Motor Control, 2015, 19, 173  -190 http://dx.doi.org/10.1123/mc.2013-0098 © 2015 Human Kinetics, Inc.

original article

Entropy of Movement Outcome in Space-Time Shih-Chiung Lai National Taipei University of Nursing and Health Sciences

Tsung-Yu Hsieh and Karl M. Newell The Pennsylvania State University Information entropy of the joint spatial and temporal (space-time) probability of discrete movement outcome was investigated in two experiments as a function of different movement strategies (space-time, space, and time instructional emphases), task goals (point-aiming and target-aiming) and movement speed-accuracy constraints. The variance of the movement spatial and temporal errors was reduced by instructional emphasis on the respective spatial or temporal dimension, but increased on the other dimension. The space-time entropy was lower in targetaiming task than the point-aiming task but did not differ between instructional emphases. However, the joint probabilistic measure of spatial and temporal entropy showed that spatial error is traded for timing error in tasks with space-time criteria and that the pattern of movement error depends on the dimension of the measurement process. The unified entropy measure of movement outcome in space-time reveals a new relation for the speed-accuracy function. Keywords: speed-accuracy, movement entropy, space-time outcome

In experimental protocols of the movement speed-accuracy relation it is typically the case that movement accuracy is interpreted in relation to a spatial criterion and, speed is the average velocity derived from the ratio of distance covered in the movement time, a property that also covaries with movement time when amplitude is fixed (Fitts, 1954; Schmidt, Zelaznik, Hawkins, Frank, & Quinn, 1979; Woodworth, 1899). There can, however, also be an explicit or implicit temporal criterion in movement tasks that provides the basis to determine a movement temporal error (Ellis, Schmidt & Wade, 1968; Kim, Carlton, Liu & Newell, 1999; Wing & Kristofferson, 1973). This potential to measure movement error in the two dimensions of space and time has led to the proposition that a space-time approach to the analysis of movement outcome variability is required, in which rather than an analysis of Lai is with the Dept. of Exercise and Health Science, National Taipei University of Nursing and Health Sciences, Taiwan. Hsieh and Newell are with the Dept. of Kinesiology, The Pennsylvania State University, University Park, PA. Address author correspondence to Tsung-Yu Hsieh at [email protected]   173

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either spatial error or temporal error separately, the measure of movement variability should be of a unified space-time property of the movement speed and accuracy relation (Hancock & Newell, 1985; Newell, 1980). Here we investigate this general proposition directly through the analysis of an information entropy (Shannon & Weaver, 1949; Scott, 1992; Williams, 1997) joint probability measure of movement space-time error variability under the manipulations of instructional speed-accuracy emphasis, target, or point aiming (Experiment 1), and movement speed-accuracy conditions (Experiment 2). The movement outcome scores or errors in space and time each have their own specific dimensions and units. For example, the amplitude of dart throwing outcome away from the target/central point can be a measure used to assess the performance of dart throwing (Radlo, Steinberg, Singer, Barba, & Melnikov, 2002); and the unit of this spatial difference could be centimeter (cm) or millimeter (mm). On the other hand, the timing of movement outcome is important in a range of actions, such as sport or music, and the unit of temporal error might be recorded in millisecond (ms). It is the case also that a small temporal error (undershoot or overshoot from the criterion time) could lead to a large spatial outcome error in a range of activities, and vice-versa. One advantage of information entropy measures is that they focus on probabilities of the outcome irrespective of the measured movement dimension (Williams, 1997). In many movement tasks, however, there is the potential of a relation between the spatial and temporal errors because each error is measured with respect to the other (Hancock & Newell, 1985; Lee, 1980; Newell, 1980). Indeed, in interceptive actions such as baseball batting, it is difficult to separate spatial and temporal error in the failure of the bat to contact the ball. Thus, even when the errors are measured independently it makes little sense to say that the batter missed the ball by 10 ms but also only by 1 mm! Here we investigate a unified spatial and temporal error measurement of the probabilistic estimates of movement outcomes that can be implemented even though the units for assessing movement error in space and time error are different. The measure is an unified space-time entropy because it considers the joint probability structure (Williams, 1997; Scott, 1992) of spatial and temporal movement outcome error. In Fitts’ paradigm (Fitts, 1954; Fitts & Peterson, 1964), the target width (W) was viewed as analogous to noise (N). For a point-aiming movement there is only a small point or dot that, in effect, when viewed in terms of Fitts’ law, can be construed as reflecting a very small target width. This does not mean, however, that there is no noise or that the noise must be small in a point-aiming task. It is well established that when aiming to a point that the biological/environmental noise still exists and contributes to a subjective “effective target width” (We) (Meyer, Abrams, Kornblum, Wright, & Smith, 1988; Schmidt et al., 1979). According to the analogy between noise and target width (Fitts, 1954; Shannon & Weaver, 1949), this We is the reflection of noise that we need to focus on in the discrete point-aiming movements and target-aiming task. The We in the current study is defined as the actual horizontal measure between the maximum and minimum of movement outcome (± 2 SD). It has been shown that probability through information entropy provides a dimensionless alternative measure of the variability of movement outcome (Lai, Mayer-Kress, Sosnoff, & Newell, 2005). Since probability is the foundation of information entropy (Shannon & Weaver, 1949; Scott, 1992; Williams, 1997), MC Vol. 19, No. 3, 2015

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the theoretical focus of the current study is on the development of a new unified joint probability measure of the spatial and temporal error of movement outcome. There are very few studies that have evaluated the performance outcome of aiming movements in terms of direct evaluations of the actual probability distribution as opposed, for example, to just assuming a normal movement error distribution. The departure from normality in the probability distribution could, however, influence the information content in movement performance (Newell & Hancock, 1984; Soukoreff & MacKenzie, 2004), especially when applying the model of Fitts’ law in aiming movements. The purpose of this study was to examine the unification of the spatial and temporal error variables of discrete movements into a single joint probabilistic space-time measure and to investigate how different task goals (point-aiming and target-aiming), movement strategies (space-emphasis and time-emphasis) and movement speed-accuracy conditions influenced this unified score in contrast to their established effects on the standard variance measures on each of the dimensions considered separately. It was anticipated that the movement variability of aiming to a point and of aiming to a target in terms of the probabilistic approach would be different from the variability measured in the traditional single dimension distributional analysis, and that each measure will be influenced by movement strategy and task goal (Carlton, 1994; Fitts, 1966; Zhai, Kong, & Ren, 2004). In general, higher probability is related to lower entropy (uncertainty) or lower variability in a system (Cover & Thomas, 1991). In Experiment 1 it was hypothesized that movement variability would be influenced by an instructional emphasis of accuracy or speed (Fitts, 1966) so that, for example, the spatial variability of the space-emphasis group would be lower than that of the time-emphasis group, and vice-versa. However, the influence of instructional emphasis on the entropy of the combined space-time measure is less certain a priori, as it has been proposed that in dual space-time criterion tasks (Hancock & Newell, 1985; Newell, 1980), participants trade spatial error for timing error, rather than speed for accuracy. This leads to the hypothesis that entropy of movement in space-time is lowest in the midrange of space-time parameter constraints (Hsieh, Liu, Mayer-Kress, & Newell, 2013) that was tested directly in Experiment 2. The demonstration of instructional and task influences on the unified spatial and temporal variability would provide support for the validity and utility of the unified spatial-temporal measure of movement outcome variability.

Experiment 1: Instructional Speed-Accuracy Strategy and Point/Target Aiming In this experiment the contrast of movement space-time entropy to the traditional spatial and temporal movement errors was made in the context of the known effects of movement speed-accuracy strategies (Fitts, 1966) and point/target aiming (Carlton, 1994) tasks. The functions for spatial error and temporal error have been shown to be different though related when considered in the unified spatial-temporal framework (Hancock & Newell, 1985; Hsieh et al., 2013; Newell, 1980). Here we contrast the separate and unified error functions for movement time and spatial accuracy through the joint probabilistic measurement of entropy of the space-time outcome in the MC Vol. 19, No. 3, 2015

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context of movement speed-accuracy instructions and point/target aiming. It was hypothesized that the joint probability measurement would provide a measure of entropy in space-time leading to a different effect from the consideration of movement speed and accuracy on timing error or spatial error considered separately.

Methods

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Participants Eighteen right-handed, healthy young adults were randomly assigned to one of three experimental groups: control with equal space and time emphasis, spaceemphasis, and time-emphasis. All participants gave their informed consent to the experimental procedures that were approved for compliance with the policy of the Institutional Review Board of Penn State University.

Apparatus and Task The apparatus in the experiment was a computer driven graphics writing tablet (Lai et al., 2005). The participants were requested to perform discrete aiming movements with specific spatial and temporal constraints. The experiment consists of two tasks: a point-aiming task and a target-aiming task. In the point-aiming task, participants were requested to draw a line from a point (2 mm in diameter) on the left of the body midline and to aim to stop at a point (2 mm in diameter) on the right of the body midline. The movement outcomes do not always fall on the target point; instead they create a certain target width that has been called an effective target width (We). The We was defined as the onedimensional distance between the maximal and minimal movement amplitudes in the point-aiming tasks. The individual’s We was used as the new target width in the target-aiming tasks (Carlton, 1994). The experimental strategy was to have subjects complete line drawing movements with a designated movement time and fixed movement amplitude to a point target. Based on the measured We on this task, a fixed-size target was constructed that encompassed 95% of the movement outcome performed in the point task. The same subjects performed the same line drawing movement, but under the target-aiming task (see Figure 1). The focus of this experiment was to investigate the unification of the spatial and temporal movement scores of the two tasks in terms of the probabilistic structures of movement outcome. This provides the structure of spatial and temporal variability in aiming movements that has been masked by using the traditional distributional approach (mean or standard deviation) of each dimension separately. To investigate the performances of point-aiming and target-aiming tasks, and since movement velocity is a determinant in the control of discrete aiming movements (Newell, Hoshizaki, Carlton, & Halbert, 1979), the average movement velocity was held constant through the combination of amplitude and movement time being fixed at 50 mm-500 ms.

Procedures The experiment required each participant to complete 500 point-aiming trials and 500 target-aiming trials in each space-time condition in two days. The individual subjective MC Vol. 19, No. 3, 2015

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Figure 1 — A schematic of: (a) the point-aiming task with the effective target width (We), and (b) the target-aiming task with the target width. A indicates amplitude. The W in Figure 1.b is determined by the individual’s We in Figure 1a.

effective target width (We) was calculated after the completion of the 500 pointaiming trials in that condition. The corresponding target width for the following 500 target-aiming tasks was determined from the individual’s We. Trials on the targetaiming task were performed next day following testing on the point-aiming task. The goal of this experiment was to test whether the task goals (point-aiming and target-aiming) and movement strategies influenced the performance of discrete aiming movements in terms of a single index that unifies spatial and temporal error. The verbal instructions to the participants provided a means to control the participant emphasis on the respective spatial and temporal criteria. A control group had the instruction of the equal emphasis on space and time. The participants in this group were instructed to aim to the point or target as accurately and consistently as possible giving equal emphasis to the space and time criteria. For the spatialemphasis and temporal-emphasis groups, participants were instructed to pay more attention to the spatial and temporal constraints, respectively. The three instructional groups participated in both point-aiming and target-aiming tasks. The initiation of MC Vol. 19, No. 3, 2015

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movement was defined by the stylus crossing the low velocity threshold of 3mm/s and staying above that threshold for 30 ms (6 frames). The stylus was to remain in contact with the tablet during the movement until the trial was completed. The trial was completed when the stylus touched the target position or the stylus came to a stop. The end of movement was defined by either the stylus leaving the tablet surface and/or the stylus’s velocity below 3 mm/s for more than 40 ms (8 frames). Participants received the feedback on the screen (temporal error and also subjects could see how far their stylus point from the target) within 2s of finishing the trial and returned the stylus back to the home position to start the next trial. The next trial started as soon as the participant was ready. The performance entropy was calculated in terms of probability based on an analysis of the frequency distribution of movement outcome. The number of bins, based on the systematic analyses, was set at 20 for both the spatial and temporal data, with the last 450 trials in each of the respective dimensions. The performance variability was the summation of the spatial and temporal measures in terms of probability. A detailed explanation of the determination of the unified space-time entropy measure is provided in Appendix A.

Data Analysis The primary independent variables in Experiment 1 were the instructional movement strategies (control, spatial and temporal emphases) and task goals (pointaiming and target-aiming). The major dependent variables included constant error, variable error, information entropy in terms of the joint probabilistic spatial and temporal measures of the discrete aiming movements. A 3 (movement strategy) × 2 (task goal) mixed ANOVAs was used to test the main effects of the movement strategies (between-subject factor) and task goals (within-subject factor) on the respective dependent variables. The Greenhouse-Geisser correction was used to correct violation of sphericity and the Bonferoni correction was applied for the post hoc comparisons, with eta square (η2) (Green & Salkind, 2003) revealing the strength of the effects.

Results Distributional Analyses Spatial Error.  Figure 2 (left column) shows the spatial constant error (CE), variable

error (VE), and information entropy of the point-aiming and target-aiming tasks as a function of instructional bias. The black and gray bars indicate the point-aiming and target aiming tasks, respectively. There was no significant interaction between instructional strategy and task goal for the three spatial dependent variables: F(2, 15) = 1.64, p > .05 for spatial CE, F(2, 15) = 0.28, p > .05 and for spatial VE. The task goal effect on spatial CE was significant, F(1, 15) = 52.51, p < .00, η2 = 0.78, but not for spatial VE, F(1, 15) = 0.73, p > .05. There were no movement strategy effect, F(2, 15) = 0.49, p > .05 for spatial CE, and F(2, 15) = 3.16, p > .05 for spatial VE. The 3 (movement strategy) × 2 (task goal) mixed ANOVAs for entropy in space showed that there was no significant interaction between instructional strategy and MC Vol. 19, No. 3, 2015

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Figure 2 — Left Column. Spatial constant error (CE), variable error (VE), and entropy of the point-aiming and target-aiming tasks as a function of the different instructional bias groups. The black and gray bars indicate the point-aiming and target-aiming tasks, respectively. Right Column. Temporal constant error (CE), variable error (VE), and entropy of the point-aiming and target-aiming tasks as a function of the different instructional bias groups. The black and gray bars indicate the point-aiming and target-aiming tasks, respectively. The error bars denote the between subject standard deviation. MC Vol. 19, No. 3, 2015

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task goal in the entropy of space, F(2, 15) = 3.38, p > .05. The task goal effect on entropy of spatial error was significant, F(1, 15) = 9.93, p < .05, η2 = 0.40, but not for the movement strategy, F(1, 15) = 1.76, p > .05.

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Temporal Error.  Figure 3 (right column) shows the temporal constant error CE,

variable error VE, and entropy of the point-aiming and target-aiming tasks as a function of the different instructional strategy groups. An interaction between movement strategy and task goal showed a significant difference for temporal CE: F(2, 15) = 7.91, p < .00, η2 = 0.62. The post hoc simple main effect analysis showed that the temporal CE of the space-emphasis group of the point-aiming tasks was significantly higher than that of the target-aiming tasks. However, the task effect was not observed for the other two groups. The interaction was not significant for temporal VE, F(2, 15) = 0.46, p > .05. Similar to the results in the spatial error analysis, the task goal effect was significant for the temporal CE, F(1, 15) = 24.61, p < .00, η2 = 0.62, but not for the temporal VE, F(1, 15) = 3.43. There were no movement strategy effect, F(2, 15) = 2.49, p > .05 for temporal CE, and F(2, 15) = 1.37, p > .05 for temporal VE. The 3 (movement strategy) × 2 (task goal) mixed ANOVAs for timing entropy showed that there was no significant interaction between instructional strategy and task goal, F(2, 15) = 1.49, p > .05. The task goal effect in timing entropy was significant, F(1, 15) = 30.62, p < .05, η2 = 0.67, also for the movement strategy effect, F(2, 15) = 7.99, p < .05, η2 = 0.52. The post hoc paired comparisons in the

Figure 3 — The unified spatio-temporal entropy of the point-aiming and target-aiming tasks as a function of the different instructional bias groups. The black and gray bars indicate the point-aiming and target-aiming tasks, respectively. The error bars denote the between subject standard deviation. MC Vol. 19, No. 3, 2015

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movement strategy showed that timing entropy in the space-emphasis group was significantly higher than time-emphasis and control groups. Entropy Analysis.  Figure 3 shows the performance variability, i.e., the unified spatiotemporal entropy, of the point-aiming and target-aiming tasks as a function of the different instructional bias groups. In the probabilistic approach to analysis the movement spatial and temporal variables were unified based on the same criterion. The 3 (movement strategy) × 2 (task goal) mixed ANOVA was conducted to test the effects of the movement strategy and task goal factors on the performance variability, which is the spatiotemporal entropy of the discrete aiming movements (see the detailed algorithms example in Appendix A). The task goal effect was significant F(2, 15) = 56.83, p < .05, η2 = 0.79, where the space-time entropy of the point aiming task was significantly higher than target aiming task, but the movement strategy effect was not significant F(2, 15) = 0.67, p > .05. In addition, there was a significant interaction between instructional strategy and task goal in the unified spatiotemporal entropy, F(2, 15) = 8.82, p < .05, η2 = 0.54. The post hoc simple main effect analysis showed that entropy in the point-aiming task was significantly higher than target aiming task in both the time-emphasis group and control group, but not in the space-emphasis group.

Experiment 2: Movement Speed-Accuracy Relation in Space-Time We investigated the speed-accuracy relation in space-time through a line drawing task that had error criteria in terms of both spatial and temporal dimensions (Hancock & Newell, 1985) in a second experiment. This provided a more direct test than that provided in Experiment 1 of the effect of instructional emphasis on either timing emphasis or space emphasis of movement. The data were taken from Hsieh et al. (2013, Experiment 2) and reanalyzed in terms of the space-time entropy measure developed here. The hypothesis was that the unified space-time entropy measure would reveal the trade-off between space and time and reveal a new movement speed-accuracy error function from those in place for the single dimensions of spatial or temporal error.

Methods Participants Twelve right-handed healthy young adults (6 males and 6 females) who did not participate in Experiment 1 were recruited for this experiment. The experimental procedures were approved by the Penn State University IRB committee and all participants signed a consent form before engaging in the experimental tests.

Apparatus A Wacom Intuos graphic tablet (Model GD-1218-R, 12 × 18 in) was connected to a PC computer (the pixel range was set at 800 × 600) and used for data collection. MC Vol. 19, No. 3, 2015

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The sample frequency was 130 Hz. A customized line drawing program was used to preset different feedback weightings in space-time and calculate the performance score for the participant immediately after each trial. The ratio of the distance moved by the cursor on the screen to the actual distance moved by the pen on the graphic board was 1:1.

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Task The task was to draw a line between two targets on the graphic tablet with a pen. The movement distance was held constant (20 cm) between the two targets (2mm in diameter), and the criterion movement time ranged from fast (250 ms), fast-middle (425 ms), middle (600 ms), middle-accurate (775 ms), to accurate (950 ms) for each condition. The line was drawn from left (starting point) to right (target point) and the computer screen was positioned at the midline in front of the participant’s body. Each participant completed 5 conditions here labeled as fast, fast-middle, middle, middle-accurate and accurate. Each space-time condition consisted of 220 trials. It took approximately 45 min to complete one space-time condition and each participant came in 5 consecutive days to complete the 5 testing conditions. The order of the 5 conditions was randomly determined for each participant. Before the start of a trial, the positions of the starting and target points were shown on the computer screen. The participants were instructed to look at the screen as they drew the line from the left point to the right point. There was no concurrent feedback provided on the screen of the past motion of the pen point, that is, no trace of the line was shown on the screen when performing the task. A performance score that was a weighted combination of the contribution of space and time to the criterion was presented on the screen immediately after the completion of each trial. Participants were instructed that their goal was to minimize this performance score (as close to 1 as possible). The start of the movement and movement stop were defined as in Experiment 1. Participants received the weighted space-time feedback score on the screen within 2 s of completing the trial and then moved the pen back to the left start point and the next trial began as soon as the participant was ready. Derivation of the Weighted Spatial and Temporal Task Criterion.  In Experiment 2, we develop and investigate a new space-time approach to weighting and decomposing the contributions of the dimensions of space and time to an integrated space-time performance score. Before the experiment proper some pilot procedures were adopted to construct the design of the experimental conditions and the weighting of the space-time performance score (Hsieh et al., 2013). Two pilot testing participants performed 300 trials of line drawing task with 2 movement time conditions: 150 ms and 1500 ms. The pilot participants were instructed to draw the line to the point target as accurately as possible under the specific movement time requirement. The pilot data showed mean timing error of 72 ms and mean spatial error of 14.86 mm for the 150 ms condition. For the 1500 ms condition, the mean timing error was 572 ms and the mean spatial error was 0.41 mm. We applied the base 10 logarithmic function to the ratios of timing error to spatial error (Log10 [72/14.86], Log [572/0.41]) as the linear interpolation function to generate the bases for the additional space-time conditions between MC Vol. 19, No. 3, 2015

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the fast and slow (accurate) conditions (0.68, 1.29, 1.91, 2.52, and 3.13). We then applied the base 10 exponential function to the logarithms to derive the weighting ratios of the space-time conditions: 1:4 (fast), 1:20 (fast-middle), 1:80 (middle), 1:320 (middle-accurate), and 1:1400 (accurate). These weighting coefficients were used in the following equation that provided a preliminary testing score:

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Preliminary Testing Score = Pt × MT + Pe × E (1)

where MT is the time between the start and stop of each movement trial; E is the Euclidean distance between the target point and the end point of each trial, Pt is the weighting parameter for MT that has an unit that is the inverse of the MT unit, Pe is the weighting parameter for E that has an unit that is the inverse of the E unit. Therefore, the values of Pt and Pe were 1:4; 1:20; 1:80; 1:320; 1:1400, respectively. Thus, for example, in the fast condition the weighting coefficients were 1 (Pt) and 4 (Pe) for the preliminary testing score (Equation 1). The criterion MT of different space-time conditions was generated by way of the linear interpolation based on the mean MT of the pilot testing conditions, 213 ms and 929 ms. The criterion MTs used in Experiment 2 were 250 ms, 425 ms, 600 ms, 775 ms, and 950 ms. We also used the SDs of the pilot testing data and linear interpolation to establish the spatial error criteria, 15 mm, 10.5 mm, 6.5 mm, 3 mm, and 0.5 mm. For the timing error criteria, one half of the SD from the pilot testing data were used in the linear interpolation procedure to prevent overlapping situations for adjacent space-time conditions (e.g., mid fast and middle conditions). The criteria timing errors of Experiment 2 were 25 ms, 38 ms, 50 ms, 62 ms, and 75 ms. These criteria were used to establish the norms of each condition to rescale the preliminary feedback scores. The norm (N) was calculated with the following equation:

N = Pt × Ct + Pe × Ce (2)

where Pt and Pe were the same weight factors according to the condition, and Ct and Ce were the criterion timing error and the criterion spatial error (e.g., Pt is 1, Pe is 4, Ct is 25 ms, and Ce is 15 mm in the fast condition). The final performance score was taken as the ratio of the preliminary testing score to the norm:

Performance Score = Preliminary testing score/N

In other words, the weighted movement time and spatial error in each condition was scaled to the criterion score (as established above) of its condition. Given the two pilot procedures described above, we established “criterion” movement time/ spatial error conditions at different points on the speed-accuracy trajectory relation. If each respective criterion was achieved then the participant would receive a performance score of 1 for that condition. Data Analysis.  The major dependent variables included constant error, variable error, and the performance variability in terms of the joint probabilistic spatial and temporal measures of the discrete aiming movements. Repeated-measures ANOVAs were used to examine the effects of the space-time task conditions on each dependent variable. The Greenhouse-Geisser correction was used to correct violation of MC Vol. 19, No. 3, 2015

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sphericity and the Bonferoni correction was used for Post hoc comparisons and eta square (h2) (Green & Salkind, 2003) were used to reveal the specific differences of group means and to test the strength of the effects, respectively.

Results

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Distributional Analyses Spatial Error.  Figure 4 (left column) shows the shows the spatial constant error (CE), variable error (VE), and information entropy of the movement aiming tasks as a function of different speed-accuracy constraints. The one way repeated measure ANOVA revealed a significant effect for condition, F(4, 44)=20.387, p < .05, ηp2 =0.65 in the spatial variable error, but not in the spatial constant error, F(4, 44)=0.267, p > .05. The post hoc paired comparisons in the spatial variable error showed that all of the conditions were significantly different from each other, except the fast from fast-middle conditions, fast-middle from accurate conditions, and middle from middle-accurate and accurate conditions. The one way repeated measure ANOVA on the spatial entropy revealed a significant effect for condition, F(4, 44)=31.477, p < .05, ηp2 =0.74. The post hoc paired comparisons in the spatial entropy showed that all of the conditions were significantly different from each other, except the fast-middle from the accurate condition, middle from both middle-accurate and accurate conditions, and middleaccurate from accurate. Temporal Error.  Figure 4 (right column) shows the temporal constant error (CE), variable error (VE), and information entropy of the movement aiming tasks as a function of different speed-accuracy constraints. The one way repeated measure ANOVA revealed a significant effect for condition, F(4, 44)=41.561, p < .05, ηp2 =0.79 in the temporal constant error, but not in the temporal variable error, F(4, 44)=1.62. The post hoc paired comparisons in the temporal constant error showed that all of the conditions were significantly different from each other, except the fast from fast-middle conditions, The one way repeated measure ANOVA on the temporal entropy revealed a significant effect for condition, F(4, 44)=6.863, p < .05, ηp2 =0.38. The post hoc paired comparisons in the temporal entropy showed that fast from accurate conditions, middle from both middle accurate, and accurate conditions were significantly different from each other. Entropy Analysis.  Figure 5 shows the performance variability, i.e., the unified spatiotemporal entropy, of the line drawing task tasks as a function of the different spatial and temporal constraints. In the probabilistic approach to analysis the movement spatial and temporal variables were unified based on the same criterion. The one way repeated measure ANOVA on the unified spatiotemporal entropy revealed a significant effect for condition, F(4, 44)=20.165, p < .05, ηp2 =0.65. The post hoc paired comparisons in the unified spatiotemporal entropy showed that all of the conditions were significantly different from each other, except the fast-middle from middle-accurate and accurate conditions, and middle from middleaccurate conditions. MC Vol. 19, No. 3, 2015

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Figure 4 — Left Column. Spatial constant error (CE), variable error (VE), and entropy of the line drawing task as a function of different spatial and temporal constraints. Right Column. The temporal constant error (CE), variable error (VE), and entropy of the line drawing tasks as a function of spatial and temporal constraints. The error bars denote the between subject standard deviation. MC Vol. 19, No. 3, 2015

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Figure 5 — The unified spatio-temporal entropy of the line drawing task at different spatial and temporal constraints. The error bars denote the between subject standard deviation.

Discussion The concept of probability is the foundation of information entropy (Cover & Thomas, 1991; Shannon & Weaver, 1949; Williams, 1997). In the direct probabilistic approach of entropy all the movement outcomes can be measured and used in the experimental analysis, without assuming how the data of movement outcome are distributed (as in the use of standard deviation for variability analysis), or being concerned that data outliers may cause systematic or random errors. Higher probability relates to lower entropy or lower variability in a system. In the current study, the entropy was defined from the individual spatial or temporal error data and the particular focus was the joint probabilities of the unified spatial and temporal movement outcome entropy. The experiments through established manipulations of speed-accuracy relations tested the hypothesis that a unified probabilistic spatial and temporal entropy measure can reflect a single index of movement variability (Hancock & Newell, 1985). The effects of movement strategies (control, space-emphasis and time-emphasis), task goals (point-aiming and target-aiming) and movement speed-accuracy relations on the unified movement spatial and temporal outcomes were examined. The goal of the study was to investigate how these common manipulations on movement speed and accuracy relate to both the standard spatial and temporal error measures considered independently (as they usually are), and a unified space-time entropy measure within the concept of movement error in space-time as outlined by Hancock and Newell (1985). MC Vol. 19, No. 3, 2015

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The findings from both experiments showed that the space-time entropy measure changed systematically in a way that is consistent with considering the combined effects of the spatial or temporal outcome measured individually. The spatial and temporal instructional movement strategy emphasis in Experiment 1 led to the same level of space-time entropy when the individual dimension (space or time) variability showed the contrasting effects of movement velocity. This effect on space-time entropy was paralleled and shown more explicitly in Experiment 2 where the graded effects of movement velocity led to the minimum of the joint probabilistic measure of entropy in the middle range. Thus, movement accuracy considered simultaneously in the joint probabilities of space and time reveals a different movement speed-accuracy function (Hancock & Newell, 1985), from those generated on the standard consideration of spatial or temporal error alone. The task goal manipulation of Experiment 1 showed a significant difference between the entropy of the point-aiming and target-aiming tasks. In the control, space-emphasis and time-emphasis groups, the entropy was higher in the pointaiming task than in the target-aiming task. Furthermore, the single point target resulted in relatively more variability of movement outcome on the respective single dimension measures (Carlton, 1994). Since the concept of probability was used in the calculation of performance variability, it was concluded that the target-aiming task led to higher movement consistency than the point-aiming task. With respect to the traditional distributional analyses, the findings showed that only variable spatial error was affected by the task goal. In the temporal domain, on the contrary, the movement strategy influenced both the temporal constant error and variable error, and an effect of task goal was found on the temporal constant error. It appeared that the movement strategy and task goal generated more constraint to movement temporal variability than to movement spatial variability. In addition, the space-emphasis instruction led to higher temporal constant error and variable error than the time-emphasis. This suggests that the movement temporal characteristics could be more easily changed compared with the spatial characteristics by the different movement strategies and different task goals. This finding is consistent with the proposition of Hancock and Newell (1985) that when there are dual spatial and temporal criteria, the performance trades spatial error for temporal error (and vice versa), rather than speed and accuracy (see also Hsieh et al., 2013; Newell, 1980). Here our Experiment 1 measured directly the joint probability of spatial and temporal scores in point-aiming and target-aiming tasks with different emphasis of accuracy instructions on space, time, or both. The performance variability in the current study has unified the spatial and temporal movement variables and no simple main effect of movement strategy was found in Experiment 1. The unified measure of spatial and temporal entropy showed coherent systematic change under the task goal and instructional bias conditions, in a way that is not reflected in the consideration of the spatial and temporal errors on an independent basis. In summary, the experiments have provided evidence for the validity of a space-time measure of movement outcome entropy in movement aiming tasks where there are implicitly or explicitly dual space and time outcome criteria. We anticipate that this approach will also be relevant in aiming tasks where a moving object needs to be intercepted in some fashion so that movement outcome is determined by being in a particular place in a particular time within a redundant solution frame of reference. MC Vol. 19, No. 3, 2015

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Appendix A Calculation of Space-Time Entropy From the Joint Spatial and Temporal Error Probabilities Here we provide an example of the calculation of the joint probability of spatial and temporal error for the point-aiming task in the time-emphasis group for a single participant. The upper left panel and upper right panel of Figure A1 show respectively the distribution of temporal error and spatial error individual trial data of the 450 trials used in the analysis of this condition. The lower left and right panel shows the 2 dimension and 3 dimension of spatio-temporal error distribution. With the 450 trials in the individual spatial and temporal dimensions 20 bins were used to cover the range of the spatial and temporal error data. Scott’s (1992) rule holds that the number of bins should be at least a × s × n -1/3, where a =3.49, s is an estimate of the standard deviation and n is number of trials. As Scott noted many authors advise that for real data sets histograms based on 5–20 bins usually suffice. To calculate the outcome entropy of discrete aiming movements, one needs to know the probabilities of the data distribution of the movement outcome. In the method for obtaining the probabilities in the experimental data, we used properties MC Vol. 19, No. 3, 2015

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of the actual frequency distribution, and we calculated the entropy (Hp) obtained with the following equation:

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Hp = ∑ Pi log2 (1/Pi) (A1) where Pi is defined in the frequency distribution, indicating the relative frequencies of data points in the ith bin (Shannon & Weaver, 1949; Williams, 1997). Here we used the concept of joint probability to calculate the joint probability of spatial and temporal error for the point-aiming task and to observe how different instructional and experimental emphases influence the values of information entropy. P (x, y) is the joint probability that can be obtained from the following equation (Williams, 1997, Eq. 27.12, p.413). P(x,y) = P(x)P(y | x) (A2)

Figure A1. The spatial (upper right panel) and temporal (upper left panel) distributions of the individual trial error data of the 450 trials of a single participant in the point-aiming task in the time-emphasis group. The distribution of the unified spatiotemporal errors is organized for 2 dimensions (lower left panel) and 3 dimensions (lower right panel with 20 × 20 bins).

MC Vol. 19, No. 3, 2015

Entropy of Movement Outcome in Space-Time.

Information entropy of the joint spatial and temporal (space-time) probability of discrete movement outcome was investigated in two experiments as a f...
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