February 2018
Volume 18, Issue 2
Open Access
Article  |   February 2018
Temporal estimation in prediction motion tasks is biased by a moving destination
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Journal of Vision February 2018, Vol.18, 5. doi:10.1167/18.2.5
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      Jonathan C. Flavell, Brendan T. Barrett, John G. Buckley, Julie M. Harris, Andrew J. Scally, Nathan B. Beebe, Alice G. Cruickshank, Simon J. Bennett; Temporal estimation in prediction motion tasks is biased by a moving destination. Journal of Vision 2018;18(2):5. doi: 10.1167/18.2.5.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

An ability to predict the time-to-contact (TTC) of moving objects that become momentarily hidden is advantageous in everyday life and could be particularly so in fast-ball sports. Prediction motion (PM) experiments have sought to test this ability using tasks where a disappearing target moves toward a stationary destination. Here, we developed two novel versions of the PM task in which the destination either moved away from (Chase) or toward (Attract) the moving target. The target and destination moved with different speeds such that collision occurred 750, 1,000 or 1,250 ms after target occlusion. To determine if domain-specific experience conveys an advantage in PM tasks, we compared the performance of different sporting groups ranging from internationally competing athletes to non-sporting controls. There was no difference in performance between sporting groups and non-sporting controls but there were significant and independent effects on response error by target speed, destination speed, and occlusion period. We simulated these findings using a revised version of the linear TTC model of response timing for PM tasks (Yakimoff, Bocheva, & Mitrania, 1987; Yakimoff, Mateeff, Ehrenstein, & Hohnsbein, 1993) in which retinal input from the moving destination biases the internal representation of the occluded target. This revision closely reproduced the observed patterns of response error and thus describes a means by which the brain might estimate TTC when the target and destination are in motion.

Introduction
Whether estimating that it is safe to exit a busy road junction or judging when to strike/catch an approaching ball, it is not unusual for the target of interest to become momentarily occluded from the actor's view, for example, by people or street furniture. Yet we anticipate that the target continues to move unseen and are able to internally represent the occluded trajectory. In the laboratory, our ability to do this has been examined using prediction motion (PM) tasks in which a target moves toward a fixed destination (usually a line perpendicular to the target's trajectory) before disappearing or passing behind an occluder (e.g., Rosenbaum, 1975; Lyon & Waag, 1995; De Lucia & Liddell, 1998; Benguigui, Ripoli, & Broderick, 2003; Bennett, Orban de Xivry, Lefevre, & Barnes, 2010b; Bosco et al., 2015). The participant's task is typically to provide a response (usually via a button press) to indicate when the now-invisible target would have reached its destination. 
It has been suggested that performance in PM tasks is influenced by imposed oculomotor strategies (Bennett, Baurès, Hecht, & Benguigui, 2010a; Makin & Poliakoff, 2011), characteristics of target motion such as velocity (Sokolov & Pavlova 2003; Baurès, Oberfeld, & Hecht, 2010; Baurès & Hecht, 2011; Bennett et al., 2010a; Zago, Iosa, Maffei, & Lacquaniti, 2010; Baurès & Hecht, 2011; Nakamoto, Mori, Ikudome, Unenaka, & Imanaka, 2015), and the duration of target occlusion before it strikes the line (Peterken, Brown, & Bowman, 1991; Baurès et al., 2010). Other factors known to influence performance during PM tasks include the duration of visible motion (Sokolov & Pavlova, 2003), the target size (Sokolov & Pavlova 2003; Battaglini, Campana, & Casco, 2013), the presence of background texture (De Lucia, Tresilian, & Meyer, 2000; Battaglini, Campana, Camilleri, & Casco, 2014), motion aftereffects (Gilden, Blake, & Hurst, 1995; Battaglini et al., 2014), stimulus-to-background contrast (Battaglini et al., 2013), and the presence of visual distractors (Lyon & Waag, 1995). In addition, prior experience of fast-interceptive tasks has been shown to influence PM performance. For example, expert baseball players mislocate suddenly disappearing targets (traversing left to right on a computer screen) as significantly further ahead than novice players (Nakamoto et al., 2015). It was suggested that this overestimation was the result of the experts' enhanced capability for motion prediction and that such domain-specific expertise may be advantageous in compensating for neural transmission and processing delays as well as for transient loss of visual information (e.g., from saccades, blinks, or target occlusion). 
In the present study, we created two novel extensions of the PM task to explore how participants respond when both the target and the destination to which it is moving are in motion. In the first experiment (Chase), the destination retreated from the target as it was approached, whereas in the second experiment (Attract) the destination and target moved toward one another. Importantly, the Chase and Attract experiments differ from those of previous studies that examined simultaneous motion prediction of two occluded targets as they approached a fixed visible destination (e.g., Oberfeld & Hecht, 2008; Baurès, Oberfeld, & Hecht, 2011). Rather than generating two time-to-contact estimates (TTC, the period from occlusion to the instant of contact) and preparing two motor responses, here the Chase and Attract tasks required a single estimate of TTC based on the relative motion between an occluded and a visible object. 
As in typical PM tasks where a target moves toward a stationary destination, we expected to find, for both Chase and Attract, changes in response times as target speed and occlusion period were varied (e.g., Peterken et al., 1991; Baurès et al., 2010; Baurès et al., 2011; Bennett et al., 2010a; Zago et al., 2010; Baurès & Hecht, 2011; Nakamoto et al., 2015). Moreover, here we also sought to determine whether having a continuously visible moving destination biases the estimation of target speed during occlusion. In addition, we examined if performance in these novel PM tasks is influenced by expertise in ball sports (i.e., domain-specific expertise). To this end, we compared male and female non-sporting controls to athletes who may often have to predict visible and occluded motion. 
Methods
Participants
Members of the Huddersfield Giants Rugby League football club (all male, n = 19), the Leeds/Bradford Marylebone Cricket Club University squad (all male, n = 24), the England's national women's cricket team (n = 16), and male (n = 29) and female controls (n = 20) participated. Controls were students at the University of Bradford who had never played ball sports at a competitive level and were not routinely engaged in ball sports. Not all participants were included in the analysis for each experiment due to the data in some trials being unsuitable for analysis because of being erroneous or unrealistic (trial exclusion criteria are detailed in “Data acquisition, processing, and analysis”). Participation numbers following data exclusion are provided in Table 1. Protocols were approved by the Committee for Ethics in Research at the University of Bradford and were in accord with the tenets of the Declaration of Helsinki. Participants gave written informed consent and reported normal or corrected-to-normal vision and no known neurological or sensorimotor deficits. 
Table 1
 
For each experiment the number of participants per group (mean ± SD) following removal of unsuitable data.
Table 1
 
For each experiment the number of participants per group (mean ± SD) following removal of unsuitable data.
Experimental setup
Participants sat in a darkened room facing a 20-in. (33.90 × 44.23 of visual angle) Sony Trinitron GDM-F520 CRT monitor (Sony Corp., Tokyo, Japan). An adjustable chin rest ensured the head was stationary and that the center of the screen was 50 cm away and parallel to the eyes. A custom-made response key was positioned on a table between the participant and the screen. Participants were seated with arms relaxed and their forearms resting on the table. A PC (Dell Latitude E6530, Intel Core i7-3540, 3 GHz CPU, 4GB RAM, 32-bit Windows 7; Round Rock, TX) presented stimuli at a mean refresh rate of 85 Hz and spatial resolution of 1,600 × 1,400 pixels using custom scripts and Psychtoolbox (version 3.0.11; Brainard 1997; Pelli 1997; Kleiner, Brainard, & Pelli, 2007) operating within Matlab (R2014a; The MathWorks Inc., Natick, MA). 
Task and stimuli
The participant's task in both experiments was to press the response key with the index finger of their dominant hand (determined using the Edinburgh Handedness Inventory; Oldfield, 1971) when the leading edge of a horizontally moving circular target was judged to collide with a vertical line (the destination) that was oriented perpendicular to the target's trajectory. Each participant completed Chase then Attract on the same day. Figure 1 shows the chronology of trials in both experiments. The visual appearance of stimuli was identical in each experiment and included a cue (white circle of radius 1 mm or 0.11°), a target (black circle of radius ∼5 mm or 0.57°) with a white circle at its center (radius 1 mm or 0.11°), and a destination [black line of 1 mm in width (0.11°) and 230 mm in length (24.7°)]. The target and destination were located midway between the top and bottom of the screen, and were presented against a gray screen background. The display whites, grays, and blacks had luminance of 65.1, 10.63, and 0.01 candelas/m2, respectively. The black arrows in Figure 1 represent the direction in which the target and line (destination) were moving and the dashed white circle represents the unseen position of the target when its leading edge would have contacted the destination. Neither the arrows nor dashed circle were visible to the participant. 
Figure 1
 
Schematic showing the trial chronology in Chase (A) and Attract (B).
Figure 1
 
Schematic showing the trial chronology in Chase (A) and Attract (B).
The start of a trial was identical in both experiments: A stationary cue appeared to the left of the screen (Figure 1A and 1B, panel 1), and the participant pressed the response key to begin. Pressing the key simultaneously replaced the cue with the target and caused the destination to appear some way to the target's right (Figure 1A and 1B, panel 2). The target and line remained stationary for a random period between 500 to 1,500 ms. Target and line behavior differed between the experiments from this point on. In Chase, the target and the destination began to move rightward (Figure 1A, panel 3), whereas in Attract the target began to move rightward whereas the destination began to move leftward (i.e., they moved toward one another, Figure 1B, panel 3). Target and destination movements always had linear, horizontal trajectories of constant speed (see Table 2). The target disappeared 500 ms after movement onset (i.e., there was always 500 ms during which target movement was visible) but the destination remained visible throughout the trial (Figure 1A and 1B, panel 4). Participants were tasked with pressing the response key at the time when they estimated the now non-visible target would have collided with the destination (Figure 1A and 1B, panel 5). A trial ended when the response key was pressed, and was followed by a new trial being presented 1,000 ms later. By altering the horizontal starting locations of the target and destination, the target occlusion period was either 750, 1,000, or 1,250 ms. The target and the destination were no closer than 12 mm or 1.4° to the edge of the screen at the start of a trial, or at the moment of collision giving a maximum possible image rendering area of 281 mm, or 31.37° wide. The horizontal starting positions of the target and the destination were randomly jittered between trials to discourage estimation of contact time based on the initial position of stimuli rather than on the observed movement. For both the Chase and Attract experiments, participants completed five repetitions for each speed condition and occlusion period combination to yield 90 trials in each experiment (6 speed conditions × 3 occlusion periods × 5 repetitions). Condition order was randomized for each participant in both experiments. No feedback on response accuracy was given for any practice or experimental trials, and participants were not instructed where to look on the screen. Neither eye movements nor eye position were recorded. 
Table 2
 
Target speed (TS) and destination speed (DS) in °/s for each experiment.
Table 2
 
Target speed (TS) and destination speed (DS) in °/s for each experiment.
Before beginning the experimental phase, participants completed a practice block of “classic” PM trials (target movement toward a stationary destination) to familiarize themselves with the apparatus and general task requirements (a full description is available in Supplementary Material). Next, they were given an explanation of the upcoming experiment and performed eight practice trials (randomly chosen from the possible conditions though without repetition of any condition). 
Data acquisition, processing, and analysis
Matlab (version R2014a) was used for data acquisition and post-experiment processing and analysis. The response key was sampled at the instant of each screen image refresh (i.e., at 85 Hz). Response error was calculated as the difference between actual contact time (of target and destination) and the instant at which the response key was pressed. The sign and magnitude of the error indicates how early (negative error) or late (positive error) a response was made. Outlying response errors were excluded using a negative cut-off of (total travel time/2 ×1) and a positive cut-off of (total travel time − view time; note that view time was always 500 ms). For example, in the 750 ms occlusion condition, the target's total travel time was 1,250 ms (750 ms occluded motion + 500 ms visible motion), giving a negative cut-off of −625 ms and a positive cut-off of 750 ms. Response errors that fell outside this range were excluded because they were seen as erroneous (or implausible) responses. Any participant with three or more of the five trials excluded for any given condition in a Chase or Attract (e.g., Chase: occlusion period of 750 and speed condition of 15|10) was removed from the data set for that task (i.e., from Chase or Attract). This resulted in the exclusion of 11 of 108 (10.2%) participants from Chase and seven of 108 (6.5%) participants from Attract. A fuller description of data exclusion rates is provided in Supplementary Material S1. Data are also available in Supplementary Material S2 and S3
Statistical analysis
Response errors were analyzed via random effects regression modelling (StataCorp LP, College Station, TX). This is an iterative process that is tolerant of missing data and attempts to find the simplest model that only includes terms with significant effects (i.e., terms that affect the data). Terms were incorporated sequentially, with their statistical significance determined using the likelihood ratio test, and provisionally retained if they returned p values of 0.1 or less. Because of the iterative nature, only terms in the final model at p ≤ 0.005 were deemed meaningful. The following terms and their interactions were explored via the aforementioned modeling approach: target speed (three levels); destination speed (three levels); occlusion period (three levels); and group (five levels). 
Results
Chase: Target moves toward a retreating line
Observation of box and whisker plots (Figure 2A) of response errors for each speed condition and occlusion period indicated several patterns in the data. Across all conditions, responses1 occurred: (1) less late and/or increasingly early as the occlusion period increased; (2) less early and/or increasingly late as the target speed increased when destination speed was held constant (e.g., 15|10 to 20|10 to 25|10, and 20|15 to 25|15); and (3) less late and/or increasingly early as destination speed increased when target speed was held constant (e.g., 20|10 to 20|15 and 25|10 to 25|15 to 25|20). 
Figure 2
 
Box and whisker plots of raw response errors including median (central line), 25th and 75th percentiles (box edges), range of data (whiskers), and outliers (dots) for (A) the Chase experiment, and (B) the Attract experiment. Data are split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray, and 1,250 ms = dark gray). Positive errors indicate late responses (responses after the target would have struck the line) and negative errors indicate early responses (responses before the target would have struck the line). For each combination of speed condition and OP, the percentage of trials considered outliers in these plots was between 0 and 2.52% in Chase, and between 0 and 2.39% in Attract.
Figure 2
 
Box and whisker plots of raw response errors including median (central line), 25th and 75th percentiles (box edges), range of data (whiskers), and outliers (dots) for (A) the Chase experiment, and (B) the Attract experiment. Data are split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray, and 1,250 ms = dark gray). Positive errors indicate late responses (responses after the target would have struck the line) and negative errors indicate early responses (responses before the target would have struck the line). For each combination of speed condition and OP, the percentage of trials considered outliers in these plots was between 0 and 2.52% in Chase, and between 0 and 2.39% in Attract.
Regression modeling indicated that response errors were unaffected by group (p = 0.44) but were significantly affected by occlusion period, target speed, destination speed, and by their interactions (p < 0.001) with the exception of the target speed × destination speed interaction. However, the proportion of the overall variance that was explained by the model when including both main and interaction terms (r2 = 0.282) was only marginally greater than when including only main terms (r2 = 0.276). Hence, we accepted the simpler (main terms only) model (p < 0.001, Table 3). Coefficients indicate that responses occurred less late and/or increasingly early as the occlusion period increased and as destination speed increased, but that responses occurred less early and/or increasingly late as the target speed increased (as suggested by Figure 2A). 
Table 3
 
Chase experiment: Output of random effects regression model. Occlusion period (OP), target speed (TS), and destination speed (DS) were treated as covariates rather than factors because linear relationships were found between these terms and response errors.
Table 3
 
Chase experiment: Output of random effects regression model. Occlusion period (OP), target speed (TS), and destination speed (DS) were treated as covariates rather than factors because linear relationships were found between these terms and response errors.
Attract: Target and line approach one another
Observation of box and whisker plots (Figure 2B) of response errors for each speed condition and occlusion period indicates two patterns in the data. Across all conditions, responses occurred: (1) less late and/or increasingly early as occlusion period increased; and (2) less late and/or increasingly early as destination speed increased (when target speed was held constant, e.g., conditions 10|10 to 10|15 to 10|20, and 15|10 to 15|15). The effects of alterations in target speed were inconsistent and appeared to depend on accompanying destination speed. For example, for the slowest destination speed (10°/s) there appeared to be no change in response errors as target speed increased (e.g., 10|10, 15|10, and 20|10) but for the middle destination speed (15°/s) the response errors indicated that responses occurred less early and/or increasingly late as target speed increased (e.g., 10|15 and 15|15). 
Regression modeling indicated that response errors were unaffected by group (p = 0.231) but were significantly affected by target speed, destination speed and occlusion period and their interactions (p < 0.001). However, the proportion of the overall variance in the data that was explained by the model when including both main and interaction terms (overall r2 = 0.101) was again only marginally greater than when including only main terms (overall r2 = 0.1). Hence, we accepted the simpler (main terms only) model (p < 0.001, Table 4). Coefficients for the main effects indicate that responses occurred less late and/or increasingly earlier as destination speed increased and as occlusion period increased. They also indicate a non-linear effect of target speed whereby an increase in target speed from 10°/s to 15°/s led to a greater change in response errors (+73 ms per deg/s change in target speed) than an increase in target speed from 10°/s to 20°/s (+38 ms per deg/s change in target speed). 
Table 4
 
Attract experiment: Output of random effects regression model. Occlusion period (OP) and destination speed (DS) and were treated as covariates rather than factors because linear relationships were found between these terms and response error. Target speed (TS) was treated as a factor.
Table 4
 
Attract experiment: Output of random effects regression model. Occlusion period (OP) and destination speed (DS) and were treated as covariates rather than factors because linear relationships were found between these terms and response error. Target speed (TS) was treated as a factor.
Discussion
Much of current understanding regarding the ability to internally represent an occluded trajectory is informed by work using the prediction motion (PM) task (Peterken et al., 1991; Lyon & Waag, 1995; De Lucia & Liddell, 1998; Benguigui et al., 2003; Bennett et al., 2010b; Bosco et al., 2015). Typically, the PM task requires a participant to estimate time to contact (TTC) of a target that moves at a particular speed and then becomes occluded or disappears as it approaches a fixed destination. Accordingly, it has been suggested that participants could estimate TTC based on information available prior to target occlusion (see De Lucia & Liddell, 1998). Here, we conducted two novel variations of the PM task in which optimal performance required participants to take account of information from both a moving target and a moving destination. Specifically, we investigated the ability to estimate TTC of a target at a destination that was either retreating from a target (Chase) or approaching it (Attract). Given the previously reported influence of prior experience on PM performance (Nakamoto et al., 2015), we also sought to determine if temporal estimation in these novel tasks differed as a function of expertise in ball sports (i.e., domain-specific expertise). To summarize our results, we found that in both Chase and Attract tasks, response errors did not vary by sporting expertise. However, there were independent influences of target speed, destination speed, and occlusion period. To aid interpretation of these findings, we present a revision to the linear model of response timing in traditional PM tasks (i.e., moving target and stationary destination) proposed by Yakimoff et al. (1987) and Yakimoff et al. (1993). This revision closely reproduces the pattern of response errors observed in our experimental data where both the target and destination are in motion. Key to this is replication of the bias in the internal representation of target speed (and thus estimated TTC) by retinal input from the observed moving destination. 
Modeling TTC estimation
In prediction motion experiments where the target moves toward a stationary destination (Peterken et al., 1991; Baurès et al., 2010) it is common to find that participants respond earlier as occlusion period increases. Yakimoff et al. (1987) and Yakimoff et al. (1993) proposed that this occurs in part because participants overestimate target speed. They suggested response times can be modeled as:  
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}\tag{1}{T_r} = \alpha \times {T_c} + \theta \end{equation}
where Tr is the response time, Tc is the actual time to contact (TTC), α is a constant representing the magnitude of overestimation of target speed and θ is a constant representing the sum of participant's internal delays (visual, neural, mechanical, etc.).  
Using the general concept that relative speed of a target that approaches a destination is misestimated during occlusion in PM tasks (see also Lyon & Waag, 1995; Makin et al., 2008), we sought to simulate the observed pattern of response errors in the Chase and Attract tasks. Importantly, in our experiments visual input from the moving target was available for only the first 500 ms of a trial, whereas visual input from the destination was available throughout. Assuming that participants solve the PM task using some form of motion extrapolation, rather than a counting strategy (De Lucia & Liddell, 1998), the implication is that participants need to generate and remember an estimate of target speed early in the trial, whereas they can continually update their estimate of destination speed throughout the trial. Accordingly, we considered whether the patterns in our data could be described by a revision to the linear model of response timing in PM tasks proposed by Yakimoff and colleagues. Specifically, we considered whether the remembered target speed was influenced by the continuously present destination speed. In other words, could the moving destination bias the internal representation of the now unseen target motion? 
First, we included terms that reflect the participants' perception of target and destination speed when they are visible:  
\begin{equation}\tag{2a}pTS = TS \times \gamma \end{equation}
 
\begin{equation}\tag{2b}pDS = DS \times \gamma \end{equation}
where pTS and pDS are the perceived speed of the target and destination, respectively, TS is the target speed, DS is the destination speed, and γ is a constant representing the magnitude of speed misestimation. At this stage, we assume that misestimation of TS and DS is minimal and hence γ is set to 1. However, this constant could change depending upon the background (e.g., coarse vs. sparse) or pattern of eye movements (i.e., fixation vs. pursuit).  
Next, we characterized participants' estimation of target speed when there is a bias induced by the continuously visible moving destination:  
\begin{equation}\tag{3}eTS = \left( {pTS\times\alpha } \right) - \left( {\left( {\left( {pTS\times\alpha } \right) - pDS} \right)\times\beta } \right)\end{equation}
where eTS is the participant's estimate of target speed, α is a constant representing a misestimate of target speed following target occlusion, and β is a constant representing the magnitude of the influence of pDS.  
To estimate the relative speed (eRS) of the target and the destination, and to account for direction of travel in Chase and Attract, we used the following:  
\begin{equation}\tag{4a}Chase\ eRS = eTS - pDS\end{equation}
 
\begin{equation}\tag{4a}Attract\ eRS = eTS + pDS\end{equation}
 
With values of γ = 1, α = 1.3, and β = 0.2, we generated an estimate of the relative speed between the moving target and destination. This was then expressed relative to the actual relative speed (aRS) to give a measure of the misestimation (αrv) during occlusion as shown in Equation 1
Having also estimated the constant θ in Equation 1 (i.e., sum of participant's internal delays) for each speed condition in Chase and Attract as the intercept in a linear regression of actual median TTC (response error + occlusion period) against occlusion period, we resolved a revised version of the linear model thus:  
\begin{equation}\tag{5}{T_r} = {\alpha _{{{rv}}}} \times OP + \theta \end{equation}
where OP (the occlusion period in Chase and Attract) takes the place of Tc from Equation 1.  
Finally, we estimated participants' response errors in Chase and Attract by subtracting our simulated Tr from the occlusion period. As can be seen by comparing the data shown in Figure 2 and Figure 3, we were able to reproduce response errors with both magnitude and sign that were a close match to the actual response errors observed in Chase and Attract. We did not test exhaustive variations of α and β in Equation 3, such as might be expected for individual participants, or allow the value of γ to stray from 1. Our goal was simply to demonstrate that, with the chosen values, our revised version of Yakimoff and colleagues' linear model was able to provide a reasonable description of the pattern of response errors in our Chase and Attract experiments. 
Figure 3
 
Modeled estimates of response errors split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray and 1,250 ms = dark gray). (A) Chase experiment. (B) Attract experiment. Estimates were generated using Equations 2 through 5 with γ = 1, α = 1.3, and β = 0.2.
Figure 3
 
Modeled estimates of response errors split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray and 1,250 ms = dark gray). (A) Chase experiment. (B) Attract experiment. Estimates were generated using Equations 2 through 5 with γ = 1, α = 1.3, and β = 0.2.
Nonlinear effects of target speed in Attract
For the Attract experiment, there was a non-linear effect of target speed whereby response errors increased (i.e. responses were made later) more when target speed increased from 10°/s to 15°/s (+73 ms) than when target speed increased from 10°/s to 20°/s (+38 ms) (Table 4). Note that because target speed was treated as a factor in our statistical modeling (due to the nonlinear effect) we do not have the comparison of response error change as target speed increase from 15°/s to 20°/s. This nonlinear effect was replicated by our model, thus suggesting it was a result of the experimental conditions tested. Why might this be the case? Equation 3 reveals that there is a greater bias of estimated target speed when the target speed is 20°/s compared to when it is 15°/s. This is because, in these conditions, the difference between destination speed and target speed is greater when the target speed is 20°/s (speed condition 20|10 = Δ10) than when it is 15°/s (speed condition 15|10 and 15|15 = Δ5 and Δ0, respectively). This means that the magnitude of overestimation of target speed is lower when the target speed is 20°/s than in both speed conditions where target speed is 15°/s (see Figure 4, dark gray dots). In other words, the unequal balancing of target and destination speeds created unequal bias on estimated target speed, which ultimately resulted in an apparent non-linear effect of target speed on response errors. In Figure 4 we also present model predictions for the overestimation of target speed for two conditions that were not tested in the Attract task. It is clear that when the difference between the target and destination speeds is the same within speed conditions (e.g., 15|10 to 20|15 and 15|15 to 20|20), target speed is overestimated to a greater extent in the higher target speed conditions. Our model predicts that had we tested these speed conditions in Attract, we could have expected linear effects of target speed on response errors (as found in Chase). 
Figure 4
 
Overestimates of target speed in three tested conditions from Attract (dark gray dots) and three untested potential conditions for Attract (light gray dots). The overestimate is the estimated target speed (eTS from Equation 3) minus actual target speed.
Figure 4
 
Overestimates of target speed in three tested conditions from Attract (dark gray dots) and three untested potential conditions for Attract (light gray dots). The overestimate is the estimated target speed (eTS from Equation 3) minus actual target speed.
Difference in destination speed effect between Chase and Attract
The magnitude of the destination speed effect was approximately three times greater in Chase (−44 ms, Table 3) than in Attract (−15 ms, Table 4). The bias of destination speed on estimated target speed, predicted by our model, goes some way to explain this effect. In Chase, the destination speed is always less than the target speed, so there is always a negative bias. However, in Attract the destination speed can be less than, greater than, or equal to the target speed, so the bias can be negative, zero or positive. If this is the source of the magnitude difference between Chase and Attract, then running the Attract experiment with an extended range of speed conditions (perhaps additionally testing 15|20, 20|15, and 20|20) should deliver a similar magnitude of destination speed effect as we found in Chase
Comparison of 15|10 and 20|10 speed conditions between Chase and Attract
We have proposed that the bias of target speed by destination speed occurs in the same way for both Chase and Attract (Equation 3). One might assume, therefore, that response errors should not differ between Chase and Attract when the speed conditions are comparable (i.e., in the 15|10 and 20|10 speed conditions that appeared in both tasks). However, it is critical to remember that our model incorporates the relative speed for each task (i.e., “eTSpDS” in Chase; “eTS + pDS” in Attract), and thereby accounts for the direction of travel of both the target and destination. This results in different estimated response errors for each task even when the speed conditions are comparable. In other words, although 15|10 and 20|10 speed conditions were used in both tasks, it is only the magnitude of the speeds that were comparable because the destination's direction of the travel was different in each task. 
No effect of sporting expertise
There are situations where sporting expertise appears to convey an advantage in predicting an object's future motion. For instance, Nakamoto et al. (2015) studied representational momentum (RM), a phenomenon where people judge an occluded moving target as being further along its path than it actually is. They found that expert baseball players exhibited greater RM than novices. The authors suggest that this may be advantageous in fast-ball sports because it could alleviate information processing delays and aid interception or avoidance of occluded targets. In the present study we found no influence of sports expertise in any of our PM tasks (i.e., there was no indication that motion prediction in sporting elites differed to that in controls, p ≥ 0.231). Why might this be case? Nakamoto et al. intended their task to simulate baseball pitching and as such used high-speed targets (10 or 15 m/s) with short presentation and occlusion times (133 ms occlusion time in both cases). They found expert/novice differences only in the highest speed condition where novices' TTC estimation was poorer. It is possible that the advantage exhibited by elite baseball players in Nakamoto et al. (2015) was related to greater reliance on anticipatory processes operating during the initial viewing period rather than improved extrapolation of an occluded trajectory. Such anticipation requirements were not a feature of the tasks used the present study. Indeed, whereas motion prediction is a general visuomotor ability that is exercised by all, “advanced” anticipatory processes may only be present in sub-populations who experience situations where such abilities are conducive to their goals. 
Study limitations
All participants completed Chase before Attract, and thus we cannot rule out the possibility of experiment order effects. However, it is relevant to note that participants never received feedback on the magnitude or direction of response errors, so there is no obvious reason why performance would change in any meaningful or systematic way following exposure to certain conditions. Also, our proposed model, which does not include any attempt to incorporate order effects, produced reasonable estimations of participants' response errors in both the Chase and Attract tasks. Therefore, it seems likely that any unaccounted effect of task and/or condition order would have had a negligible influence on the observed pattern of results. This could be confirmed in a future study that combines conditions of the Chase and Attract tasks into a single experiment with randomized trial order. 
Summary
Extending upon the classic prediction motion task, we determined response errors in two novel prediction motion tasks in which the target's destination could move either away from (Chase) or towards an approaching target (Attract). We found that, irrespective of participants' experience of playing sport, response errors became less late or increasingly early as the occlusion period or destination speed increased, and increasingly early or less late as target speed increased. We presented a revision of Yakimoff and colleagues' (1987, 1993) linear model of response timing in PM tasks, which closely reproduced the pattern of response errors observed in our experimental data. Future work could perhaps consider other ways in which our results (or results from tasks like Chase and Attract) may be modeled, and what biological mechanisms may underlie TTC judgments in these tasks. 
Acknowledgments
This study was funded by the Biotechnology and Biological Sciences Research Council (grant references BB/J018163/1, BB/J016365/1, and BB/J018872/1). The authors declare that they have no conflict of interest. Special thanks for participating go to the England and Wales Cricket Board, the England Women's cricket team, the Leeds/Bradford MCCU, and the Huddersfield Giants Rugby League Football Club. Desney Richards (Technical Director of Research Solutions) produced the button box used in these experiments. 
Commercial relationships: none. 
Corresponding author: Brendan T. Barrett. 
Address: University of Bradford, School of Optometry and Vision Science, Bradford, West Yorkshire, UK. 
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Footnotes
1  Though response errors are shown in Figure 2, we refer to the time of the response in relation to the actual TTC for ease of understanding. Response times are the sum of the occlusion period and the response error.
Figure 1
 
Schematic showing the trial chronology in Chase (A) and Attract (B).
Figure 1
 
Schematic showing the trial chronology in Chase (A) and Attract (B).
Figure 2
 
Box and whisker plots of raw response errors including median (central line), 25th and 75th percentiles (box edges), range of data (whiskers), and outliers (dots) for (A) the Chase experiment, and (B) the Attract experiment. Data are split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray, and 1,250 ms = dark gray). Positive errors indicate late responses (responses after the target would have struck the line) and negative errors indicate early responses (responses before the target would have struck the line). For each combination of speed condition and OP, the percentage of trials considered outliers in these plots was between 0 and 2.52% in Chase, and between 0 and 2.39% in Attract.
Figure 2
 
Box and whisker plots of raw response errors including median (central line), 25th and 75th percentiles (box edges), range of data (whiskers), and outliers (dots) for (A) the Chase experiment, and (B) the Attract experiment. Data are split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray, and 1,250 ms = dark gray). Positive errors indicate late responses (responses after the target would have struck the line) and negative errors indicate early responses (responses before the target would have struck the line). For each combination of speed condition and OP, the percentage of trials considered outliers in these plots was between 0 and 2.52% in Chase, and between 0 and 2.39% in Attract.
Figure 3
 
Modeled estimates of response errors split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray and 1,250 ms = dark gray). (A) Chase experiment. (B) Attract experiment. Estimates were generated using Equations 2 through 5 with γ = 1, α = 1.3, and β = 0.2.
Figure 3
 
Modeled estimates of response errors split by speed condition and occlusion period (OP; 750 ms = pale gray, 1,000 ms = gray and 1,250 ms = dark gray). (A) Chase experiment. (B) Attract experiment. Estimates were generated using Equations 2 through 5 with γ = 1, α = 1.3, and β = 0.2.
Figure 4
 
Overestimates of target speed in three tested conditions from Attract (dark gray dots) and three untested potential conditions for Attract (light gray dots). The overestimate is the estimated target speed (eTS from Equation 3) minus actual target speed.
Figure 4
 
Overestimates of target speed in three tested conditions from Attract (dark gray dots) and three untested potential conditions for Attract (light gray dots). The overestimate is the estimated target speed (eTS from Equation 3) minus actual target speed.
Table 1
 
For each experiment the number of participants per group (mean ± SD) following removal of unsuitable data.
Table 1
 
For each experiment the number of participants per group (mean ± SD) following removal of unsuitable data.
Table 2
 
Target speed (TS) and destination speed (DS) in °/s for each experiment.
Table 2
 
Target speed (TS) and destination speed (DS) in °/s for each experiment.
Table 3
 
Chase experiment: Output of random effects regression model. Occlusion period (OP), target speed (TS), and destination speed (DS) were treated as covariates rather than factors because linear relationships were found between these terms and response errors.
Table 3
 
Chase experiment: Output of random effects regression model. Occlusion period (OP), target speed (TS), and destination speed (DS) were treated as covariates rather than factors because linear relationships were found between these terms and response errors.
Table 4
 
Attract experiment: Output of random effects regression model. Occlusion period (OP) and destination speed (DS) and were treated as covariates rather than factors because linear relationships were found between these terms and response error. Target speed (TS) was treated as a factor.
Table 4
 
Attract experiment: Output of random effects regression model. Occlusion period (OP) and destination speed (DS) and were treated as covariates rather than factors because linear relationships were found between these terms and response error. Target speed (TS) was treated as a factor.
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