Errors of Analysis in Distance Assessment: Effects of Vertical on Horizontal Components

Abstract

(a) Participants indicated, for pairs of circles whose locations varied on the horizontal and vertical axes of a frontal plane, whether the horizontal distance between the circles exceeded a target horizontal distance. The error rate depended on the vertical as well as the horizontal distance between the circles. (b) Participants indicated, for pairs of circles that were varying horizontal (or vertical) distances and a constant vertical (or horizontal) distance apart in a frontal plane, whether the horizontal (or vertical) distance between them matched a target horizontal (or vertical) distance. Incorrect “match” responses were more likely if the horizontal (or vertical) distance between the circles was less than as opposed to greater than the target distance. The results suggest that distance judgments for pairs of stimuli varying on the horizontal and vertical axes are based on the overall distance between the stimuli, with the relevant axis given more weight than the irrelevant axis in assessment of the distance. The results do not support the view that that such distance judgments are based on the relevant distance between the stimuli, with the relevant and irrelevant axes being erroneously interchanged on some iterations of the assessment process.

Keywords

distance perception attention dimensional interaction dimension weighting reference frame dimensional analysis

The human visual system is quite accurate at assessing the distance between pairs of stimuli in a frontal plane. The power function obtained in scaling such distances has an exponent close to 1 (Stevens, 1975), and the system is capable of detecting small differences between such distances (Klein & Levi, 1985; Wilson, 1986). Less is known about the process by which the visual system analyzes the distance between a pair of frontal stimuli into horizontal and vertical components. For example, consider the circles shown in the left panel of Figure 1. How accurately does the visual system assess the horizontal or vertical distance between these circles? And how does the visual system perform the analysis?

Figure 1.
Stimuli for Distance Classification Task of Experiment 1.

The questions are of practical and theoretical interest. For example, allocentric memory representations are thought often to be organized in terms of horizontal and vertical axes (McNamara, 2013; Rump & McNamara, 2013). Encoding the locations of objects in such reference frames often requires analyzing the distances between objects into horizontal and vertical components. As another example, recent models assume that pointing responses are directed to Cartesian locations, where such locations are most simply represented as Cartesian distances from current Cartesian locations (Biess et al., 2007).

Perceptual analysis such as is required here has long been of interest in psychology and has been typically studied with the perceptual classification task (Garner, 1974; Kemler Nelson, 1993; Melara & Marks, 1990; Wagemans et al., 2012). In this task, the participant classifies stimuli in terms of their values on a “relevant” perceptual dimension. The critical question is whether dimensional interaction occurs—whether the values of the stimuli on “irrelevant” dimensions influence classification performance. Pairs of dimensions that show dimensional interaction are called integral; pairs of dimensions that do not show dimensional interaction are called separable (Maddox, 1992).

Two of the explanations that have been given for dimensional interaction will be important here. The explanations differ in how deeply they involve information about irrelevant dimensions in the perceptual classification process. Under the Exemplar-Based Random Walk Model, the irrelevant and relevant dimensions jointly articulate the similarity space that grounds performance of the classification. Thus, both relevant and irrelevant information are involved at a fundamental level (Little et al., 2016; Nosofsky & Palmeri, 2015). A stimulus is classified with respect to a particular dimension on the basis of its relative similarity to the exemplars of classes that have been distinguished on this dimension. Similarity is assessed in terms of the space mentioned earlier, with its dimensions weighted to reflect their importance for the classification. Dimensional interaction occurs to the degree that irrelevant dimensions are given weight in the similarity assessment. Under the dimensional similarity hypothesis, in contrast, information about irrelevant dimensions is involved in perceptual classification because the dimensions of the stimulus space are imperfectly articulated (Jones & Goldstone, 2013; Ons et al., 2011). Dimensional interaction occurs to the degree that information regarding the irrelevant dimension is erroneously accessed for the test stimuli (Melara, 1992; Melara & Day, 1992; Melara et al., 1993). Thus, irrelevant information is less fundamentally involved here. The contrast between the preceding two explanations is highlighted in a recent summary of perceptual analysis research that characterizes the two explanations as emerging from traditions focused on divided and selective attention, respectively (Algom & Fitousi, 2016).

The analysis of frontal distance has been addressed only indirectly in previous work. Previous work has explored the classification of spatial position in terms of horizontal and vertical components (Garner & Felfoldy, 1970). This work does not directly generalize to the analysis of distance, however, because position and distance may dissociate (Abrams & Landgraf, 1990). Previous work has also explored the classification of rectangles in terms of width and height (Macmillan & Ornstein, 1998). Again, generalization is problematic because rectangles may be perceived in terms of higher order dimensions such as size and shape (Krantz & Tversky, 1975).

Recently, the analysis of frontal distance has been explored more directly (Dopkins, 2005; Dopkins & Galyer, 2018). The test items were pairs of small circles whose locations varied on the horizontal and vertical axes of a computer screen. Under speed stress, participants indicated whether the horizontal (or vertical) distance between the pairs of circles was greater than a target distance. The error rate in this distance classification task depended on the irrelevant as well as the relevant distance between the circles. Thus, the pattern resembled dimensional interaction. Because the axes of a spatial reference frame are not strictly psychological dimensions, the authors spoke of axis interaction.

The present study generalized axis interaction to unspeeded tasks, assessed the impact of such interaction on the precision and bias of distance judgments, and tested two hypotheses regarding its underlying mechanism. According to the Overall Distance Hypothesis (ODH), derived from the Exemplar-Based Random Walk Hypothesis, the human visual system is incapable of directly assessing the horizontal or vertical distance between two stimuli that vary on both axes; the system is only capable of assessing the overall Euclidean distance between the stimuli (the possibility is later considered and discounted that the system assesses the overall city block distance between two stimuli). The system indirectly assesses horizontal or vertical distance by differentially weighting the relevant and irrelevant axes, so as to reduce the contribution of irrelevant distance, before assessing overall distance. Axis interaction occurs if the contribution of irrelevant distance is not reduced to zero (fuller instantiations of the hypotheses are given in Appendix).

According to the Axis Interchange Hypothesis (AIH), derived from the dimensional similarity hypothesis, the human visual system is capable of directly assessing the horizontal or vertical distance between two stimuli that vary on both axes. However, the relevant and irrelevant axes are erroneously interchanged on some iterations of the sequential sampling process in which distance is assessed (Ratcliff, 2014). Axis interaction occurs because of this interchange. The ODH and the AIH derive from very different approaches to dimensional interaction, as were articulated earlier.¹ Choosing between them may have implications for an understanding of distance assessment, as is discussed later.

Experiment 1

Experiment 1 asked whether axis interaction occurs in the unspeeded classification of frontal distances, whether such interaction affects the precision and/or bias of unspeeded classification, and whether the ODH or AIH best accounts for such interaction. On each trial, two small circles appeared on a computer screen with each of the circles occupying one of a 6 × 6 evenly spaced horizontal-by-vertical array of locations (see Figure 1). Participants indicated whether the circles were less than three (<3) or three or more (≥3) horizontal positions apart in terms of the location array. The task was stated in terms of horizontal position in the location array to make clear that responses were to be based on horizontal distance alone. More concretely, a horizontal position was the horizontal distance between the centers of test circles in adjacent columns of the location array; for example, circles appearing in the first and second columns of the array were one horizontal position apart. The reference for horizontal orientation was the frame of the computer screen. To reinforce this reference, a test circle array, comprising circles in all 36 tested locations was presented at the beginning of each trial block. Participants were instructed to strive for accuracy rather than speed. To be sure that the participants understood the task, feedback was provided following errors. Notice that the experimental task required participants to discriminate pairs of circles that were greater than and less than the average horizontal distance (2.5 positions) apart. Participants are quite capable of performing this sort of discrimination, in which the standard is implicit and held in memory (Morgan, 1992; Morgan et al., 2000; Westheimer & McKee, 1977).

If axis interaction and an effect of irrelevant distance were observed, the plan was to ask whether this interaction affected the precision and/or bias of the classification and to pit the ODH against the AIH as accounts of the results. The ODH held that the irrelevant distance effect would be larger for <3 than for ≥3 pairs. This followed because responses would depend on the overall Euclidean distance between the test circles and because the Pythagorean theorem dictated that values of overall Euclidean distance for adjacent levels of irrelevant vertical distance would differ by larger amounts for <3 than for ≥3 pairs. In contrast, the AIH held that the irrelevant distance effect would not differ in size for <3 and ≥3 pairs. This followed because the same levels of irrelevant vertical distance would be erroneously interchanged with horizontal distance for <3 and ≥3 pairs.

Method

Participants

Thirty-two undergraduates (21 females) participated for credit in a psychology class. A GPower analysis based on an effect size from pilot work indicated that this sample would provide power >.95 for testing the ODH prediction that the effect of irrelevant distance would be larger for <3 than for ≥3 pairs (Faul et al., 2007). As for all experiments in the study, participants had normal or corrected-to-normal vision, participated in only one experiment, were naïve as to the purpose of that experiment, and gave written consent in accordance with the guidelines of the George Washington University Institutional Review Board, #091626.

Design

The design roughly equated the numbers of test pairs for the different combinations of horizontal and vertical position. One by-product of this approach was that low levels of horizontal distance (for which a negative response was appropriate) were tested more frequently than high levels (for which a positive response was appropriate). Thus, the design fostered a conservative response bias. The test pairs for a given participant were created as follows: (a) 180 pairs were created with horizontal distance equal to 0 by crossing each of the 6 horizontal positions in the array with the 30 ordered pairs of vertical position that could be formed by sampling twice, without replacement, from the 6 vertical positions in the array; (b) 180, 144, 108, 72, and 36 pairs were created with horizontal distance equal to 1, 2, 3, 4, and 5 by (i) forming the 15 unordered pairs of horizontal position that could be formed by sampling twice, without replacement, from the 6 horizontal positions in the array, where the resulting unordered pairs comprised 5, 4, 3, 2, and 1 pair(s) for which horizontal distance was equal to 1, 2, 3, 4, and 5, respectively and (ii) crossing each of these horizontal position pairs with the 36 ordered pairs of vertical position that could be formed by sampling twice, with replacement, from the 6 vertical positions in the array. The test pairs for each participant were presented in a different random order.

Stimuli

The stimuli were presented on the screen of a PC computer. The test circles were drawn in back pixels on the white screen. Each circle was 3 mm in diameter. Circles in horizontally and vertically adjacent locations of the 6 × 6 location array were separated by 8 mm from edge to edge. The location array was centered on the screen. The inner edge of the screen frame, the primary horizontal/vertical reference, measured 34 × 27 cm. Each test circle was therefore an average of 11.85 cm from the primary horizontal reference and 15.35 cm from the primary vertical reference. The test circle array* that was presented at the beginning of each block extended 5.8 cm on the horizontal and vertical axes.

Because participants sat approximately 60 cm from the computer screen, each test circle subtended a visual angle of approximately 0.29 degrees, circles in adjacent locations of the location array were separated by an angle of approximately 0.76 degrees, the inner edges of the screen frame subtended angles of approximately 31.64 and 25.36 degrees, each test circle was an average of approximately 11.28 degrees from the primary horizontal reference and 14.58 degrees from the primary vertical reference, and the test circle array subtended an angle of approximately 5.53 degrees on the horizontal and vertical axes.

Procedure

Participants viewed the stimuli binocularly, under standard fluorescent lighting. At the beginning of each trial, “Ready” appeared at the center of the computer screen. When participants pressed the space bar of the computer, “Ready” disappeared, and a pair of circles appeared. Participants indicated, with the “B” and “N” keys, respectively, whether the test circles were less than 3 or 3 or more horizontal positions apart in terms of the location array. The trials were presented in 30 blocks of 24 trials apiece. The test circle array was presented at the beginning of each block.

Results and Discussion

The data and analysis code for this and all experiments of the study are available through the Open Science Framework. The error rate in the task was examined as function of the distance between the test circles (a) on the horizontal axis, which was relevant to the classification and (b) on the vertical axis, which was irrelevant to the classification. The error rate for <3 pairs increased with increases in the relevant and irrelevant distance between the test circles. The error rate for ≥3 pairs decreased with increases in the relevant and irrelevant distance between the test circles. The effect of irrelevant distance was informally noted to be greatest when the relevant distance was closest to the average distance (the critical distance for the judgment; see Table A1 of the Appendix for the complete results of the analysis). Although the response-time data were not of primary interest, no speed-accuracy trade-off was present; the correlation of response time and error rate across participants was positive, t(31) = 16.87, p < .0001 (average coefficient: .53). The present pattern, an instance of axis interaction, has previously been observed for speeded judgments of horizontal and vertical distance and of distance along the axes of an oblique frame (Dopkins & Hoyer, 2015; Dopkins & Sargent, 2014; Varner et al., 2014). Here, the pattern is observed for unspeeded judgments of horizontal distance.

To find out how the observed axis interaction affected precision and bias in the task, the probability of a “≥3” response was plotted for each participant as a function of relevant distance at the different levels of irrelevant distance (see Figure 2 for plots aggregating across participants). Under the assumption that the resulting functions were cumulative normal, the z score method of Gescheider (2013) was used to compute, for each participant, the difference threshold as an index of precision and the point of subjective equality as an index of bias (it would not have made sense to plot an entire psychometric function, including guess and lapse rates (Wichman & Hill, 2001). To illustrate the problem with this, although the probability of a “≥3” response for Relevant Distance 0 increased with increases in irrelevant distance, under the ODH, this was not because the guess rate increased but rather because the irrelevant distance, and thus the overall distance, between the test circles increased. Across participants, the difference threshold increased, F(1, 31) = 4.35, mean square error (MSE) = 0.13, p < .05, and the point of subjective equality decreased with increases in irrelevant distance, F(1, 31) = 17.76, MSE = 0.48, p < .001 (see Figure 3). By implication axis, interaction decreased precision and induced a less conservative bias in the discrimination task.

Figure 2.
Experiment 1: Mean Probability, Across Participants, of a “≥3” Response as a Function of the Relevant Horizontal and Irrelevant Vertical Distance Between the Test Circles.

Figure 3.
Mean Values, Across Participants, of Difference Threshold and PSE.

To compare the size of the irrelevant distance effect for <3 and ≥3 pairs, the error rate in each participant’s data was regressed on irrelevant distance for the two sets of pairs. Across participants, the average coefficient for the error-rate data was larger for <3 (.34) than for ≥3 (.19) pairs, t(31) = 3.05, p < .005. By implication, the irrelevant distance effect was larger for <3 than for ≥3 pairs, supporting the prediction of the ODH but not the AIH. To assess the impact of practice on the irrelevant distance effect, the size of the effect, as given by the regression coefficients for <3 and ≥3 pairs, was compared in the first and second halves of the experiment. Neither the effect for <3 pairs, t(31) = 0.33, nor the effect for ≥3 pairs, t(31) = 0.13, differed in the two halves of the experiment.

To further pit the ODH against the AIH, the data for each participant were fit in terms of the two hypotheses. The appendix gives details of the model fits. The ODH fit the data better than did the AIH. Across the individual participant fits, the average value of G was lower for the ODH (118) than for the AIH (132), t(31) = 3.59, p < .0001.

Finally, the decreases in precision and conservatism of bias that were observed as a function of irrelevant distance are consistent with the ODH.

The precision result follows because, as dictated by the Pythagorean theorem, the values of overall Euclidean distance that corresponded to adjacent levels of relevant distance in Experiment 1 differed less as the level of irrelevant distance increased. The bias result follows because, under the ODH, bias in the task of Experiment 1 was determined by the decision criterion, which was a value of overall Euclidean distance, and because, as the level of irrelevant distance increased, a given value of overall Euclidean distance corresponded to decreasing levels of relevant distance.

Experiment 2

Axis interaction induced a bias in the classification task of Experiment 1. Bias of this sort often causes nonveridical perception, giving rise to what are sometimes called visual “illusions” (Morgan et al., 1990). The bias effect of Experiment 1 is difficult to interpret in these terms given that bias was also influenced by the relative numbers of <3 and≥3 pairs. Experiment 2 asked more directly whether the bias induced by axis interaction causes nonveridical perception. In addition, the experiment again pitted the ODH against the AIH.

The experiment used a distance matching rather than a distance classification task. On each trial, participants saw (a) a target segment that ran parallel to the horizontal base of the computer screen and (b) two test circles whose location differed on the horizontal and vertical axes of the screen (see left panel of Figure 4). Participants indicated whether the test distance—the horizontal distance between the test circles (the distance between X and Y in the left panel of Figure 4)—matched the target distance—the distance between the end points of the target segment. Notice that, as a side benefit, the target segment provided a very clear reference with respect to horizontal orientation. As in Experiment 1, the task was unspeeded.

Figure 4.
Left panel: Experiments 2 and 3: Stimulus configuration. Only the test circles and the target segment were presented to participants. Middle and right panels: Experiments 2 and 3: Test distances corresponding to a given target distance.

For each target distance, the test distance varied through 11 levels, with one of the levels being Equivalent to the target distance, with five of the levels being Smaller than the target distance, with five of the levels being Larger than the target distance, and with the Smaller and Larger test distances being equally different from the target distance. The vertical distance between the test circles was held constant across the experiment (see middle panel of Figure 4). Interest focused on incorrect “match” responses. The experiment asked whether bias induced by axis interaction caused a higher, nonveridical, rate of these responses to Smaller than to Larger items.

The ODH predicted this result by the following rationale: Responses will be based on the overall Euclidean distance between the test circles. In the assessment of this distance, the horizontal and vertical axes will be differentially weighted so as to reduce the contribution of irrelevant vertical distance. The results of Experiment 1 (see Appendix) suggest that differential weighting will not completely remove vertical distance from consideration. Because all pairs of test circles will be separated by a constant vertical distance, the test distance for all pairs of circles will be overestimated. Because the test distance will be, respectively, less than and greater than the target distance for Smaller and Larger items, the test distance will be perceived as being more nearly equivalent to the target distance for Smaller than for Larger items.

In contrast, the AIH predicted that the rate of incorrect “match” responses would not differ for Smaller and Larger items, by the following rationale: Responses will be based on the horizontal distance between the test circles, with vertical distance being erroneously exchanged for horizontal distance on some iterations of the distance assessment process so that responses will be based to some degree on vertical as opposed to horizontal distance. Under the design of the experiment, the vertical distance between the test circles will be held constant across Smaller and Larger items. As a consequence, responses will be erroneously based on the same vertical distances for Smaller and Larger items. Thus, the test distance will be perceived as equivalent to the target distance with equal likelihood for Smaller than Larger items.

Method

Participants

The 32 participants (18 females) were drawn from the same pool as was used in Experiment 1. A GPower analysis based on an effect size from pilot work indicated that this sample size would provide a power >.95 for testing the ODH prediction.

Design

In each item, a target distance was paired with a test distance (see middle panel of Figure 4). Five target distances were used: 94, 109, 124, 139, and 154 mm. For each target distance, an Equivalent item was created by setting the test distance equal to the target distance. Each of these items was tested 20 times (the number of Equivalent items was kept relatively small because the plan was to compare the Smaller and Larger items to one another rather than to the Equivalent items). For each target distance, five Smaller items were created by setting the test Distance 6, 12, 18, 24, and 30 mm smaller than the target distance. Each of these items was tested four times. For each target distance, five Larger items were created by setting the test Distance 6, 12, 18, 24, and 30 mm larger than the target distance. Each of these items was tested four times. The vertical distance between the test circles was held constant at 50 mm across all items.

Stimuli

If the test segment* is defined as delineating the test distance, the leftmost point of the test segment (X in the left panel of Figure 4) was always located 50 mm to the right of and 50 mm above the leftmost point of the target segment. The test circles were the same as for Experiment 1. Because participants sat approximately 60 cm from the computer screen, each 6 mm increment added to or subtracted from a target distance in the process of creating a test distance subtended a visual angle of approximately 0.57 degrees.

Procedure

At the beginning of each trial, the target segment was presented. When participants pressed the space bar of the computer, the two test circles were presented. Participants were instructed to push the “B” key if the horizontal distance between the test circles was the same as the distance between the end points of the target segment and the “N” key if it was not. Participants were encouraged to respond accurately and received feedback after making errors. In other respects, the procedure followed Experiment 1.

Results and Discussion

The probability of a “match” response varied with the test distance (see Figure 5), F(10, 310) = 19.97, MSE = 0.14, p < .0001, showing a significant quadratic trend, F(1, 31) = 158.54, MSE = 0.44, p < .0001. The probability of a “match” response did not vary with target distance, F(4, 124) = .72, MSE = 0.06, p = .585. A planned comparison showed that the probability of a “match” response was greater for Smaller than for Larger items, t(31) = 4.82, p < = .00005. The Smaller/Larger difference in the probability of a “match” response did not differ in the first and second halves of the experiment, F(1, 31) = 3.07, MSE = 0.006, p = .09. No speed-accuracy trade-off was present; across participants, the correlation of response time and error rate was positive, t(31) = 3.78, p < .001 (average coefficient: .19).

Figure 5.
Experiment 2: Mean Probability of a “Match” Response as a Function of Test Distance.

The higher probability of a “match” response for Smaller than Larger items suggests that the test distance (the horizontal distance between the test circles) was more likely to be mistaken for the target distance if it was smaller than as opposed to larger than the target distance. By implication, participants overestimated the horizontal distance between the test circles. The lack of an effect of target distance on the probability of a “match” response argues against any Roelof-like localization effects. The key fact here is that the target segment extended across different portions of the screen for different target distances. If localization effects influenced the outcome, one would have expected these effects to have effects of different magnitude for different target distances.

To confirm the forgoing interpretation of the results of Experiment 2, several follow-up studies were run. On each trial of Experiment 2A, a pair of the test circles from Experiment 2 and a horizontal distance scale was presented. Participants assessed the horizontal distance between the test circles in terms of the scale. Across participants (N = 20) and items, the mean of the assessed horizontal distances was 137.44 mm (standard error of the mean [SEM] across participants = 4.75). This was significantly greater than the mean of the actual horizontal distances, which was 124 mm. In short, horizontal distance was overestimated. In Experiment 2B, 20 participants performed the task of Experiment 2 with all stimuli being rotated 90 degrees clockwise so that the critical judgments concerned vertical distance. The probability of (incorrect) “match” responses was again higher for Smaller than for Larger items, t(19) = 3.97, p < .001, implying that the present results hold regardless of whether the horizontal or vertical axis is relevant to the match.

An alternative interpretation of the present results proposes that participants performed the task on the basis of the relevant horizontal distance between the test circles and that they estimated the horizontal distance by projecting (a) the segment connecting the test circles onto (b) the horizontal axis implied by X and Y in the left panel of Figure 4. Under this interpretation, the results reflect the fact that participants underestimated the angle implied by the segment connecting the test circles and the horizontal axis implied by X and Y. Because participants underestimated this angle, they overestimated the horizontal distance between the test circles. Experiment 2C tested this interpretation. Twenty participants were presented with the test circles, as in Experiment 2, and asked to estimate the size of the angle formed by the implicit segment connecting the circles and the horizontal axis of the computer screen. Across participants and item types, the mean of the assessed angle was 32 degrees (SEM across participants = 3.81). This was significantly greater than the mean of the actual angle, which was 24 degrees. These results offer no support for the proposed interpretation.

The difference between the probability of an (incorrect) “match” response for Smaller and Larger items confirms the prediction of the ODH and argues against the AIH, which predicted no such difference. In addition, the results show that axis interaction can induce bias that causes nonveridical distance assessment. Asked to assess the horizontal distance between the test circles, participants apparently overestimated this distance consequent to such interaction.

Experiment 3

But can we be sure that this overestimation reflects axis interaction such as is proposed under the ODH? Perhaps horizontal distance was assessed correctly, and performance was biased by a factor independent of that assessment. One possible such factor is the overall distance between the test circles. More concretely, participants may have correctly assessed the horizontal distance between the test circles but been biased by the overall distance between the circles to respond as if they had overestimated the horizontal distance. Experiment 3 tested this possibility by replicating the results of Experiment 2 while equating the average overall distance in the Smaller and Larger conditions.

Participants performed the same task as in Experiment 2 with a different stimulus set. Each item again comprised a target segment and two test circles, with one circle located downward and to the right relative to the other circle. Rather than remaining constant across trials as in Experiment 2, the vertical distance between the test circles was negatively correlated with the horizontal distance between them: In the Equivalent item for a given target distance, the test circles were the target distance apart on the horizontal and vertical axes. Each Smaller item for that target distance was created by shifting the rightmost test circle of the Equivalent item the same amount horizontally closer to and vertically more distant from the leftmost test circle. Each Larger item for that target distance was created by shifting the rightmost test circle of the Equivalent item the same amount horizontally more distant from and vertically closer to the leftmost test circle (see right panel of Figure 4).

The ODH predicted a higher probability of (incorrect) “match” response for the Smaller than for the Larger items. As in Experiment 2, this followed because responses in the task would be based on the overall Euclidean distance between the test circles and because vertical distance would be given nonzero weight in the assessment of this overall distance so that the test distance would be overestimated. Because the test distance would be, respectively, smaller than and larger than the target distance for the Smaller and the Larger items, the test distance would be perceived as more nearly equivalent to the target distance for the former than the latter items (see right panel of Figure 4).

However, the ODH predicted a smaller difference between the Smaller and Larger items of Experiment 3 than was observed in Experiment 2. Recall that the vertical distance between the test circles was constant across the Smaller and Larger items in Experiment 2. In Experiment 3, in contrast, the vertical distance would be, respectively, larger and smaller for the Smaller and Larger items than it had been for the Smaller and Larger items of Experiment 2. As a result of these differences, the overall Euclidean distance between the test circles, and thus the probability of a “match” response, would differ less for the Smaller and Larger items of Experiment 3 than it had for the Smaller and Larger items of Experiment 2 (see right panel of Figure 4).

At the same time, the AIH predicted no difference in the probability of a “match” response for the Smaller and Larger items. In the Equivalent item for a given target distance, the test circles would be the target distance apart on the horizontal and vertical axes. The Smaller and Larger items for that target distance would have five vertical distances that were the same amount, respectively, smaller and larger than the target distance (see right panel of Figure 4). Thus, the vertical distance between the test circles would, on average, be same amount different from the correct target distance for the Smaller and the Larger items. Thus, if vertical axis was interchanged with the horizontal axis, a “match” response would be equally likely for the Smaller and Larger items.

Finally, the probability of a “match” response would not differ for the Smaller and Larger items if the results reflected only bias from overall distance. This followed because the average overall distance between the test circles would be equivalent for the Smaller and Larger items. This, in turn, followed because the difference between the Smaller and Equivalent items in terms of horizontal and vertical distance would match the difference between the Larger and Equivalent items in terms of vertical and horizontal distance (see right panel of Figure 4).

Method

Participants

The 32 participants (17 females) were drawn from the same pool as in Experiment 1. A GPower analysis based on an effect size of .5 indicated that this sample size would provide power >.85 for testing the ODH prediction.

Design

Five target distances were used: 80, 82.5, 85, 87.5, and 90 mm. For each target distance, an Equivalent item was created by setting the horizontal and vertical distance between the test circles equal to the target distance. Each of these items was tested 40 times. Five Smaller items were created by moving the rightmost test circle of the Equivalent Item 3, 6, 9, 12, or 15 mm at the same time to the left and downward. Each of these items was tested 8 times. Five Larger items were generated by moving the rightmost test circle of the Equivalent Item 3, 6, 9, 12, or 15 mm at the same time to the right and upward. Each of these items was tested 8 times (see right panel of Figure 4).

Stimuli

The test circles were the same as for Experiments 1 and 2. Because participants sat approximately 60 cm from the computer screen, each 3 mm increment added to or subtracted from a target distance to create a test distance subtended a visual angle of approximately 0.28 degrees.

Procedure

The procedure was the same as for Experiment 2.

Results and Discussion

The probability of a “match” response varied with the test distance (see Figure 6), F(10, 310) = 28.03, MSE* = 0.08, p < .0001, showing a significant quadratic trend, F(1, 31) = 90.62, MSE = 0.15, p < .0001. The probability of a “match” response varied with target distance, F(4, 124) = 15.30, MSE = 0.03, p < .0001, showing significant decreasing linear, F(1, 31) = 26.78, MSE = 0.66, p < .0001, and quadratic, F(1, 31) = 9.62, MSE = 0.18, p < .005, trends. A planned comparison showed that the probability of an (incorrect) “match” response was higher for Smaller than for the Larger items, t(31) = 2.14, p < .05. The Smaller/Larger difference in the probability of “match” response did not vary with the target distance, F(4, 124) < 1. The Smaller/Larger difference also did not differ in the first and second halves of the experiment, F(1, 31) = 1.109, MSE = 0.001, p = .30. No speed-accuracy trade-off was present. Across participants, the correlation of response time and error rate was positive, t(31) = 4.41, p < .001 (average coefficient: .26).

Figure 6.
Experiment 3: Mean Probability of a “Match” Response as a Function of Test Distance.

The higher probability of a “match” for the Smaller than the Larger items confirms the prediction of the ODH over that of the AIH. At the same time, the results are difficult to explain in terms of bias from overall distance, which predicted no difference between the probability of a “match” response for the Smaller and Larger items. The lack of an interaction between the effect of target distance and the Smaller/Larger difference argues against the role of Roelof-like localization effects in the difference. Again, the key fact is that the target segment extended across different portions of the screen for different target distances. If localization effects influenced the Smaller/Larger difference, one would have expected these effects to vary in their impact for different distances. The fact that the Smaller/Larger difference did not vary with target distance suggests that localization effects did not influence this difference.

General Discussion

When participants assessed the horizontal distance between pairs of stimuli in nonspeeded classification and the horizontal or vertical distance between pairs of stimuli in nonspeeded matching, performance depended on the irrelevant as well as the relevant distance between the stimuli. The irrelevant distance effect occurred in the context of error feedback, which, one would think, would have motivated participants to reduce the effect to the extent that they were able. Thus, one might ask how large the effect would have been in the absence of feedback. In fact, however, the effect differed in size across the two halves of none of the experiments and thus showed no evidence of being subject to reduction through feedback. By implication, the effect reflects a fairly constant aspect of distance assessment.

Axis interaction reduced the precision of distance discrimination and induced bias toward the overestimation of relevant distance. The effect on bias is of greatest practical interest in that it presumably affects distance assessment even when precise discrimination is not required. The clearest indication of this effect came from follow-up Experiment 2A in which participants directly overestimated the horizontal distance between stimuli whose locations differed on the horizontal and vertical axes.

To explain axis interaction in distance assessment, the ODH proposes that assessment responses are based on the overall distance between the test stimuli. The AIH proposes that the relevant and irrelevant axes are erroneously interchanged on some iterations of the assessment process. The present results support the ODH over the AIH. In Experiment 1, the effect of irrelevant distance was larger for <3 than for ≥3 pairs. In Experiments 2 and 3, the probability of an incorrect “match” response was greater for Smaller than Larger items. Both of these results were predicted by the ODH but not the AIH. In addition, the ODH fit the data from Experiment 1 better than did the AIH.

In supporting the ODH, these results suggest that the human visual system does not acknowledge the axes of physical space in assessing frontal distance; the system is evidently incapable of directly assessing anything but the overall distance between two stimuli. In pointing out this incapacity, the ODH highlights a fundamental fact about distance assessment that is implicit in some prior accounts. For example, several writers have proposed that the distance between two stimuli is assessed on the basis of the visual angle between the stimuli (Foley et al., 2004). Because the visual angle is stated in terms of polar coordinates, it does not acknowledge the horizontal and vertical axes of a reference frame. Thus, under this view, as under the ODH, the horizontal and vertical distance between two frontal stimuli cannot be directly assessed.

The ODH suggests a means by which this limitation is overcome. The horizontal or vertical distance between two stimuli can be indirectly assessed to the degree that the horizontal and vertical axes are differentially weighted so as to reduce the contribution of irrelevant distance in the assessment of overall distance. The horizontal and vertical axes, though not strictly psychological dimensions, behave here as such. Because the present axis interaction resembles dimensional interaction, one would say that the horizontal and vertical axes behave as integral dimensions. Although the capacity of attention for reliably selecting integral dimensions has been questioned in some past work, this capacity has recently been affirmed (Jones & Goldstone, 2013) and is reaffirmed here.

Although the present study explored reference frames corresponding to the axes of physical space, the results may generalize to reference frames of other sorts. In previous work, axis interaction was observed in a task that followed Experiment 1 except that the location array was tilted relative to the horizontal axis of space, and participants were required to assess distances relative to the tilted “horizontal” (Varner et al., 2014). Experiments are planned asking whether support for the ODH extends to this sort of classification task.

The present results speak more generally to the process by which the human visual system assesses distance between pairs of frontal stimuli. Two possibilities have been distinguished regarding that process (Watt, 1992). An additive process bases the assessment on the number of instances of a standard increment lying between representations of the stimuli. A subtractive process bases the assessment on the difference between the positions of the stimuli in a localization system. Versions of both of these ideas have been proposed in past work (Burbeck & Yap, 1990; Craven & Watt, 1989; Levi & Klein, 1990; Levi et al., 1988; Morgan & Regan, 1987). One could argue that the AIH assumes a subtractive process. Under the account, the visual system has access to the positions of the test stimuli on the relevant and irrelevant axes. Thus, in supporting the ODH over the AIH, the present results suggest that distance is assessed through an additive rather than a subtractive process.

In conclusion, the results demonstrate that observers make systematic errors assessing the horizontal or vertical distance between pairs of stimuli varying on the horizontal and vertical axes of a frontal plane. The results suggest that these errors reflect the visual system’s incapacity for directly assessing anything but the overall distance between two stimuli.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Stephen Dopkins

Note

Appendix

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