better to change the height dimension. Similarly, if we want to make an object smaller without making it appear smaller, we are better off changing the width dimension. This conceptualization could potentially inform our understanding of other visual illusions such as that documented by Beck, Emanuele, and Savazzi (2013), where they document that taller objects are regarded as thinner, and thinner objects as taller. Indeed, their experiments tap into a background contextual variable, namely, that we are holding area constant. These kinds of heuristics probably emerged from observing individual categories, for example, people. Thus, we expect most people in the category to be about the same size. Thus, if someone is taller, they are likely to be thinner, that is, the background assumption is that humans are approximately the same size, blue whales are approximately the same size, chihuahuas are approximately the same size, and so on. Thus, any variation in one dimension is likely to generate the sense of a change on another dimension to keep the overall size the same. 

Note also that these results suggest that area assessments are not necessarily a completely bottom-up, perceptually driven phenomenon. There are also top-down influences from comparison processes that yield context effects that further influence the bottom-up perceptual processes. In other words, it appears that, on encountering a stimulus, bottom-up and top-down processes are capable of producing outputs that then have to either engage in a race to be the final output or be integrated in some manner to produce a final output. Delineating these processes might be a fruitful line of inquiry to pursue. 

Future research may choose to investigate whether the perception of changing dimensions is linearly scaled once the dimension crosses JND. For instance, Experiment 1 suggests that elongation bias exists in a narrow region where one cardinal dimension has crossed a JND but the other has not, resulting in the illusion of a taller and equally wide rectangle. However, once both cardinal dimensions cross their JND thresholds, the illusion breaks, and people may even become more regressive in their judgments resulting in smaller elongation biases (as seen in Experiment 1) or perhaps even reversals (as also seen in Experiment 1). Also, future research may develop this research to generalize our findings to object perception in a three-dimensional context; the current study only used stimuli that revealed the total surface areas of the objects across all studies. In this vein, a recent study (Sereno, Robles, Kikumoto, & Bies, 2020) has found that the context influences how the size of the dimension (e.g., width) is perceived. Also, although the current study investigated the JND hypothesis with static stimuli, future research may choose to investigate the JND account in the scenario that context provides nonrigid and rigid motion, as in the bar–cross–ellipse illusion (Caplovitz & Tse, 2006). Furthermore, we investigated the process of comparing areas visually, but did not consider other sensory inputs. By comparison, previous research found that the elongation bias was reversed when vision was blocked and only tactile information was provided (Krishna, 2006), suggesting reversed JNDs in the sensory modality. 

Conclusion

Previous literature has provided a more bottom-up, perceptually driven approaches to explaining area assessment, focusing primarily on how individuals utilize and integrate dimensions (e.g., height, width). Our research adds a more top-down, comparison-based approach to this process by investigating the dynamics of perceiving a single dimension as well as the contextual effect of the area comparison process. This interplay of bottom-up and top-down processes appears to have the potential to inform other aspects of visual information processing as well. With this in mind, the present study contributes to our understanding of not only the psychological process of elongation bias but also how humans compare and estimate the area of objects. 

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. 

Commercial relationships: none. 

Corresponding author: Dongeun Kim. 

Email: [email protected]. 

Address: Department of Marketing and Organizational Communication, Grenon School of Business, Assumption University, 500 Salisbury Street, Worcester, MA 01609, USA.