Position biases in sequential location selection: Effects of region, choice history, and visibility of previous selections

Ronald Hübner

doi:10.1371/journal.pone.0276207

. 2022 Oct 14;17(10):e0276207. doi: 10.1371/journal.pone.0276207

Position biases in sequential location selection: Effects of region, choice history, and visibility of previous selections

Ronald Hübner ^1,^*

Editor: Alessandro Pluchino²

PMCID: PMC9565670 PMID: 36240249

Abstract

In some situations, e.g., when filling out lottery tickets, it can be advantageous to select random locations. However, people usually have difficulties with this, because they are biased by preferences for certain regions, such as the center of an area. According to ideas from art theory, the preferred regions reflect the hidden structure of perceptual forces within an area. In the present study, these structures were investigated and modeled under different conditions for areas with square and rectangular shape. The general task was to sequentially place a number of dots at random locations in an area by clicking with the computer mouse at corresponding positions on the screen. Whereas in a single-dot condition each dot had to be placed in an empty area, the previously placed dots remained visible in a multiple-dots condition. In three experiments it was found that dots were preferentially placed at the center, the diagonals, and the principal axes. This preference was more pronounced in the single than in the multiple-dot condition. Moreover, sequential analyses revealed that dot placing was not only planned in advance, but that the participants also agreed to some extent in their sequential selections, which produced surprisingly similar sequential spatial patterns across participants, at least for the first dots. Altogether, the results indicate that people have great difficulties with the random selection of locations. Their selections are strongly affected by the attraction of specific regions, by previous selections, and by sequential habits.

Introduction

In everyday life, we sometimes encounter situations, where we must select one of several locations. If the locations differ in relevant aspects, then the selection might be easy and based on rational grounds. But what if the locations are subjectively or even objectively equivalent? Assume, for instance, we want to grab one of the equivalent packs of the same product on a shelf in a supermarket or mark the cells on a lottery sheet. How should we proceed? A suitable strategy for choices amongst objectively equivalent options is to choose randomly [1, 2]. However, this is not what people usually do. Rather, they often choose specific positions with a higher probability than others. In menu choice, for instance, items at the beginning or end of a menu are favored [3], although mainly for vertical arrangements [4]. Bar-Hillel [5] explained this edge preference by assuming that, even though menu items are normally presented simultaneously, they are mentally processed sequentially, and that this leads to the well-known serial position effect [6].

Whereas position biases in menu choice are relatively harmless, they can have severe negative consequences in other areas. In eyewitness identification, for instance, choosing a suspect from a simultaneous live lineup, or from a two-dimensional photo array largely based on his or her spatial position can lead to misidentification with all its negative consequences [7]. Thus, for preventing such faulty decisions, knowledge about position biases is very important.

Interestingly, different from menu selection, no edge preference occurs in eyewitness identification. Rather, besides a bias to choose from the top row of photo arrays [7], edge aversion is observed [7, 8]. The latter bias also occurs in multiple-choice questions, where not only test makers but also test takers have the strong tendency to prefer central locations [9]. How can these contrasting results be explained? Bar-Hillel [5] noticed that the latter two domains involve competitive social interactions with others that resemble those in hide-and-seek games and assumed that this leads to strategic decisions. The decision maker tries to guess the location chosen by the hider and chooses one of the better hiding places, i.e., the central positions.

However, preference of central positions is not only observed for choices in competitive social interactions, but also in more simple situations such as grabbing a box from the shelf [10] or selecting a seat [11]. Christenfeld [10] supposed that the central-position bias in choice behavior reflects a simple rule applied automatically to minimize mental effort. Later, Bar-Hillel, Peer [12] argued that not only mental effort is minimized in perceptual-motor tasks, but also physical effort. They assumed that the central-position bias is due to reachability, i.e., central positions are chosen because they are easiest to reach and require less motor fine tuning.

These considerations show that the origin and details of the central-position bias in simple tasks without competitive social interaction remain largely unknown. A promising approach could therefore be to investigate situations that are as simple as possible. All tasks considered so far require choosing one of several items in an array. As a consequence, it is possible that additional biases from the features of the items or array contaminate pure position biases [13]. If the items are numbers, for instance, as in lottery or roulette choices, those considered as most representative [14] are often preferred [15]. Therefore, a more direct access to position biases might be possible by examining situations where a location in an empty area has to be chosen. That even an empty area has some hidden spatial structure has already been proposed by Arnheim [16, 17]. He assumed that such a structure results from attentional and/or perceptual processes, and cause that people prefer certain regions. A simple method to make the hidden structure of an area visible is to ask persons to place a dot at a random location within this area. The pattern of superimposed dots from many persons then reveals the hidden structure. This method, however, involves the perception and production of randomness, which are research topics in its own right.

The perception and production of randomness

The question of how persons perceive and produce randomness is of general interest in areas such as psychology, economy, and art theory. To investigate this issue, different experimental subjective-randomness tasks have been developed. For instance, persons had to judge the randomness of patterns [e.g., 18], to select random positions within an array [e.g., 19], or to generate a random pattern [e.g., 20]. Moreover, different layouts have been used. Some researchers investigated the perception or generation of random sequences, mostly with symbols representing some kind of events [e.g., 21]. Others specified one- or two-dimensional areas and asked participants to randomly place dots freely within the empty area [e.g., 22], or to randomly mark cells in a grid [e.g., 19].

In the present study the production of randomness is mainly used as method for investigating position biases in two-dimensional arrays. For this objective, participants were asked to randomly place dots in a rectangular or square area. The rationale behind this method is based on results showing that persons have great difficulties with the perception and production of randomness. Therefore, their location choices should largely be affected by the hidden structure of the given area. It can be expected that people preferentially place dots in locations that are attractive in some sense [23]. Accordingly, the choice behavior should reveal the distribution of regional attraction in a given area. In addition, for uncovering such distributions, it was also of interest in the present study to investigate processes of sequential location selection. However, before reporting more details on the present aim, a short overview of related and relevant topics and results will be given.

Sequences of events

Sequences of events occur in many areas such as gambling, stock market, or sports. For predicting future events, it is essential to know whether the underlying generation process is purely random or involves some structured mechanism. Accordingly, an important question is to what extent people can discriminate random from non-random sequences [24]. The evidence from corresponding studies suggests that the ability to discriminate is rather low. Binary sequences, for instance, are judged as random, when they alternate frequently, but not so often that they show a pattern [25, 26]. The tendency to expect frequent alternation is known as negative recency effect [27] and related to the gamblers fallacy [e.g., 28, 29]. Yet, a random process can produce long sequences of identical events, which, however, are usually not perceived as random. Rather, our intuition is that the properties of a random process, i.e., equiprobability and irregularity, should also be present in short sequences. This intuition leads to what Kahneman and Tversky [14] have called the representativeness heuristic. An observed sequence of events is considered random if it is similar to the mental prototype of randomness.

Falk and Konold [21] proposed an alternative account. In their implicit encoding theory, they assume that people judge the degree of randomness of a sequence based on the ease of which it can be encoded. The more difficult a sequence can be encoded or remembered, the more random it seems to be. In a similarly manner, Hahn and Warren [30] argue that biases and misperceptions of randomness may actually reflect limits of the human short-term memory and/or attention span. Additionally, however, they assume that the biases also reflect genuine aspects of our statistical environment [see also 31]. Recently, Gronchi and Sloman [32] provided evidence for the notion that both heuristics are used. People use the representativeness heuristic for evaluating the randomness of a sequence, but the encoding strategy for evaluating its regularity.

Similar biases as in the perception of randomness have also been observed for the production of randomness. In corresponding tasks participants must imagine a random process such as throwing a coin or rolling a dice and generate an appropriate symbolic sequence [e.g., 21, 26, 27, 31]. It turns out that participants tend to over-alternate between the possible outcomes, which is compatible with the representativeness heuristic. However, rather than attributing this bias to an intuitive misunderstanding of randomness, Wiegersma [33] assumed that the over-alternation results from controlling the automatic tendency for repeating the last response. Meanwhile there are several theories for explaining the perception and production of randomness in relation to binary sequences [for an overview see 24].

Random sequences can also affect the position bias. In full-length multiple-choice tests, for instance, examinees seem to assume that the positions of correct answers are balanced, and, therefore try to produce a balanced position sequence on their own [9]. Thus, repeated location selections can not only be affected by edge aversion, but also by the tendency of producing a representative sequence.

Marking cells in a grid

A popular method for investigating the perception and production of spatial randomness is to have people mark cells in one- or two-dimensional grids, which is related to producing binary sequences. Ross and Weiner [34], for instance, asked their participants to produce a random pattern by marking either 16 or 64 cells in a 20×20 grid. A comparison with the prediction of a homogeneous Poisson process revealed a significant deviation for the 64-cells condition. Specifically, the produced patterns were too regular. Interestingly, such patterns are liked more than real random ones, which often look more clustered [e.g., 35].

For the 16-cells condition, Ross and Weiner [34] observed no significant deviation from a Poisson process. However, this statistical result might be due to the small number of participants in this condition. As mentioned, specific locations are usually preferred when only few locations must be marked or selected. In Roulette [5] and Lottery Games [36], for instance, the central region is preferred, while outer regions are avoided. Even though there is also some effect of representative or personally meaningful numbers such as their birthdate, age, or postal code [37], the common overall preference becomes visible if the patterns of many persons are superimposed.

Similar results have been observed in hide-and-seek games [19, 38]. In the study of Falk, Falk [19], for instance, participants had to mark one or three cells in a 5×5 grid. They chose the locations either from the perspective of competition or from that of cooperation. Furthermore, there were two control groups. One had to choose locations randomly, while the other was instructed to produce a pleasant-looking pattern. Interestingly, superimposing the marked locations of all participants in a condition did not only reveal that the distributions were inhomogeneous, but also that they were similar across conditions. If the columns and rows are labeled from A to E and 1 to 5, respectively, then the cells C3 (center) and B4 (one step left and up from the center) were mostly preferred.

Free placement of dots

As mentioned, Arnheim [16, 17] proposed that even empty spatial areas (frames) have some hidden structure. In rectangular areas, for instance, the axes, diagonals, and the center are regions of high attraction. Palmer and his coworker provided empirical support for this conjecture [39, 40]. They presented a circle at various positions in an empty area and asked their participants to rate `how well the circle fits within the area at that position’. As a result, the mean ratings were highest at the center and fall off monotonically with the distance from it. Moreover, mean ratings were also increased along the principal axes, and the diagonals. A more indirect approach for revealing hidden forces in a spatial area is to ask people to place a single dot or multiple dots at random location within that area.

Single dots

One of the first studies, in which a single dot had to be placed inside an outline area, was Psotka [41]. He used several shapes, including square and rectangle, which were drawn on a sheet of paper. The participants were instructed to place a dot inside each shape either ‘at random’ or ‘in the first place that comes to mind’. Superimposing the individual dots revealed that they were preferentially placed on or near to the diagonals of the shape. A similar result was later found by Lisanby and Lockhead [22], who gave their participants a sheet of paper with a square or rectangle drawn on it and asked them to place a single dot within the frame. They observed that the locations near the diagonals and the center were preferred. Furthermore, more dots were placed in the upper half of the area than in the lower half. Accordingly, the distributions of superimposed dots were different from that produced by a homogeneous random process. Interestingly, Lisanby and Lockhead [22] found similar results when they instructed their participants to place a dot at an aesthetically pleasing location. From this coincidence they concluded that art, aesthetics, and subjective randomness are strongly related. Furthermore, they proposed a two-step process for the selection of random locations: Persons first select an aesthetic location, which is then adjusted to appear unpredictable.

Multiple dots

In some studies, participants had to place multiple dots within an area. Gemelli and Alberoni [42], for instance, asked their participants in their first experiment to draw 20 dots on a rectangular sheet of paper by simulating randomly falling drops of water. They observed that at least some participants placed the dots irregularly all over the paper without forming subgroups, similarly to what has been observed for marking cells [e.g., 34]. However, in a later study by Dudley [43], where participants had to place ten dots randomly in a square area, the pattern of superimposed dots shows less irregularity. Rather, the center was preferred while the outer regions were avoided.

Thus, it seems that the spatial distribution of dots depends on how many dots must be positioned within an area. For one or few dots, people prefer the center and the diagonals, whereas many dots are distributed more evenly across the area. That many dots are more evenly distributed might be due to the fact that the visible dots make an inhomogeneous distribution more evident, and people try to avoid global inhomogeneities. Alternatively, one can assume that visible dots have a repelling effect, which prevents clustering [17]. Such an account is similar to sequential inhibition [44], i.e., the hypothesis that an already placed dot inhibits the placement of another dot in its near neighborhood, an effect analogous to the negative recency effect in producing random sequences [28].

The present study

The general aim of the present study was to use subjective randomness for investigating position biases in two-dimensional areas. For this objective three experiments were conducted in which participants were asked to place dots at random locations within a given area. Based on previous results, it was expected that people are unable to do so, because a hidden structure of perceptual forces in an area makes regions differently attractive for selection. Up to now, the distribution of regional attraction was mostly investigated in experiments in which a single or multiple dots had to be placed with a pencil on a blank sheet of paper [e.g., 22, 41–43].

The present study went further and not only collected data with computers to minimize possible effects of reachability, but also examined sequential effects and the preference order of regions. For this objective, participants had to randomly place a dot in an empty area on the screen not just once, but several times in a row. For comparison, there was also a multiple-dots condition. More specifically, the first and last experiment included two basic experimental conditions. In a single-dot condition, each of several dots had always to be placed in an empty area, whereas in a multiple-dots condition the already placed dots remained visible. Furthermore, the number of dots and the form of the area were varied across experiments.

As we have seen, if participants have to place a single dot in an empty area, they prefer to choose a place near the center or diagonals. Therefore, a more specific question was whether this preference also lasts across several dots in a row. If so, we should observe a similar pattern for each sequential dot superimposed across participants. If not, we might observe a structured pattern for the first dot but a more widely distributed pattern for the subsequent dots. For the multiple-dots condition it was expected that, due to the persistent visibility of the placed dots, the effect of the hidden perceptual structure is generally reduced.

As a general method, the individual dots or dot patterns were superimposed across participants. It was then tested whether they could have been generated by a homogeneous spatial Poisson point process [44]. Such a process is characterized by two key properties: Homogeneity and independence. Homogeneity means that the points occur at each location with equal probability, while independence requires that the outcome in one region of space has no effect on the outcome in other regions.

To examine the sequential processes of location selection in detail, one method was to connect the subsequent dots of each participant by straight lines, as in the study of Gemelli and Alberoni [42]. These researchers observed that the individual trajectories formed broken lines with mostly acute angles. Here, the length and direction of the paths between subsequent dots were also analyzed.

As we will see, the participants were largely unable to randomly select locations. Rather, they were strongly affected by the distribution of regional attraction. Moreover, they followed, at least to some extent, a common sequential selection strategy.

Experiment 1

The first experiment consisted of a single-dot condition (10S), in which one dot had to be placed ten times in a row at a random location within an empty square area. In view of the results reported in the literature, it was expected that the hidden perceptual structure of the empty area strongly affects the selection of location, i.e., that specific regions such as the center and the diagonals are salient and more attractive for placing dots than others. Accordingly, the dots should cluster in these regions. In addition, there was a multiple-dots condition (10M) that was similar to the single-dot condition, except that already placed dots remained visible. Because the visibility of the dots makes inhomogeneities more apparent, it was expected that the dots in condition 10M are more evenly distributed across the area than those in condition 10S.

Method

Participants

Eighty-three persons (33 males, mean age 23.6, SD = 8.12), mostly students from all disciplines of the University of Konstanz, were recruited via a local online-system (ORSEE, Greiner, 2015) for the online experiment. As incentive, each participant had the chance to win one of eight vouchers worth 4 €. The study was performed in accordance with the ethical standards of the Declaration of Helsinki (1964) and was approved by the ethics committee of the University of Konstanz (Confirmation 27/2020). Participants were informed that they are free to withdraw from the study at any point in time without any negative consequences. Informed consent was obtained from all participants by check-marking a box on the informed-consent page before the actual experiment started.

Procedure

When the participants started the online experiment, they were asked to use a notebook or desktop computer. After confirming that this was the case, the actual experiment began with a short instruction that informed about the task and continued by requiring the participants to work through two conditions. Condition 10S consisted of ten trials. On each trial the participants had to place a single black dot (radius 5 px) at a randomly chosen location within a screen-centered empty white canvas (400 x 400 px) by clicking on the corresponding location with the computer mouse. The participants were instructed to imagine that the canvas is a sheet of paper on which a raindrop was falling on a random spot. There was no time limit. Participants could forward to the next trial by clicking on a corresponding button. In condition 10M, the task was also to randomly place 10 dots on the canvas, while the already placed dots remained visible. The order of conditions was randomized across participants. Altogether, the participants spent about 5 minutes on the study.

Results

Spatial dot distribution

The dot patterns for the two conditions, superimposed across participants, are shown in Fig 1. For comparison, a random-dot pattern (10R) is also shown. It was produced by uniformly distributing 830 dots over the area. As can be seen, the human-generated dots are not distributed homogeneously. Rather, they cluster along the principal axes, the diagonals, and at the center. As expected, this characteristic is more pronounced for condition 10S than for 10M. To test the attraction of the diagonals, the same procedure as in Lisanby and Lockhead [22] was applied. That is, the coordinates of the dots were first folded around the imaginary vertical axis and then around the horizontal axis that bisect the area, so that all dots are finally located in the lower left quadrant. If the diagonals attract the placement of dots, then there should be a positive correlation between the x and y coordinates. This was indeed the case. The correlation for condition 10S was r = 0.234, t(828) = 6.91, p < 0.001, and for 10M r = 0.197, t(828) = 5.78, p < 0.001. Moreover, testing across the individual correlations revealed that the mean correlations differed between the conditions (.199 vs. .142), t(82) = 1.34, p < .05 (one-sided). This indicates, as predicted, that the clustering along the diagonals was reduced in condition 10M.

A further result is that the participants placed more dots in the upper half of the area than in the lower one, which was the case for condition 10S (463 vs. 367), t(82) = 3.12, p < 0.01, as well as for 10M (451 vs 379), t(82) = 3.00, p < 0.01.

To make the underlying structure of the superimposed dot patterns more visible, the intensity functions of the corresponding point processes were estimated by means of the R [45] function “density.ppp” from the library “spatstat” [44]. The standard deviation (sigma) of the smoothing kernel was set to 20. Fig 1 shows the obtained intensity maps (column Intensity). The intensity reflects the average number of dots per unit area, which is one pixel (px) in the present case. If the participants had placed the dots in accordance with a homogeneous Poisson process, then the intensity should vary like that of the random-dot pattern (10R). This was obviously not the case. Rather, the intensity is concentrated along the diagonals, somewhat along the axes, but mostly at the center, while it is rather low in the outer regions.

To support this interpretation, the program “quadrat.test” [44] was applied to compute χ² tests of complete spatial randomness (CSR). Such a test is based on quadrat counts, for which the area of a pattern is divided into equally sized cells. For the present analyses, a 5x5 grid was chosen. As result, the test was highly significant for both the 10S pattern, χ²(24) = 185, p < 0.001, and the 10M pattern, χ²(24) = 145, p < 0.001, indicating that they were probably not produced by a homogeneous Poisson process. In contrast, the test for the 10R pattern was not significant, χ²(24) = 21.3, p = 0.754.

Modeling regional attraction

The data and estimated intensities clearly show that the dots were not placed randomly across the area. Rather, specific regions were preferred. If one assumes that the intensity maps in Fig 1 roughly reflect the regional preferences, then they might be used to construct a formal model of regional attraction. A possible approach is demonstrated next. By following Arnheim [16, 17], it is assumed that the center, the diagonals, and the main axes are regions of high attraction. The attraction of each component, i.e., the horizontal axes, the vertical axes, the center, the major diagonal, and the minor diagonal, can then be represented by a corresponding bivariate normal distribution with density function f(x, y). For all functions, the means of the variables X and Y are fixed to the center of the area, i.e., to μ_x = w/2 and μ_y = h/2, where w denote the width of the area in px, and h its height. Because the shape of the density function is largely determined by the correlation between X and Y, it is convenient to define the covariance matrix in terms of the variances σ²_x and σ²_y, and the correlation ρ between X and Y. Thus, we have 5x3 = 15 free parameters that must be determined for the five component distributions. To reduce their number, several simplifying assumptions are made by introducing three model parameters α, β, and γ. All density functions and how their free parameters are replaced by h, zero, or by one of the model parameters can be seen in Table 1.

Table 1. Density functions for the different regions and corresponding parameter values in terms of the three model parameters α, β, and γ, and the width (w) and height (h) of the area.

Regions	Function	Parameters
		μ _x	μ _y	σ²_x	σ²_y	ρ
Horizonal axis	f _h	w/2	h/2	α	h	0
Vertical axis	f _v	w/2	h/2	h	α	0
Center	f _c	w/2	h/2	β	β	0
Major diagonal	f _j	w/2	h/2	α	Β	γ
Minor diagonal	f _i	w/2	h/2	α	β	−γ

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In a further step, the density functions are used for constructing three component maps of attraction for the area of 400x400 px, one map M_C = f_c for the center, one map M_A = f_h + f_v for the axes, and one map M_D = f_j + f_i for the diagonals. After scaling each component map so that its sum equals one, they are combined to a master map M of spatial attraction by multiple linear regression:

M = a_{0} + a_{D} M_{D} + a_{A} M_{A} + a_{C} M_{C .}

(1)

Eq (1) is a formal model of spatial attraction in a rectangular area. This model was fitted to the data of condition 10S by means of the R program “optim” [45]. It searched for values of the three model parameters that maximize the correlation between the empirical intensity map and the master map of attraction. The obtained parameters are α = 7.933976x10⁶, β = 8.522916x10⁶, and γ = 0.999. The corresponding maps of attraction for the three regional components (Center, Diagonals, and Axes) can be seen in Fig 2.

Fig 2 — The range of values in the three maps are 6.0849x10^-5 to 6.1131x10^-5, 2.3756x10^-10 to 2.7611x10^-4, and 2.0420x10^-25 to 4.8591x10^-04, respectively.

The corresponding regression parameters are: a₀ = −1.178, a_D = 26.8, a_A = 6.08, and a_C = 19334. The resulting master map of attraction can be seen Fig 1 (in column Attraction Model). Its correlation with the empirical intensity map is r = .867. It should be noted that the component maps also correlate. The correlation between the map of diagonals with that of the axes is r = .0559, that between the map of diagonals and the map of the center is r = .205, and that between the map of the axis and the map of the center is r = .515.

Applying the model to condition 10M revealed as model parameters α = 3.5612x10⁴, β = 4.39661x10⁵, and γ = 0.999. The regression coefficients are a₀ = −31.9, a_D = 22.2, a_A = 0.638, and a_C = 5.22049x10⁵. The corresponding master map of attraction can also be seen in Fig 1. Its correlation with the empirical intensity map is r = .883.

For demonstrating the regional attraction for a random process, the corresponding master map is also shown in Fig 1. It is of course homogeneous.

Path lengths and directions

Until now, we have analyzed and modeled the final dot patterns. To get an insight into the processes involved in the sequential selection of locations, the direct paths between successive dots were visualized by straight lines for each participant, as in Gemelli and Alberoni [42]. The corresponding patterns resulting from superimposing all lines across participants are shown in Fig 3. The blue gradient of the lines reflects their length. By considering the patterns, it becomes obvious that not only the dots were frequently placed near the diagonals, but also that the direct path between successive dots often run along or parallel to the diagonals, at least for condition 10S. Moreover, the perceived mean darkness of the line patterns indicates that the paths for condition 10S are longer than for 10M.

Fig 3 also shows the histograms of the path lengths. As can be seen, the distributions for both experimental conditions are positively skewed. However, the skewness is greater for condition 10M (skewness: .553) than for 10S (skewness: .258). Because the greater skewness is due to a higher proportion of shorter paths, the mean path length for condition 10M (169 pixels, SD = 84) is also significantly shorter than that for 10S (188 pixels, SD = 95), t(82) = 3.54, p < 0.001. Nevertheless, the skewness is smallest (.175) for the random-dot pattern. A computer simulation of 10⁵ random lengths within the square area revealed a skewness of .185. This demonstrates that random length is not distributed symmetrically in such an area.

To also consider the distribution of path directions in combination with path length, the two measures were plotted analogous to a so-called wind-rose diagram (the used R plot function was a modified version of a function provided by Andy Clifton, see https://stackoverflow.com/users/2514568/andy-clifton), which is shown in Fig 4. If the directions were equally distributed, then 4.17% of the paths should occur in each of the 24 direction bins. However, a simulation with 10⁵ x 748 lines with random length and direction, categorized in the same way as the empirical ones, revealed that the directions are not distributed equally. Rather, the percentages vary systematically across bins from 4.06% to 4.34%. They are highest for the four diagonal directions (45°,135°,225°,315°). The corresponding percentages for the observed directions in condition 10S are 5.09%, 4.69%, 6.02%, and 5.22%. The simulated critical value for a significance level of .05 is 5.49 for all directions. Thus, only lines with a direction around 225° (down left) occurred significantly more frequent. For condition 10M the corresponding percentages are 5.09%, 4.28%, 5.09%, and 5.22%, of which none is significant.

If we consider the four axial directions (0°, 90°, 180°, 270°), then their simulated percentage in the corresponding bins is 4.24%, respectively. The observed ones for condition 10S are 8.03%, 4.55%, 2.95%, and 4.02%. Because the critical value is the same as for the diagonal directions, we can conclude that only the directions in the 0° (to right) bin occurred significantly more frequent. For condition 10M the percentages are 8.03%, 4.82%, 4.15%, and 5.76%. Thus, directions close to 0° were again significantly more frequent, but also those close to 270° (down).

Concerning the lengths, it can again be seen that in condition 10M more paths are of short or medium length, compared to 10S. Moreover, most of the longest paths in condition 10M have a direction close to the diagonal directions of 225° or 315°.

Sequence of locations

To extend the analyses of the sequential selection process of locations beyond that of pairs of consecutive dots, the intensities were also plotted separately for each of the ten selections in the two experimental conditions. As can be seen in Fig 5, most participants placed the first dot near the minor diagonal in the top-left quadrant. The second dot was then placed more towards the center. This sequence holds for both conditions, which indicates that, at least at the beginning, most participants followed a common temporal pattern to place the dots. Not surprisingly, at later steps the selected locations largely differ between participants.

For analyzing how persistently the dots were placed near the diagonals, the correlations of the two-times folded coordinates of the dots were computed separately for each of the ten sequential superimposed dot patterns. The result is shown in panel A of Fig 6. As can be seen, the correlation for the first dot is relatively high for both conditions (10S: r = .423; 10M: r = .569). For the next dot it is considerably lower. Later, the correlation rises and falls again. In condition 10S the maximum correlation (.448) was reached at dot 4. Overall, it seems that the correlation does not vary randomly across the sequence of dots but rise and fall like a damped wave line, where the wavelength is slightly larger for condition 10S.

Fig 6 — Correlations of the two-times folded coordinates (panel A) and p-values of the quadrat (CSR) tests (panel B) for the successive ten superimposed dot patterns in the two experimental conditions in Experiment 1. The dashed line in the upper panel indicates the critical value for a significance level of .05, i.e., values above this line are significant at this level. In the lower panel the dashed line indicates p = .01.

Inspecting correlations of the dot coordinates is rather selective because they only detect the preference of the diagonals, and only if the individually preferred location near the diagonals varies to some extent across participants for a given dot in the sequence. If participants commonly prefer the center or a specific corner, then the correlation remains relatively small. Therefore, to examine to what extent the dots have generally been placed at commonly preferred locations in the sequence, a CSR test was computed for each of the sequential dot patterns. The dots in each pattern were summarized by the nine quadrats of a 3×3 grid. Given the relatively small number of dots in the sequential patterns, a finer grid was not applicable. As mentioned, the CSR test provides the probability p that the pattern is compatible with a homogeneous Poisson process. The obtained p values for the two experimental conditions are shown in panel B of Fig 6. Even if we choose a significance level of p = .01, it can be concluded for condition 10S that the participants placed the first four dots in locations also preferred by many of the other participants (p for dot number 5 is .027). Only later they selected less commonly preferred locations. In condition 10M, the desynchronization already started after the first two dots (p for dot number 3 is .023).

Discussion

In this experiment the participants were instructed to simulate rain by randomly placing 10 dots in a white square area on the screen. In a single-dot condition each dot had to be placed in an empty square area, whereas in a multiple-dots condition the already placed dots remained visible. The general question was how the resulting dots are distributed in the different conditions. From the literature it is already known that location selection is biased by the hidden structure of the area [22, 41]. For a square area this means that the regions near the diagonals, axes, and center are highly attractive for being selected. This attraction was also observed in the present experiment. In addition to placing more dots in the upper half of the area than in the lower one, the participants placed dots frequently in the mentioned regions, while outer zones were largely avoided (see Fig 1). As expected, the regional attractions were more pronounced in the multiple-dots conditions, i.e., when the already placed dots remained visible.

The preference for the diagonals was also confirmed by the significant correlations of the two-times folded coordinates of the dots. Compared to those reported in Lisanby and Lockhead [22], though, they were much smaller, even in the single-dot condition (0.23 vs. 0.77). However, the 195 participants in that study had to place only one single dot. If we consider only the pattern for the first dot in the present experiment, then the correlation increases to .42 in condition 10S, and even to .57 in 10M. Possible reasons for why these correlations are higher than the average ones will be discussed below. At the moment it is important to note that the dot patterns for both experimental conditions deviate significantly from complete spatial randomness, i.e., from patterns generated by a homogeneous spatial Poisson process. An example of a pattern (10R) that does not significantly deviate is shown in the last row of Fig 1.

The underlying structure of the dot patterns is roughly reflected by the intensity maps (column Intensity in Fig 1). Obviously, the highest intensity is at the center, especially for the single-dot condition. Moreover, for this condition there is also an additional large peak on the minor diagonal. Interestingly, the two highest peaks correspond to the positions C3 and B4 in Falk, Falk [19], respectively, which were also marked most frequently in their 5×5 grid. As expected, the regional bias is somewhat reduced when previously placed dots remain visible.

Thus, the overall results support the notion that people are largely biased in selecting random locations within an area, especially when there are no other visible dots. In the modeling section it has been shown how the regional attraction of the center, the diagonals, and the axes might be recovered from the intensity maps of the different conditions. The obtained master maps of regional attraction are shown in Fig 1 (column Attraction Model). By considering the parameter values of the individual component maps (see Fig 2) it is obvious that the center map plays a major role. Interestingly, different from what one might have expected, the center map is less responsible for the high peak of attraction at the center, but more for the attraction at some distance to the center that is neither explained by the attraction of the diagonals nor by that of the axes. This especially holds for the multiple-dots condition. It should be noted, though, that the relation between the regression coefficients should be interpreted with caution. The large value for the center map of attraction is mainly due to the broad distribution. The applied scaling, so that its overall attraction sums up to one, leads to rather small values, which then require a large amplification by the corresponding regression coefficient.

A further result is that not only the dot patterns of the two conditions differ, but also the planning and realization of the dot sequences. This is shown by analyzing the sequential selections. When pairs of successive dots were connect by lines for each participant, as in Gemelli and Alberoni [42] and the superimposed lines of all participants are considered (Fig 3), then it is obvious that there are more shorter lines for the multiple-dots condition. As a consequence, the corresponding histogram of path length is more positively skewed, compared to that for the single-dot condition. Usually, the distribution of random line lengths approximates a symmetric triangular distribution [46]. However, given the restrictions of a square area, where longer paths are possible between the corners, this is not the case. A corresponding computer simulation revealed that for the present case the distribution of the random path length is positively skewed.

By considering Fig 3 it can also be seen that not only the dots were placed near the diagonals, but that the paths between successive dots also often run along them, or at least in parallel. How the path directions are distributed in each condition can be seen in the ‘wind-rose’ diagrams (Fig 4). For the single-dot condition there was a high percentage of paths directed around 225° (down left) and 0° (to right). In contrast, for the multiple-dots condition, frequent paths occurred only in bins around two axial directions 0° and 270 (down). Thus, disproportionately many paths with a diagonal direction occurred merely in the single-dot condition, whereas paths directed from left to right occurred frequently in both conditions.

The differences in path length and direction further indicate that the sequential selection process of locations differed between the two conditions. They suggest that in the single-dot condition the participants tried to simulate a spatial random process, whereas in the multiple-dots condition a spatial random pattern of dots was planned for which the realization of a random sequence of locations played a subordinate role. It should be noted that, different from Ross and Weiner [34], the participants were not explicitly instructed to also select a random sequence of locations. However, it is likely that they, at least for the single-dot condition, implicitly assumed that locations of subsequent dots should be independent.

In addition to the pairwise spatial relation between successive dots, the superimposed dot patterns were also analyzed separately for each sequential dot. To examine how persistently the dots were placed near the diagonals, the correlations of the two-times folded coordinates of the dots were computed for each of the ten dots. As can be seen in panel A of Fig 6, for both conditions the correlation for the first dot is significantly high, but then it dops below significance for the second dot. Overall, it seems that the correlations rise and fall across the dot sequence like a damped wave, which indicates a negative recency effect. The participants first selected a location near a diagonal, but next some location off the diagonals. Later dots were again placed near the diagonal and so on.

A surprising result is that the first dot was commonly placed in a rather restricted area. If we consider Fig 5, then it is obvious that most participants placed the first dot along the minor diagonal in the upper-left quadrant, which holds for both experimental conditions. Thus, whereas the overall pattern, in particular that for the single-dot condition, looks similar to that observed in other studies where only one single dot had to be placed, this is not the case for the patterns of the first dot alone. This indicates that the number of dots to be placed strongly influenced the location of the first dot. The fact that the participants largely agreed to start near the minor diagonal in the left upper quadrant, indicates that the sequence of dot locations was planned in advance, at least to some extent.

To examine whether this performance depended on the order of the conditions, which was randomized, both orders were also analyzed separately. Of the 35 participants who started with the single-dot condition, 20 (57%) placed their first dot in the upper left quadrant in that condition, and 21 (60%) did this in the subsequent multiple-dots condition. Of the 48 participants who started with multiple-dots condition, 35 (73%) placed the first dot in the upper left quadrant in that condition, and 23 (48%) did this in the subsequent single-dot condition. This demonstrates that in both conditions there was a strong tendency to place the first dot near the minor diagonal in the upper left quadrant irrespective of the condition and their order.

The fact that the correlation of the two-times folded coordinates of the dots was nevertheless relatively high already for the first dot, although only a small region of the diagonals was used, shows that a high correlation has to be interpreted with care. Placing many dots along one half of one diagonal already leads to the same high correlation as when the dots were evenly scattered along both diagonals. Folding the dot coordinates two times makes the difference in performance disappear. Thus, a high correlation does not guarantee that the dot pattern looks like an oblique cross.

The correlation measure is also unsuitable for detecting preferred regions other than those along the diagonals, like the axes or the center. Thus, an absent correlation in a two times folded dot pattern does not imply the absence of a commonly selected region. To examine for each sequential dot patten whether such regions were present, it was tested to what extent the patterns are compatible with a homogenous Poisson process. The p-values for the intensities in Fig 5 are shown in panel B of Fig 6. As can be seen, in the single-dot condition the participants placed at least the first four dots in locations also preferred by other participants. Only then they selected less commonly preferred locations. In the multiple-dots condition desynchronization already started after the first two dots. This indicates that the first dots were systematically placed at commonly preferred locations, although not necessarily at the diagonals. Only then less commonly preferred locations were selected. In the multiple-dots condition, less common locations were already selected somewhat earlier, which demonstrates that the visibility not only affected the overall patterns but also the strategies of location selection.

Experiment 2

After basic results on position biases in simple selection conditions were obtained in Experiment 1, this experiment served for examining two further issues. First, in an extended multiple-dots condition the effects of an increased number of dots, compared to Experiment 1, should be examined. As we have seen, if ten dots have to be placed, they cluster in central regions, while outer areas are largely ignored. This might change with more dots. As mentioned, Ross and Weiner [34] required to mark 64 cells in a 20×20 grid and observed rather regular patterns. Similar results were observed for the free placement of 20 dots by Gemelli and Alberoni [42]. In the present experiment the participants had to place 30 dots in a multiple-dots (30M) condition. It was expected that the dots are now more evenly distributed across the whole area. This does not necessarily mean that the patterns are compatible with a homogeneous Poisson process, because they might be too regular now. As can be seen in Fig 1 (10R), a homogeneous Poisson process does produce clusters and empty spaces, but unsystematically.

The second aim of this experiment was primarily methodological in scope. In Experiment 1, the randomness of the superimposed dot patterns was assessed by testing for complete spatial randomness, i.e., whether they are compatible with a homogeneous spatial Poisson process. However, by doing this, an important property of the Poisson process was ignored. Such a process does not only produce dots at random locations, but also a random number of dots, which follows from its two key properties mentioned in the Introduction, homogeneity and independence [44]. Thus, by requiring the placement of a fixed number of dots in an area, the latter assumption was violated. Furthermore, the violation also holds for the random-dot pattern in Fig 1 (10R). By producing a fixed number of uniformly distributed dots, a binomial point process was simulated rather than a Poisson point process [44]. This raises the question whether the violation of independence might lead to invalid conclusions in the present study. The fact that the CSR test was not significant for the random pattern suggests that it does not. Nevertheless, this issue should be tested more directly in the present experiment. Therefore, a specific multiple-dots condition (XM) was included, where the participants were free to place as many dots as the wanted. This allowed them to simulate a spatial Poisson process. Moreover, because the superposition of independent Poisson processes is also a Poisson process, it is now possible that the superimposed dot pattern is indeed compatible with a spatial Poisson process. Accordingly, it is justified to assess its homogeneity with the CSR test.

For condition XM it was also of interest to see over which range the number of produced dots would extend. To prevent any systematic influence on that range, all participants started with that condition.