The bisection point across variants of the task

Abstract

Bisection tasks are used in research on normal space and time perception and to assess the perceptual distortions accompanying neurological disorders. Several variants of the bisection task are used, which often yield inconsistent results, prompting the question of which variant is most dependable and which results are to be trusted. We addressed this question using theoretical and experimental approaches. Theoretical performance in bisection tasks is derived from a general model of psychophysical performance that includes sensory components and decisional processes. The model predicts how performance should differ across variants of the task, even when the sensory component is fixed. To test these predictions, data were collected in a within-subjects study with several variants of a spatial bisection task, including a two-response variant in which observers indicated whether a line was transected to the right or left of the midpoint, a three-response variant (which included the additional option to respond "midpoint"), and a paired-comparison variant of the three-response format. The data supported the model predictions, revealing that estimated bisection points were least dependable with the two-response variant, because this format confounds perceptual and decisional influences. Only the three-response paired-comparison format can separate out these influences. Implications for research in basic and clinical fields are discussed.

Fig. 1

Indecision model and predictions on observed performance in two variants of the landmark task under three scenarios (columns). a Psychophysical functions μ mapping physical space onto perceptual space. The function has the form of Eq. 1, with α = 0 and β = 0.9 (left and right columns), so that the perceptual midpoint and physical midpoint match (i.e., the zero crossing of μ is at x = −α/β = 0), or with α = 0.9 and β = 0.9 (center column), so that the perceptual midpoint is not at the physical midpoint (i.e., the zero crossing of μ is at x = −α/β = −1). b Distributions of perceptual position (curve) given by Eq. 2 for a configuration in which the transecting bar is at x = 1, yielding the mean perceptual position indicated by the solid lines in row a. Also shown is the decision space with boundaries at S = δ₁ and S = δ₂, which partitions the continuum into intervals associated with the judgments indicated at the top. Judgments are not affected by decisional bias if δ₁ = −δ₂ (left and center columns, where δ₁ = −1.3 and δ₂ = 1.3); otherwise (right column, where δ₁ = −2.2 and δ₂ = 0.4), a decisional bias is involved whereby the strength of evidence, |δ₁|, required for a “left” judgment differs from (is larger than, in this illustration) the strength of evidence, |δ₂|, required for a “right” judgment. c Ranges of psychometric functions Ψ (probability of “right” response as a function of position of the transecting bar) that could be observed when observers are asked to report a “left”/“right” judgment. The mathematical form of Ψ is given by Eq. 3. If observers invariably report “center” judgments as “right” responses (i.e., ξ = 1; red arrows in row b), Ψ as plotted in red obtains; if they always report “center” judgments as “left” responses (ξ = 0; blue arrows in row b), Ψ as plotted in blue obtains; the black curves plot the resultant Ψs for intermediate cases (left to right, ξ = .75, .5, and .25). Of all these functions, only the 50 % point on the one for which ξ = .5 reflects the true perceptual midpoint, provided that the observer does not have any decisional bias. d Psychometric functions arising in a ternary variant of the landmark task, whose mathematical forms are given by Eqs. 4. Each curve represents the probability of the response printed with the same color in the upper part of the panel. response bias no longer affects these psychometric functions, because “center” judgments are reported separately (black curves); however, decisional bias (right column) displaces the ensemble just as a true perceptual shift (center column) does. Color is available only in the online version

Fig. 2

Indecision model and predictions for a paired-comparison variant of the ternary landmark task under four scenarios (columns). a Psychophysical functions μ, identical to those in Fig. 1a and reflecting either veridical perception (first and third columns) or a perceptual shift (second and fourth columns). b Distributions of the decision variable given by Eq. A5 for a pair in which the test configuration is presented in the first interval with the transecting bar at x = 1, whereas the standard configuration is presented in the second interval and transected at the physical midpoint (x = 0). Also shown is the decision space with boundaries at D = δ₁ and D = δ₂, analogous to that in Fig. 1b. Here, judgments are also not affected by decisional bias if δ₁ = −δ₂ (first and second columns), and are affected by them otherwise (third and fourth columns). (c) Analogous to row b, but for the case in which the test configuration is displayed after the standard. (d) Psychometric functions for each type of response according to order of presentation. Psychometric functions for correct responses (“Interval 1” responses when the test is presented first, and “Interval 2” responses when the test is presented second) are shown in black/gray; psychometric functions for incorrect responses (“Interval 2” responses when the test is presented first and “Interval 1” responses when the test is presented second) are shown in red/pale red; psychometric functions for “I can't tell” responses (under both orders of presentation) are shown in blue/pale blue. The psychometric functions do not differ across presentation orders if there is no decisional bias (first and second columns), and they do differ if there is decisional bias (third and fourth columns). In either case, the ensemble of psychometric functions has a vertical axis of bilateral symmetry at the true perceptual midpoint—that is, at the zero crossing of μ in row a. Color is available only in the online version

Fig. 3

Schematic diagram of trial timings in method-of-single-stimuli (MSS) and paired-comparison (PC) variants of the landmark task. The fixation cross and the stimuli are not drawn to scale. In the MSS variants (left side), the actual transecting position in each trial was dictated by the applicable staircase, and the response requested could be binary (in MSS-2R) or ternary (in MSS-3R). In the PC-3R task (right side), in which a ternary response was requested, the interval in which the standard configuration transected at the center was presented (Interval 1 in this illustration), and the locations at which the test configuration in the other interval was transected were dictated by the applicable staircase in that trial

Fig. 4

Tracks from sample staircases used to collect data from Observer 1 (left panels) and psychometric functions fitted to the data (right panels). Only a subset of 12 staircases is plotted for the MSS-2R task (a), the MSS-3R task (b), and the PC-3R task with the test presented in either the first interval (c) or the second interval (d). The binned data used to fit the psychometric functions in the right column come from the total number of staircases used in each case. The major color conventions are as in Figs. 1 and 2, except that the use of pale/dark shades has been altered to enhance visibility. Color is available only in the online version

Fig. 5

Results for 13 observers. The central bundle of open circles indicates the bisection settings at the corresponding offsets (negative ordinates indicate positions to the left of the physical midpoint, and the horizontal dashed line indicates the physical midpoint); the radius of each circle indicates the number of settings at that particular offset. The superimposed cross-like red sketches indicate the average settings (horizontal segment) and the width of the interval spanning ±1 SD from the average. Color circles on the left for each observer indicate the bisection points (BPs) estimated separately from each variant of the landmark task; the colored ring on the right indicates the BP estimated through the joint fit of the model to data from all variants (see the inset). The results for Observer 11 in the landmark tasks are omitted (see Appendix C). Observers 1–7 were experienced, and the remaining observers were inexperienced paid volunteers. Color is available only in the online version

Fig. 6

Data and fitted psychometric functions in each variant of the landmark task for five representative observers (rows). The rightmost column plots aggregated data across the two presentation orders in the PC-3R task, and also the average of the psychometric functions fitted for each separate order. The color conventions are as in Figs. 1 and 2, including the use of paler shades for the subset of trials in which the test was presented second (fourth column). Color is available only in the online version

Fig. 7

Scatterplots of estimated α (top row) and estimated β (bottom row) across variants of the landmark task. Estimates from Observer 11 are excluded (see Appendix C). The inset at the top left of each panel shows the value of the concordance correlation coefficient ρ_c and the p value associated with the Bradley–Blackwood (1989) test for equality of means and variances. The sketches near the bottom and left axes in each panel indicate the mean and standard deviation of the data along the corresponding dimension. Color is available only in the online version

Fig. 8

Scatterplots of estimated bisection points (BPs, top row) and estimated difference limens (DLs, bottom row) across variants of the landmark task. Estimates from Observer 11 are excluded (see Appendix C). The inset at the top left of each panel shows the value of the concordance correlation coefficient ρ_c and either the p value associated with the Bradley–Blackwood (1989) test for equality of means and variances (top row) or the p value associated with a paired-samples t test (bottom row). The sketches near the bottom and left axes in each panel indicate the mean and standard deviation of the data along the corresponding dimension. The stray, colored data point plotted in each panel was excluded from all computations (see the text). Color is available only in the online version

Fig. 9

Data and psychometric functions fitted jointly to the data from all variants of the landmark task for the observers in Fig. 6 (rows). Graphical conventions are as in Fig. 6. Color is available only in the online version

Fig. 10

Scatterplots of estimated bisection points (BPs, top row) and estimated difference limens (DLs, bottom row) from separate (ordinate) versus joint (abscissa) fits of the model to data from each variant of the landmark task. Estimates from Observer 11 are excluded (see Appendix C). The graphical conventions are as in Fig. 8. Reported p values are those associated with the Bradley–Blackwood (1989) test for equality of means and variances. Color is available only in the online version

Fig. 11

Widths and locations of the interval of uncertainty (the range from δ₁ to δ₂) under each variant of the landmark task, as estimated from the joint fit. Numerals at the top are estimated values of ξ (response bias) in the MSS-2R task for each observer. The results for Observer 11 are omitted (see Appendix C). Note that the decision variable is defined differently for MSS and PC variants of the landmark task, and thus the respective widths of the interval of uncertainty are incommensurate. Color is available only in the online version

Fig. 12

Scatterplots of estimated bisection points (BPs) in the manual bisection task (abscissa) versus estimated BPs from the separate fits of the model to data from each variant of the landmark task (ordinate). These data were plotted in another form in Fig. 5. Estimates from Observer 11 are excluded (see Appendix C). The graphical conventions are as in Fig. 8, and reported p values are those associated with the Bradley– Blackwood (1989) test for equality of means and variances. Color is available only in the online version

Fig. 13

Indecision model and predictions for a length-comparison variant of the landmark task with binary and ternary response formats under four scenarios (columns). a Psychophysical functions μ_L (light green) and μ_R (dark green) mapping physical length onto perceived length in each visual hemifield. The two functions are identical in the absence of perceptual distortions (first and second columns), but they differ when perceptual space is distorted in one of the hemifields (third and fourth columns). Vertical–horizontal segments indicate the physical and perceived lengths of the two segments of a sample 60-unit line transected two units to the left of its midpoint (i.e., the length of the left segment is 28 units, whereas that of the right segment is 32 units). b Distribution of the decision variable (difference in perceived lengths of the two segments) for the sample case illustrated in row a. Also shown are the decision boundaries δ₁ and δ₂, which are symmetrically placed in the absence of decisional bias (first and third columns), but asymmetrically placed when there is decisional bias (second and fourth columns). c Psychometric functions that may be observed under the binary response format, according to the response bias with which the observer guesses when undecided (“I can't tell” judgments). The bisection point and difference limen of the observed psychometric functions are uninterpretable. Graphical conventions are as in Fig. 1. d Psychometric functions for each type of response in the ternary response format (see the legends). The ensemble of psychometric functions is laterally shifted by both perceptual distortions and decisional bias. Color is available only in the online version

Fig. 14

Tree diagrams describing the sequences of events mapping unobservable judgments onto the observed responses in the MSS-2R (left) and MSS-3R (right) tasks. The starting point at the left of each diagram is the unobservable judgments, which occur with probabilities given by the applicable equations. Once the judgment is made, misreports occur with probabilities given by the ε parameters. In the MSS-2R task (left diagram), misreports result in a response opposite to the “left” or “right” judgment originally made, whereas “center” judgments render an arbitrary “left” or “right” response at random and according to the response bias parameter ξ. In the MSS-3R task (right diagram), misre-ports also render a response different from the judgment that was made, but, since two error responses are possible, which one is given is determined by additional parameters κ

Fig. 15

Data and psychometric functions fitted separately under each variant of the landmark task for the observers not shown in Fig. 6

Fig. 16

Data and psychometric functions fitted jointly to all variants of the landmark task for the observers not shown in Fig. 9; Observer 11 is excluded for reasons discussed in the text