Measuring the Difficulty of Steering Through Corners - Semantic Scholar
CHI 2006 Proceedings • Menus
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Measuring the Difficulty of Steering Through Corners Robert L. Pastel Computer Science Department Michigan Technology University Houghton, MI, 49931 [email protected] ABSTRACT
The steering law [4],
The steering law is intended to predict the performance of cursor manipulations in user interfaces, but the law has been verified for only a few path shapes and should be verified for more if it is to be generalized. This study extends the steering law to paths with corners. Two experiments compare the movement times of negotiating paths with corners to straight paths with the same width and movement amplitude. The experimental results show a significant effect on the movement times due to the corners, extending far into the legs of the path’s corner. Modeling the results using resource theory, a cognitive theory for divided attention, suggests that steering through corners is two simultaneous tasks: steering along the legs of the corner and aiming at the corner.
MT = a + b ∫ ds W ( s) , C
predicts the time to move a cursor through a prescribed path (called a tunnel in the papers of Accot and Zhai [2,3] ), C, with width, W(s), along the path. Again a and b are empirically determined regression coefficients. Accot and Zhai have demonstrated the generality of the law for different input devices [3], scales [2] and paths. The steering law defines the difficulty of a steering task by the index of difficulty, ID S = ∫ ds W ( s) , C
Author Keywords
ACM Classification Keywords
H5.2. Information interfaces and presentation (e.g., HCI): User Interfaces – Theory and Methods. INTRODUCTION
Currently, user interface designers have only two theories that describe cursor manipulation on the computer; Fitts’ law and the steering law. Fitts’ law [1, 7, 11], (1)
predicts the mean movement time, MT, for a user to move the cursor to a target that is a distance A away and has characteristic size W. The coefficients a and b are empirically determined regression constants. Fitts’ law has been found to be very robust and applicable to a variety of tasks. [10, 12, 1] Fitts’ law defines the difficulty of an aiming or acquisition task by the index of difficulty, IDF = log2(A/W +1),
(4)
which we will call the steering index of difficulty, IDS. For a straight path with length A and constant width W the steering index of difficulty is IDS = A/W. (Actually, IDS=A/(Wln2) but convention omits the 1/ln2 factor.) The form of the straight path steering law is qualitatively similar to Fitts’ law. The difference in the functional dependence of A/W for the measurement time between the steering law for straight paths and Fitts’ law, linear versus logarithmic, represents the increased difficulty of confining the cursor to the path. The similarity between Fitts’ law and the steering law is not coincidental. Accot and Zhai derived the steering law as the limit of a sequence of infinitesimal Fitts tasks with only lateral constraints. Because Fitts’ law is robust in the constraining direction [1, 4], the integral in the steering law transforms the aiming constraints in the free motion of the Fitts task to the lateral constraints in a steering task.
Steering law, Fitts’ law, Menu navigation, Gesturing.
MT = a + blog2(A/W +1),
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(2)
which we will call the Fitts index of difficulty, IDF. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. CHI 2006, April 22–27, 2006, Montréal, Québec, Canada. Copyright 2006 ACM 1-59593-178-3/06/0004...$5.00.
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Ahlström [5] has studied the MT for selections in cascading pull-down menus. He proposed that negotiating through pull-down menus is a sequence of tasks, Fitts tasks and steering tasks. Because vertical motion in menu hierarchies is through wide paths, he approximates the index of difficulty for the vertical part of the task, IDV, as a Fitts task, IDV = IDF. Because the horizontal motion is generally through more constrained paths, he models the horizontal part of the task, IDH, as a steering task, IDH = IDS. He also statistically models the two tasks with the same index of performance by adding the two index of difficulties, IDT=IDV +IDH. Considering that the fitted models reported in the literature for the two tasks generally have different slopes, 160 ms/bit for Fitts tasks [1, 11] and 48 ms/bits for
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steering tasks [3] (after correcting the slope for the omitted 1/ln2 factor in the IDS), the assumption may not always be valid. This paper demonstrates that negotiating around corners and consequently through menu hierarchies are not so straightforward. Ahlström demonstrates a correlation of MT with IDT, and recommends that “deeper analyses of the exact cursor paths…will help to further refine the model.” Although this paper does not study specific cursor paths, it does study in detail the MT to negotiate corners for a variety of path widths and corner angles.
effects parameters. Standard F-tests can determine the significance of terms in the fixed-effects model. A mixedeffect analysis can increase the significance of terms by accounting for the participants’ variations. We demonstrate that this study does not require this increase of significance by performing both the analysis on the mean MT and the mixed-effects analysis for all experiments. In addition the results from the analysis on the mean MT can be compared directly with earlier steering experiments. We examine the participants’ variances, which provide a measure for understanding differences between users, and should be an important quantity in HCI design and theory.
PROCEDURE AND ANAYLSIS
All experiments were performed on generic PCs equipped with 17-inch monitors (1280x1024 pixels resolution) and optical mice. The paths were displayed in random order. The participants were instructed to click in the green circle (dark grey in Figure 1) and move as quickly as possible to the red circle (light grey in Figure 1) while remaining in the black area. Figure 1 also shows as light grey points sampled cursor locations for a participant’s trial, which were collected in 16msec intervals. Timing started after moving into the black region and terminated when the cursor entered the red circle. Accuracy in negotiating the path was measured as the percentage of sampled points in the black area to the total sampled points. If the accuracy for a trial is below 98%, the participant is required to repeat the trial, and the trial is recorded as a failure. We feel that lowering the accuracy requirement to 98% better reflects real life negotiating menu hierarchies.
(a) Wide path
EXPERIMENT 1: DIFFERENT ANGLE CORNERS
In this experiment we study the effects of different corner angles on the MT and compare the times with straight paths. Intuitively the paths with corners can be negotiated by one of two modes; “cutting off the corner,” and “stop and go.” The sampled cursor positions in Figure 1 illustrate the two modes of steering through 90° paths. We believe that the narrow paths will encourage negotiating the paths by “stop and go,” and wider paths will encourage negotiating the paths by “cutting off the corner.” This experiment tests participants on paths of two different widths, 20-px and 80-px. Design
Thirteen experienced computer users volunteered to participate in the experiment, average age 21 years. Only one of the participants was left handed. Straight paths and three different corner angles, 45°, 90°, and 135°, were studied. The straight paths were studied in both the horizontal direction (to the right) and vertical direction (downwards). The first leg of all corner paths started toward the right and the second leg downward. The path length, A, is measured along the path’s centerline and between the red and green circles. The narrow paths have width, W = 20 px, and path length, A, and index of difficulty, IDS (IDS = A/W), A = 50 px, IDS = 2.5; A = 100 px, IDS = 5; and A = 150 px, IDS = 7. The wide paths have width, W = 80 px, and path lengths and IDS, A = 200 px, IDS = 2.5; A = 400 px, IDS = 5; and A = 600 px, IDS = 7. The participants negotiated all 30 paths twice in a single sitting.
(b) Narrow path
Figure 1: 90° paths; (a) 80-px wide path, “cutting off the corner” mode, (b) 20-px wide path, “stop and go” mode. The dots are the sampled cursor positions.
Straight Paths Results
As expected, the effect of IDS is significant on steering time, F2, 247 = 27.5, p < 0.0001. Also the path width is significant, F1, 247 = 43.7, p < 0.0001. There is no distinction between vertical and horizontal paths, F1, 248 = 0.5, p = 0.47, in this experiment. This result differs from the results of Dennerlein et al. [6]; they propose that the dependence of MT on path direction may be due to the differences in joint kinematics required for vertical and horizontal motion. We would expect that at smaller movement amplitudes the difference in MT due to direction would diminish. For small IDS, IDS ≤16, the results of Dennerlein et al. would give a much smaller difference in the slopes of the regressions for the vertical and horizontal directions. The interaction plots for the mean movement times, Figure 2, illustrate these
We perform both the analysis on the mean MT for the interaction plots and mixed-effects analysis [15, 16] on the raw MT for the linear models. Mixed-effects analysis extends single level analysis by specifying fixed and random effects. The fixed-effects parameters are the typical parameters specified in a single level analysis. The analysis uses both the random effects model parameters and the structure of the variance-covariance matrix for the parameters to specify the variations between the groups, participants in our experiments. Our analysis uses the same model for both fixed and random effects and a diagonal variance-covariance matrix, restricting the random effects to only the variances of the random
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(b) Width - ID Interaction
Figure 2: Straight paths interaction plots, (a) direction and IDS on mean MT, and (b) path width and ID on the mean MT.
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Figure 3: MT against IDS for the path directions and widths, (a) narrow paths, 20 px, and (b) wide paths, 80 px. Vertical path data shifted to right, and lines are the fitted model.
75% of the failed path negotiations were performed on the same path, 20 px wide and IDS = 7.5. The failure rate does have a trend with path width; the error rate is 8% and 1% for path widths 20 px and 80 px respectively. The lower failure rate in our study compared to the study of Accot and Zhai [4, 2] may be due to our relaxed criteria for successfully negotiating a path, 98% accuracy. Also our participants are young and very experienced computer users.
Corner Paths Results
The effect of IDS is significant, F2, 375 = 122, p < 0.0001. Also the effect of the corner angle is significant, F2, 375 = 76, p < 0.0001, and less so the effect of width, F2, 375 = 6.0, p = 0.014. The interaction of path width and corner angle with IDS are much less significant, F2, 375 = 2.6, p = 0.07, and F4, 375 = 2.3, p = 0.05. The interaction plots for the mean MT, Figure 4, illustrate these effects. The corner angle-IDS interaction plot, Figure 4(a), shows a clear trend of mean MT with the path’s corner angle. Paths with 45° corners have consistently the lowest average MT.
Mean MT (ms)
Term Width Estim. Std. Btw. Part. (Units) (px) Value Error Std. Dev. 20 Intercept 72 26 45 (ms) 80 126 22 2 20 Slope 34 5 11 (ms/bit) 80 43 5 13 Residual Std. Dev.: 96 ms Deg. of Free.: 255
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Table 1: Straight paths’ estimated values, standard errors, and between-participant standard deviations.
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Figure 3 shows all the collected MT against IDS for both path widths and path directions. To enhance visibility the vertical path data are artificially shifted to the right. Also shown in Figure 3 is the fitted model. The model includes only the path width and ID factors, and separate slopes and intercepts for the two path widths. Table 1 lists the estimated values for the fitted coefficients, standard errors, and between-participant standard deviations. The residual standard deviation is 96 ms, and the first and third quartiles of the standardized withinparticipant residuals are -0.61 and 0.52, respectively. The large between-participant standard deviation for the vertical paths’ intercepts suggests that the outliers are associated with specific participants. The between-participant standard
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deviation of the slopes is larger than the standard error, and implies a significant slope dependency on the participants. Some participants negotiate the paths consistently faster than other participants. Neither 95% confidence intervals for the slopes, (24 ms/bit, 44 ms/bit) for the narrow paths and (32 ms/bit, 53 ms/bit) for the wide paths contain the results from Accot and Zhai [3], 69.6 ms/bit for steering straight paths using a mouse. Our participants are expert computer users, and we can expect the lower slopes that occur with practice. The difference between the slopes for the horizontal and vertical paths is ≈25% and is significant.
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effects. The direction-IDS interaction plot, Figure 2(a), shows the insignificance of direction on the MT in this IDS range. The effect of path width is illustrated by Figure 2(b). The wider paths’ mean MTs are consistently higher than the narrow paths. In addition the ID-MT slope is steeper for the wider paths than for the narrow paths. We find the same trend in the ID-MT slope for path width in our second experiment. We will not attempt to determine the cause for the slope dependency on path width, and will analyze our results factored by path width.
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Figure 4: Corner path interaction plots, (a) corner angle and ID on the mean MT, (b) path width and IDS on the mean MT.
Paths with 90° corners have consistently the highest mean MT. The effect of the corner angle appears to be primarily on the intercept, but paths with 90° corners have significantly steeper IDS -MT slope. The lower mean MT for paths with 135° corners than for paths with 90° corners is probably due to the smaller deviation required to negotiate the path. We believe the
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cause of the lower mean MT for paths with 45° corners is biomechanical. If for 45° corners we model the participant’s limb as a cantilever, loaded during the approach to the corner and unloaded while exiting, then the deviation in negotiating the corner is no more than for a 135° corner. Also, negotiating the 45° corner benefits from the release of muscle tension developed during the approach to the corner. The effect of path width is illustrated by the width-IDS interaction plot, Figure 4(b). Again the wide paths have greater IDS -MT slopes than the narrow paths, but unlike the straight paths the intercept is less for the wide paths causing the crossing of the lines.
and intercepts for the two path widths and corner angles. Table 2 lists the estimated values for the coefficients, standard error, and between-participant standard deviations. The residual standard deviation is 131 ms, and the first and third quartiles of the standardized within-participant residuals are -0.56 and 0.48, respectively. The large between-participant standard deviation for the intercept and small between-participant standard deviation for the slopes of the narrow paths with 45° corners implies that outliers across the IDS are associated with specific participants. The between-participant standard deviations of the slopes are larger than their standard errors implying that some participants consistently negotiate corners faster than other participants. None of the 95% confidence intervals for the slopes of paths with corners contain their respective slopes for the straight paths. We conclude that the effect of the corner is significant. Term (units)
Width Angle Estim. Std. Btw. Part. (px) Value Error Std. Dev. 45 82 46 64 20 90 243 43 0.05 Intercept 135 198 44 0.37 (ms) 45 22 45 50 80 90 27 44 0.03 135 112 43 0.06 45 58 8 0.006 20 90 72 10 21 Slope 135 53 10 19 (ms/bit) 45 67 8 5 80 90 120 12 32 135 70 10 19 Residual Std. Dev.: 131 ms Deg. of Free.: 381
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The failure rate for negotiating paths with corners is dependent on the path width, F1, 10 = 5.6, p = 0.04, but less significantly on the movement amplitude, F1, 10 = 3.7, p = 0.08, or corner angle F1, 10 = 4.2, p = 0.07. The rate of failure for paths with corners is greater than for the straight paths, 12% and 6% for path widths 20 px and 80 px. The trend of the failure rate with respect to corner angle is 4% for paths with 45° corners, and 11% for paths with 90° and 135° corner angles. Apparently 45° corners are easier to negotiate than 90° corners or even 135° corners.
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Table 2: Paths with corners’ estimated values, standard errors, and between-participant standard deviations.
The relative difference of the intercepts between the two path widths decrease by ≈70%, ≈90%, and ≈40% for corner angles 45°, 90°, and 135°; the narrow paths have larger intercepts. While the relative difference of the slopes between the two path widths increase by ≈10%, ≈40%, and ≈20% for corner angles 45°, 90°, and 135°; the narrow paths have smaller slopes. Also, apparent from the models in Figure 5(b), the MT for short wide corner paths is nearly the same as the MT for the short wide straight paths. We believe that the similarity of MT, and decrease in intercept for the wider paths are manifestations of “cutting off the corner,” which allows participants to negotiate wide corners with shorter movement amplitude than straight paths. The slope increases because “cutting off the corner” has less effect on the longer paths. The paths with corner angles 45°, and 90° benefit more from “cutting off the corner” than paths with 135° corners.
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Figure 5: MT vs. IDS for paths’ corner angles and widths, (a) narrow paths, (b) wide paths; 90° and 135° corner angle data are shifted right. Bold solid, dashed and dotted lines are models; model from the straight paths are lower solid thin lines.
The corner model diverges from the straight path models for both widths. The slope of the narrow paths with corners has increased by ≈40% (averaged over the three corner angles) from the straight narrow path slopes, while the average wide path slopes increased slightly more, ≈45%. There is little variation of the slope increase due to corner
Figure 5 shows the collected MT against IDS for both path widths and corner angles. To enhance visibility the 90° and 135° corner angle data are progressively shifted to the right. Also shown in Figure 5 is the fitted model. The model includes the path width, corner angle and ID factors; separate slopes
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slopes for the mean MT increase nearly linearly from 33 ms/bit for path width 20 px to 65 ms/bit for the 120-px wide path. 800
angle, but the paths with 90° corners have larger increases for both the narrow and wide paths, ≈50% and ≈65%, respectively. The increase of slopes suggests that the IDS for paths with corners have additional factors determining their difficulties.
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The comparison of straight and corner paths in Experiment 1 suggests that the difficulty of negotiating a corner is not solely dependent on IDS = A/W. Because the paths with 90° corners have the largest slope increase, we presume studying paths with 90° corners will best illuminate the effect of a corner on the MT. Also 90° cursor manipulations are very common in user interfaces, for example negotiating a pull-down menu hierarchy. Experiment 2 focuses on 90° corners and attempts to reveal the additional difficulties of steering through corners by studying the MT over a greater range of IDS, and path widths.
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Figure 6: Straight path factored mean MT for width and IDS. Design
The error rate for the corner paths is dependent on both the path width and movement amplitude, F1, 35 = 25, p < 0.0001, F6, 35 = 14, p < 0.0006, respectively. The percent failure negotiating paths are 6%, 4%, 4% and 1% for path widths 20 px, 40 px, 60 px, and 80 px respectively. No errors were performed on the 100-px and 120px wide paths. Except for the 20-px wide paths the failures were for larger IDS, IDS ≥ 8.
Fourteen experienced computer users volunteered to participate in this experiment. The average age was 23, and all participants were right handed. Six different path widths were studied, W = 20 px, 40 px, 60 px, 80 px, 100 px, 120 px, for both straight paths and paths with 90° corners. A fully crossed design of path widths with path length is not possible because of screen limitation at the wider paths. Table 3 lists all tested path dimensions for straight paths and paths with 90° corners. The straight paths were tested only in the horizontal direction, towards the right, and the paths with 90° corners started downward and then toward the right. The participants negotiated the 90 paths twice in a single seating. W (px) 20 IDS 1.25* 2.5 4 6 8 10 12 16
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Width Estim. (px) Value 20 32 40 45 Intercept 60 57 (ms) 80 72 100 62 120 53 20 32 40 34 Slope 60 40 (ms/bit) 80 45 100 48 120 64 Residual Std. Dev.: 145 ms
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Std. Btw. Part. Error Std. Dev. 19 2 19 2 31 91 26 53 26 2 50 3 5 16 4 11 5 15 5 15 7 16 14 29 Deg. Free.: 1028
Table 4: Straight path parameters; estimated values, standard errors, and between-participant standard deviations.
Figure 7 shows the collected data with the mixed-effects analysis models; the primary purpose of the figure is to show the distribution of the individual MT with the fitted models. Table 4 lists the estimated values for the coefficients, standard error, and between-participant standard deviations. The residual standard deviation is 145 ms, and the first and third quartiles of the standardized within-participant residuals are -0.39 and 0.28, respectively. The large between-participant standard deviations of the intercepts for paths 20-px and 60-px wide suggest that the outliers in the mid IDS range are associated with specific participants. The between-participant standard deviations for the slopes are significantly larger than their standard error, and imply a significant variation in the speed of
Table 3: Straight and corner path lengths. Lengths and widths are in pixels. *tested only for straight paths, #tested only corners. Straight Paths Results
The effects of both path width and IDS are significant, F6,1028 = 50, p < 0.0001, and F6,1028 = 59, p < 0.0001, respectively. The factored mean MT for IDS and path width are shown in Figure 6. Except for the 60-px wide path, the mean MT have regular increments with IDS. The linear regressions of the mean MT against IDS for each path width have high correlation, r2 > 0.98, except for path width 60 px which has r2 = 0.96, as implied by the regular vertical increments of the mean MT with IDS in Figure 6. The
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corners follows the same trend with width as the failure rate for straight paths, and the failure rate is twice the failure rate for straight paths. This implies that the paths with corners are significantly more difficult to negotiate.
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participants negotiating the paths. Only the 95% confidence intervals for the 120-px wide paths contain the results from Accot and Zhai [3].
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Figure 9: 90° path data with fitted model, bold lines. Lower light weight lines are the straight path models.
Figure 7: Straight path data with fitted mixed-effects model 90° Corner Paths Results
Figure 9 shows the collected data with the fitted models from the mixed-effects analysis (bold lines). Also shown for comparison are the fitted models for the corresponding width straight paths (lower and shallower thin lines). Table 5 lists the estimated values for the coefficients, standard errors, and between-participant standard deviations. The residual standard deviation is 239 ms, and the first and third quartiles of the standardized within-participant residuals are -0.40 and 0.30, respectively. The large between-participant standard deviations of the intercepts for the 20-px and 40px wide paths imply that outliers in the mid IDS range are associated with specific participants. The betweenparticipant standard deviation for the slopes is the same magnitude as for the straight paths.
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Width Estim. (px) Value 20 205 40 185 Intercept 60 67 (ms) 80 87 100 82 120 30 20 85 40 75 Slope 60 94 (ms/bit) 80 98 100 105 120 118 Residual Std. Dev.: 238 ms
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The effects of both path width and IDS are significant, F6, 1001 = 181, p < 0.0001, and F6, 1136 = 173, p < 0.0001, respectively. The factored mean MT for IDS and path width are shown in Figure 8. Except for the 20-px wide paths, the increments of MT with IDS are regular. The linear regressions of the mean MT on IDS have high correlations, r2 = 0.99, for all path widths, as implied by the regular increments of MT with IDS in Figure 8. Unlike the straight paths the fitted regression line for 20-px wide path has a larger slope than the 40-px wide path, 85 ms/bit and 75 ms/bit respectively. The slopes of the fitted lines for larger width paths increase with width to 120 ms/bit for the 120-px wide path.
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Figure 8: 90° paths’ factored mean MT for IDS and path width
The failure rate for negotiating paths with 90° corners is dependent on the path widths, F1, 34 = 181, p < 0.0001, but is not dependent on movement amplitude, F1, 34 = 0.02, p = 0.87. The percent failure negotiating paths with corners are 12%, 8%, 3%, 2%, 1%, and 1% for path widths 20 px, 40 px, 60 px, 80 px, 100 px and 120 px respectively. The failure rate for paths with
Std. Btw. Part. Error Std. Dev. 59 165 41 57 38 0.7 38 0.7 45 1 62 1 8 27 5 13 6 17 6 18 8 21 13 21 Deg. Free.: 1001
Table 5: 90° corner paths’ parameter; estimated value, standard error, and between-participant standard deviations.
Only the 95% confidence interval of the slope of the 120-px wide paths with corners, (93 ms/bit, 142 ms/bit), contains the slope for the corresponding width straight paths. The
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can explain an increase in steering time for narrow paths (“stop and go” mode) due to deceleration and acceleration at the corner. Deceleration and acceleration are confined to the corners’ proximity, increasing the model’s intercept. But we have also observed increases in the slopes for corners at the higher IDS of Experiment 2. This implies that the effect of the corner extends far into the legs, and modeling steering through a corner as steering two straight paths is not adequate. Model Derivation
Negotiating a path with a corner requires continuous awareness of the corner; so that changing the cursor’s direction is synchronized at the corner. This synchronization is fundamentally an aiming or acquisition task, in other words a Fitts task. We propose that steering through the corner requires the participant to simultaneously perform two coupled tasks: steering along a straight path and synchronization at the corner. The local form of the steering law [4] for a path width W is: vS = W/τS,
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where vF is the Fitts velocity, s is the distance to the target, and τF is the participant’s Fitts time constant. The law implies that participants move the cursor slower as it approaches the target. The two laws impose simultaneous constraints on the actual velocity, v, meaning v ≤ vF, vS. How the actual velocity is constrained is unknown, but a simple formula is to add the constraints in parallel,
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where vS is the steering velocity, and τS is the participant’s steering time constant. The law restricts the steering velocity by the lateral constraints of the path. A local form for Fitts’ law has many variations [10], one such law is:
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difference between the results from the straight and corner models in this experiment qualitatively agree with the difference found in Experiment 1. The slopes for paths with corners increase approximately 50 ms/bits from the estimated slopes for the straight paths, and the intercept differences decrease with path width. Figure 10 quantitatively shows the difference in slopes and intercepts between paths with corners and the straight paths. The line in Figure 10(a) is a linear regression of the slope differences, r2 = 0.23. The 0.07 ms/px slope is not significant, F1, 4 = 1.28, p = 0.3, implying that there is no path width dependency on the slope difference for straight paths and paths with corners. The trend for the slope difference in experiment 1 between straight paths and paths with 90° corners is much larger, 0.65 ms/px. This discrepancy in the trend of the slope difference between the two experiments may be due to the difference in the range of the measured IDS for the 80-px wide paths, (2.5 bits, 7 bits) for experiment 1 and (2.5 bits, 16 bits) for experiment 2. The effect of the corner would decrease with the length of the legs of the corner. The intercept difference should be more invariant to the range of observed IDS. The intercept difference, Figure 10(b), generally decreases with path width. The linear regression for the intercept difference, r2 = 0.80, has a slope of -1.9 ms/px. The slope is significant, F1, 4 = 15.8, p = 0.016. The 95% confidence interval for the slope, (3.2 ms, -0.58 ms), contains the trend from experiment 1, -2.8 ms/px.
(7)
Expression (7) can be by derived from a cognitive theory for divided attention, resource theory [17, 18]. This is not the first study to use resource theory to model the performance of cursor manipulation as a dual task; Gopher and Navon [8, 13] illustrated the fundamental principles of resource theory in a study for two dimensional tracking. When two tasks are structurally similar, they interfere with each other by demanding resources from the same processor with limited capacity. The performance of a single task, performed alone, is typically data limited, meaning limited by the data transfer rate. The performance of a steering task through a constant width straight path should be proportional to the average velocity, v =A/MT, and can be normalized by the steering velocity, Equation (5),
Figure 10: Corner and straight paths coefficient difference, (a) slopes’ differences, (b) intercepts’ differences. CORNER STEERING MODEL
In summary, we have observed that steering though a corner increases the slope in the fitted model, and there is minimal effect due to the angle of the corner. Surprisingly, paths with 45° corners are negotiated faster than 90° and 135° corners. We will ignore the effect of the corner angle because the existence of a corner of any angle has a profound effect on the steering time. The ≈50 ms/bit increase in the slopes for paths with 90° corners at all widths implies that the effect of the corner extends far into the legs of the corners. The effect of “cutting off the corner” is manifested by the smaller intercepts for the models of wider paths. “Cutting off the corner” reduces the time to steer the proximity of the corner by shortening the path length and reducing deceleration and acceleration. Modeling steering through corners as steering two straight paths
pS = v/vS.
(8)
Similarly as Accot and Zhai [4], we will associate the instantaneous velocity with the average velocity. We recognize perfect performance, pS = 1, as the data limited performance. Norman and Bobrow [14] propose using a
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performance-resource function (PRF) to describe the relationship between performance and resource allocation. The only requirement on the PRF is that it should be nondecreasing. A horizontal asymptote in the PRF implies that the task is data limited after sufficient resource has been allocated to the task. We propose that there is no data limited region in the PRF for a steering task and that the PRF is linear, in other words, pS = rS, where rS is the fraction of resource allocated to the steering task.
words the time to negotiate a path with a corner can be modeled as a the sum of a steering and Fitts terms, MTC = a + bIDS + cIDF.
where a, b, and c are experimentally determined regression coefficients. Model - Experiment Comparison
The Fitts index of difficulty, IDF = log2(A/2W +1), in the model for corner steering, Equation (13), is nonlinear in A/W =IDS, the steering index of difficulty. This suggests that a regression analysis on the MT for paths with corners using the model in Equation (13) can possibly discriminate the Fitts term from the steering term. The regression analysis found that the IDF dependency is not significant, F6, 995 = 0.3, p = 0.94. The lack of significance does not invalidate Equation (13) as a corner steering model; it can be due to residual error in the data obscuring the nonlinearity. Even in the presence of residual data error, the linear part of the Fitts term can contribute to the increase in the slopes for the regressions. In other words, the fitted constant for the slope of MT with respect to IDS, b, is equal to the sum of a factor due to steering without the corner and a factor from the linear part of the Fitts term. Because we observe different IDS ranges for the path widths, the linear fraction of the Fitts term, IDFSlope, should be estimated for each path width as the slope from a linear regression on the set of points
We define the instantaneous Fitts performance as the fraction of progress to the target, p F = ∆s v F ∆t = v v F ,
(9)
where ∆s is the progress to the target in time ∆t, and vF ∆t is the progress that perfect performance would achieve. Again we assume that the PRF is linear without a data limited region, pF = rF, where rF is the fraction of resource allocated to the Fitts task. Several Fitts studies suggest that the PRF does not have a data limited asymptote. The inclusion of even a minimal additional demand on the participants’ resource, such as dragging rather than clicking [12] and selecting the target during the aiming process [9], degrades the Fitts performance. When the two tasks are performed simultaneously, they can deplete the processor’s resource and the performance becomes resource-limited. The performance for negotiating paths with corners degrades because the Fitts task associated with the synchronization at the corner demands resource, rF, leaving only rS = 1 – rF resource for the steering task. The steering and Fitts performance must satisfy, pS = rS = 1 – rF = 1 – pF.
{(IDi, log2(IDi /2 + 1)},
(10)
b F = (bC − bS ) ID FSlope ,
IDC = ∫ ds v( s) C
,
(15)
where bC and bS are the slopes from the linear regressions for paths with corners and straight paths, respectively. We should also correct the slopes for the omitted 1/ln(2) factor in the steering index of difficulty for constant width paths. For the second experiment, the procedure above generates 6 estimates for bF. According to our steering corner model, the Fitts term contribution to the slope should be dependent only on A/W and independent of the width. We can consider the 6 estimates, bF, as representing a random sample of estimates for the fitted slope of a hypothetical Fitts experiment. The mean of the six bF, 227 ms/bit, differs from the fitted slope found in the literature [1, 11], 160 ms/bit, t5 = 4.9, p = 0.004. The conditions of this hypothetical Fitts experiment are significantly different from the Fitts experiments in the literature. Also some tasks are not accounted for in our model, for example while negotiating the approach leg to the corner the participants must glance at different locations; this adds additional demands on the resource [17]. Because changing direction and initiating steering along the exit leg of the corner may require centering the cursor better than clicking on a button, the effective Fitts target size of the corner may be smaller. Also changing the cursor movement direction in itself may be an additional task. Acceleration in the exit leg
Equation (7) can be integrated to get the form of the index of difficulty to negotiate a path with a corner, IDC,
C
(14)
where IDi = Ai /Wi are the IDS observed for that path width. Then the Fitts term contribution to the slope, bF, at each path width can be estimated from the experiment by
Equation (10) is the same formula for the actual velocity as Equation (7). We have shown that adding simultaneous constraints in parallel is consistent with interference between two tasks with linear PRF. The addition of data limited regions in the PRF will change the relationship between the performances, and Equation (10) will not be the same as Equation (7). But small variations from linearity in the PRFs will not substantially change the equation for the actual velocity.
= ∫ (τ S W + τ F s )ds
(13)
(11)
= τ S ( A W ) + τ F ln( A W )
where the Fitts term is integrated only along the leg approaching the corner and the steering term is integrated along both legs. We identify the first term as the steering index of difficulty, IDS, for the path without a corner and identify the second term as the Fitts index of difficulty, IDF, for a target width, W, and initial separation, A/2. In other
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CONCLUSIONS
increases movement time, but most of the effect should be to increase the intercept of the regression. All these unaccounted effects would tend to yield a larger bF.
The steering law is potentially a useful quantitative law in human-computer interaction research and design, especially for modeling gestures and cursor manipulations. But the law should be evaluated for each path shape. We have found that the slope of the fitted model is significantly larger for paths with corners compared to straight paths. Surprisingly, the slope is less for paths with 45° and 135° corners than paths with 90° corners. The paths with 45° corners have the lowest MT. The implication to user interface designers is that gestures with corners should be designed with acute angles rather than 90° angles. Because participants can negotiate wide corners by “cutting off the corner,” the wide paths with corners at small IDS have smaller movement times than corresponding straight paths. This implies that gestures with wide corners or large tolerances are significantly easier to articulate. Interface designers should expect that users will round off corners while gesturing or negotiating menu hierarchies, and should allow for large tolerances at the corners. Many user interfaces on Windows and Mac platforms allow the user to round off the corner while transitioning between menu levels. Because the difference in the slopes of the fitted models between straight paths and paths with corners persists to large IDS, up to IDS ≈ 16, we conclude that the effect of the corner is more than deceleration and acceleration at the corner. The user interface designer should be aware of the additional difficulty due to the corner, even in long gestures. Modeling steering through the corner as two tasks, steering through a straight path and a Fitts task associated with the corner, furnishes a simple formula with familiar terms for the difficulty of negotiating paths with corners,
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The six bF estimates have standard deviation 34 ms/bit, which is not that large because the bF statistics is a difference statistics of bC and bS. The Q-Q plot for the bF samples, Figure 11(a), has small deviations for the extreme quantiles from the line through the first and third quartiles implying that the bF distribution is close to normal. The residual plot, Figure 11(b), identifies that the first quantile is the estimate for the 120-px wide paths. This estimate is suspect because the paths are clearly in the region of “cutting off the corner,” wide paths at small IDS.
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(a) Q-Q plot for est. Fitts's slopes
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Figure 11: Diagnostic plots for estimated Fitts slopes
We can construct a semi-empirical model for corner steering by adding the estimated Fitts contribution for the slope to the straight path slopes. Figure 12 compares the semi-empirical model for the mean of the calculated bF, 227 ms/bit, and the value found in the literature, 160 ms/bit, to the slopes for the corner data. Note that the slopes have been reduced by ln(2) to correct for the omitted factor in IDS. The model deviates significantly from the data only at 40-px wide paths and 120px wide paths. The 120-px wide path datum is suspect because it is clearly in the “cutting off the corner” mode, and the 40-px wide path with a 90° corner datum appears to be abnormally low. Comparison of the models for the two bF shows that the model is sensitive to bF.
IDC ~ IDS + IDF.
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The plus symbol in Relation (16) should not be taken literally as the sum of the two indices of difficulty; Relation (16) only expresses that the total index of difficulty is directly proportional to both the Fitts and steering indices of difficulty. A statistical model should use two factors, but because the nonlinearity of IDS in IDF can be obscured by the error in the data, the effect of IDF is to increase the slope of the regression with IDS. Our results show that the two effects are always present in paths with equal width legs from 20-px to 120-px, and that steering through a corner with equal width legs cannot be approximated as solely a Fitts task even in wide paths or as solely a steering task even in narrow paths. A study of steering through corners with unequal width legs could determine if steering through corners with narrower exit legs can be approximated by a Fitts task. If our model is correct, the effect of the corner for paths with long approach legs, IDS > 16, becomes proportionally less, but verifying this result is confounded by other effects, such as directional dependency [6] and scaling dependency [2] in the steering index of difficulty.
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(16)
The power of Relation (16) is that it allows the HCI designer to apply results from Fitts experiments to cursor manipulations with corners. For example refining Fitts’ law for bivariate pointing, Accot and Zhai [1] found that increasing the aspect
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Figure 12: Comparison of dual task model and 90° corner data
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REFERENCES
ratio of the target, W/H, where W is the amplitude tolerance, H is the directional tolerance, decreases the MT to the target. In light of Accot and Zhai’s results and our understanding of the difficulties involved with corner steering, we can explain the apparent difficulty of negotiating menus that do not allow rounding off corners. Typical menu hierarchies, curve (a) in Figure 13, require the user to negotiate a wide path, large directional tolerance, H, to a corner with small amplitude tolerance, W. Because the Fitts task of the corner is amplitude dominate, the larger directional tolerance does not alleviate the Fitts task making negotiating the corner difficult, and effects the negotiation all along the leg. Increasing menu hierarchy performance requires increasing the width of the corner in both directions. When the exit leg is wider then the entrance leg, Accot and Zhai’s results suggest that the MT for negotiating menus can be reduced. Examples of menu hierarchies with wide exit legs are pull-down menus from the tool bar of applications, curve (b) in Figure 13. Negotiating to the pull-down menu is easier than the corresponding command selection in the pull-downed menu, curve (a) in Figure 13. The exit leg does not need to widen much. Accot and Zhai found that increasing W/H beyond 3 did not reduce the MT significantly.
1. Accot, J., and Zhai, S., Refining Law Models for Bivariate Pointing, In Proc.CHI2003 ACM Press (2003), 193-200. 2. Accot, J., and Zhai, S., Scale Effects in Steering Law Tasks, In Proc. CHI2001, ACM Press (2001), 1-8. 3. Accot, J. and Zhai, S., Performance Evaluations of Input Devices in Trajectory-based Tasks: An application of the Steering Law, In Proc. CHI99, ACM Press (1999), 466-472. 4. Accot, J., and Zhai, S., Beyond Fitts’ Law: Models for Trajectory-Based HCI Tasks, In Proc. CHI97, ACM Press (1997), 295-302. 5. Ahlström, D., Modeling and Improving Selection in Cascading Pull-down Menus Using Fitts’ Law, the Steering Law and Force Fields, In Proc. CHI2005, ACM Press (2005), 61-70. 6. Dennerlein, J.T., Martin, D.B. and Hasser, C., ForceFeedback Improves Performance for Steering and Combined Steering-Targeting Tasks, In Proc. CHI2000, ACM Press (2000), 423-429. 7. Fitts, P.M., The Information Capacity of the Human Motor System in Controlling the Amplitude of Movement, Quarterly Journal of Psychology, (1954) 47, 381-391. 8. Gopher, D., and Navon, D., How is Performance Limited: Testing the Notion of Central Capacity, Acta Psychologica (1980) 46, 161-180.
(b)
(a)
9. Hoffmann, E.R. and Lim, J.T.A., Concurrent Manualdecision Tasks, Ergonomics, (1997) 40:3, 293-318. 10. Langolf, G.D., Chafflin, D.B., and Foulke, J.A., An Investigation of Law Using a Wide Range of Movement Amplitude, J. Motor Behav.,(1976) 8:2,113-128. 11. MacKenzie, S. and Buxton, W., Extending Fitts’ Law to Two-dimensional Tasks, In Proc. CHI92, (1992), 219-223. 12. MacKenzie, S., Sellen, A. and Buxton, W., A Comparison of Input Devices in Elemental Pointing and Dragging Tasks, In Proc. CHI91, ACM Press (1991), 161-166.
Figure 13: Corner paths while negotiating a menu hierarchy
This paper is only a first study of steering through corners. The study of steering through paths with different corner leg widths would have a direct application to negotiating menu hierarchies. Studying the cursor velocity through the paths could add more validity to modeling the effect of the corner as a Fitts task, and could extract the acceleration along the corner’s exit leg. Extending the theory of steering through corners to longer legs would require comparison of straight paths in both the horizontal and vertical direction.
13. Navon, D., and Gopher, D., Task Difficulty, Resources, and Dual-task Performance, in Attention and Performance, edited by Nickersor, R.S., Erlbaum, NJ, 1980, 297-315. 14. Norman, D.A.and Bobrow, D.G., On Data-limited and Resource-limited Processes, Cognit. Psych., 7, 44-64, 1975. 15. Pinheiro, J.C. and Bates, D.M., Mixed-Effects Models in S and S-PLUS, Springer, New York, NY, 2000. 16. Venables, W.N. and Ripley, B.D., Modern Applied Statistics with S, Springer, New, NY, 2002.
ACKNOWLEDGMENTS
The author wishes to thank the many colleagues who contributed to this study, especially the department chairperson, Dr. Linda Ott, for providing resources for this study and input during the analysis, and the students who developed the test platform: Joseph Vaillancourt, Scott Gross, Jacob Champlin, and Robert Miller. Also the author thanks all the participants who volunteered for the tests.
17. Wickens, C.D. and Hollands, J.G., Engineering Psychology and Human Performance, third edition Prentice Hall, Upper Saddle River, New Jersey, 2000. 18. Wickens, C.D., and Lilu, Y., Codes and Modalities en Multiple Resources: A Success and A Qualification, Human Factors, 1988, 30, 5, 599-616.
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Measuring the Difficulty of Steering Through Corners Robert L. Pastel Computer Science Department Michigan Technology University Houghton, MI, 49931 [email protected] ABSTRACT
The steering law [4],
The steering law is intended to predict the performance of cursor manipulations in user interfaces, but the law has been verified for only a few path shapes and should be verified for more if it is to be generalized. This study extends the steering law to paths with corners. Two experiments compare the movement times of negotiating paths with corners to straight paths with the same width and movement amplitude. The experimental results show a significant effect on the movement times due to the corners, extending far into the legs of the path’s corner. Modeling the results using resource theory, a cognitive theory for divided attention, suggests that steering through corners is two simultaneous tasks: steering along the legs of the corner and aiming at the corner.
MT = a + b ∫ ds W ( s) , C
predicts the time to move a cursor through a prescribed path (called a tunnel in the papers of Accot and Zhai [2,3] ), C, with width, W(s), along the path. Again a and b are empirically determined regression coefficients. Accot and Zhai have demonstrated the generality of the law for different input devices [3], scales [2] and paths. The steering law defines the difficulty of a steering task by the index of difficulty, ID S = ∫ ds W ( s) , C
Author Keywords
ACM Classification Keywords
H5.2. Information interfaces and presentation (e.g., HCI): User Interfaces – Theory and Methods. INTRODUCTION
Currently, user interface designers have only two theories that describe cursor manipulation on the computer; Fitts’ law and the steering law. Fitts’ law [1, 7, 11], (1)
predicts the mean movement time, MT, for a user to move the cursor to a target that is a distance A away and has characteristic size W. The coefficients a and b are empirically determined regression constants. Fitts’ law has been found to be very robust and applicable to a variety of tasks. [10, 12, 1] Fitts’ law defines the difficulty of an aiming or acquisition task by the index of difficulty, IDF = log2(A/W +1),
(4)
which we will call the steering index of difficulty, IDS. For a straight path with length A and constant width W the steering index of difficulty is IDS = A/W. (Actually, IDS=A/(Wln2) but convention omits the 1/ln2 factor.) The form of the straight path steering law is qualitatively similar to Fitts’ law. The difference in the functional dependence of A/W for the measurement time between the steering law for straight paths and Fitts’ law, linear versus logarithmic, represents the increased difficulty of confining the cursor to the path. The similarity between Fitts’ law and the steering law is not coincidental. Accot and Zhai derived the steering law as the limit of a sequence of infinitesimal Fitts tasks with only lateral constraints. Because Fitts’ law is robust in the constraining direction [1, 4], the integral in the steering law transforms the aiming constraints in the free motion of the Fitts task to the lateral constraints in a steering task.
Steering law, Fitts’ law, Menu navigation, Gesturing.
MT = a + blog2(A/W +1),
(3)
(2)
which we will call the Fitts index of difficulty, IDF. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. CHI 2006, April 22–27, 2006, Montréal, Québec, Canada. Copyright 2006 ACM 1-59593-178-3/06/0004...$5.00.
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Ahlström [5] has studied the MT for selections in cascading pull-down menus. He proposed that negotiating through pull-down menus is a sequence of tasks, Fitts tasks and steering tasks. Because vertical motion in menu hierarchies is through wide paths, he approximates the index of difficulty for the vertical part of the task, IDV, as a Fitts task, IDV = IDF. Because the horizontal motion is generally through more constrained paths, he models the horizontal part of the task, IDH, as a steering task, IDH = IDS. He also statistically models the two tasks with the same index of performance by adding the two index of difficulties, IDT=IDV +IDH. Considering that the fitted models reported in the literature for the two tasks generally have different slopes, 160 ms/bit for Fitts tasks [1, 11] and 48 ms/bits for
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steering tasks [3] (after correcting the slope for the omitted 1/ln2 factor in the IDS), the assumption may not always be valid. This paper demonstrates that negotiating around corners and consequently through menu hierarchies are not so straightforward. Ahlström demonstrates a correlation of MT with IDT, and recommends that “deeper analyses of the exact cursor paths…will help to further refine the model.” Although this paper does not study specific cursor paths, it does study in detail the MT to negotiate corners for a variety of path widths and corner angles.
effects parameters. Standard F-tests can determine the significance of terms in the fixed-effects model. A mixedeffect analysis can increase the significance of terms by accounting for the participants’ variations. We demonstrate that this study does not require this increase of significance by performing both the analysis on the mean MT and the mixed-effects analysis for all experiments. In addition the results from the analysis on the mean MT can be compared directly with earlier steering experiments. We examine the participants’ variances, which provide a measure for understanding differences between users, and should be an important quantity in HCI design and theory.
PROCEDURE AND ANAYLSIS
All experiments were performed on generic PCs equipped with 17-inch monitors (1280x1024 pixels resolution) and optical mice. The paths were displayed in random order. The participants were instructed to click in the green circle (dark grey in Figure 1) and move as quickly as possible to the red circle (light grey in Figure 1) while remaining in the black area. Figure 1 also shows as light grey points sampled cursor locations for a participant’s trial, which were collected in 16msec intervals. Timing started after moving into the black region and terminated when the cursor entered the red circle. Accuracy in negotiating the path was measured as the percentage of sampled points in the black area to the total sampled points. If the accuracy for a trial is below 98%, the participant is required to repeat the trial, and the trial is recorded as a failure. We feel that lowering the accuracy requirement to 98% better reflects real life negotiating menu hierarchies.
(a) Wide path
EXPERIMENT 1: DIFFERENT ANGLE CORNERS
In this experiment we study the effects of different corner angles on the MT and compare the times with straight paths. Intuitively the paths with corners can be negotiated by one of two modes; “cutting off the corner,” and “stop and go.” The sampled cursor positions in Figure 1 illustrate the two modes of steering through 90° paths. We believe that the narrow paths will encourage negotiating the paths by “stop and go,” and wider paths will encourage negotiating the paths by “cutting off the corner.” This experiment tests participants on paths of two different widths, 20-px and 80-px. Design
Thirteen experienced computer users volunteered to participate in the experiment, average age 21 years. Only one of the participants was left handed. Straight paths and three different corner angles, 45°, 90°, and 135°, were studied. The straight paths were studied in both the horizontal direction (to the right) and vertical direction (downwards). The first leg of all corner paths started toward the right and the second leg downward. The path length, A, is measured along the path’s centerline and between the red and green circles. The narrow paths have width, W = 20 px, and path length, A, and index of difficulty, IDS (IDS = A/W), A = 50 px, IDS = 2.5; A = 100 px, IDS = 5; and A = 150 px, IDS = 7. The wide paths have width, W = 80 px, and path lengths and IDS, A = 200 px, IDS = 2.5; A = 400 px, IDS = 5; and A = 600 px, IDS = 7. The participants negotiated all 30 paths twice in a single sitting.
(b) Narrow path
Figure 1: 90° paths; (a) 80-px wide path, “cutting off the corner” mode, (b) 20-px wide path, “stop and go” mode. The dots are the sampled cursor positions.
Straight Paths Results
As expected, the effect of IDS is significant on steering time, F2, 247 = 27.5, p < 0.0001. Also the path width is significant, F1, 247 = 43.7, p < 0.0001. There is no distinction between vertical and horizontal paths, F1, 248 = 0.5, p = 0.47, in this experiment. This result differs from the results of Dennerlein et al. [6]; they propose that the dependence of MT on path direction may be due to the differences in joint kinematics required for vertical and horizontal motion. We would expect that at smaller movement amplitudes the difference in MT due to direction would diminish. For small IDS, IDS ≤16, the results of Dennerlein et al. would give a much smaller difference in the slopes of the regressions for the vertical and horizontal directions. The interaction plots for the mean movement times, Figure 2, illustrate these
We perform both the analysis on the mean MT for the interaction plots and mixed-effects analysis [15, 16] on the raw MT for the linear models. Mixed-effects analysis extends single level analysis by specifying fixed and random effects. The fixed-effects parameters are the typical parameters specified in a single level analysis. The analysis uses both the random effects model parameters and the structure of the variance-covariance matrix for the parameters to specify the variations between the groups, participants in our experiments. Our analysis uses the same model for both fixed and random effects and a diagonal variance-covariance matrix, restricting the random effects to only the variances of the random
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(b) Width - ID Interaction
Figure 2: Straight paths interaction plots, (a) direction and IDS on mean MT, and (b) path width and ID on the mean MT.
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75% of the failed path negotiations were performed on the same path, 20 px wide and IDS = 7.5. The failure rate does have a trend with path width; the error rate is 8% and 1% for path widths 20 px and 80 px respectively. The lower failure rate in our study compared to the study of Accot and Zhai [4, 2] may be due to our relaxed criteria for successfully negotiating a path, 98% accuracy. Also our participants are young and very experienced computer users.
Corner Paths Results
The effect of IDS is significant, F2, 375 = 122, p < 0.0001. Also the effect of the corner angle is significant, F2, 375 = 76, p < 0.0001, and less so the effect of width, F2, 375 = 6.0, p = 0.014. The interaction of path width and corner angle with IDS are much less significant, F2, 375 = 2.6, p = 0.07, and F4, 375 = 2.3, p = 0.05. The interaction plots for the mean MT, Figure 4, illustrate these effects. The corner angle-IDS interaction plot, Figure 4(a), shows a clear trend of mean MT with the path’s corner angle. Paths with 45° corners have consistently the lowest average MT.
Mean MT (ms)
Term Width Estim. Std. Btw. Part. (Units) (px) Value Error Std. Dev. 20 Intercept 72 26 45 (ms) 80 126 22 2 20 Slope 34 5 11 (ms/bit) 80 43 5 13 Residual Std. Dev.: 96 ms Deg. of Free.: 255
0
Table 1: Straight paths’ estimated values, standard errors, and between-participant standard deviations.
45 angle 90 angle 135 angle
Figure 3 shows all the collected MT against IDS for both path widths and path directions. To enhance visibility the vertical path data are artificially shifted to the right. Also shown in Figure 3 is the fitted model. The model includes only the path width and ID factors, and separate slopes and intercepts for the two path widths. Table 1 lists the estimated values for the fitted coefficients, standard errors, and between-participant standard deviations. The residual standard deviation is 96 ms, and the first and third quartiles of the standardized withinparticipant residuals are -0.61 and 0.52, respectively. The large between-participant standard deviation for the vertical paths’ intercepts suggests that the outliers are associated with specific participants. The between-participant standard
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deviation of the slopes is larger than the standard error, and implies a significant slope dependency on the participants. Some participants negotiate the paths consistently faster than other participants. Neither 95% confidence intervals for the slopes, (24 ms/bit, 44 ms/bit) for the narrow paths and (32 ms/bit, 53 ms/bit) for the wide paths contain the results from Accot and Zhai [3], 69.6 ms/bit for steering straight paths using a mouse. Our participants are expert computer users, and we can expect the lower slopes that occur with practice. The difference between the slopes for the horizontal and vertical paths is ≈25% and is significant.
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effects. The direction-IDS interaction plot, Figure 2(a), shows the insignificance of direction on the MT in this IDS range. The effect of path width is illustrated by Figure 2(b). The wider paths’ mean MTs are consistently higher than the narrow paths. In addition the ID-MT slope is steeper for the wider paths than for the narrow paths. We find the same trend in the ID-MT slope for path width in our second experiment. We will not attempt to determine the cause for the slope dependency on path width, and will analyze our results factored by path width.
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Figure 4: Corner path interaction plots, (a) corner angle and ID on the mean MT, (b) path width and IDS on the mean MT.
Paths with 90° corners have consistently the highest mean MT. The effect of the corner angle appears to be primarily on the intercept, but paths with 90° corners have significantly steeper IDS -MT slope. The lower mean MT for paths with 135° corners than for paths with 90° corners is probably due to the smaller deviation required to negotiate the path. We believe the
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cause of the lower mean MT for paths with 45° corners is biomechanical. If for 45° corners we model the participant’s limb as a cantilever, loaded during the approach to the corner and unloaded while exiting, then the deviation in negotiating the corner is no more than for a 135° corner. Also, negotiating the 45° corner benefits from the release of muscle tension developed during the approach to the corner. The effect of path width is illustrated by the width-IDS interaction plot, Figure 4(b). Again the wide paths have greater IDS -MT slopes than the narrow paths, but unlike the straight paths the intercept is less for the wide paths causing the crossing of the lines.
and intercepts for the two path widths and corner angles. Table 2 lists the estimated values for the coefficients, standard error, and between-participant standard deviations. The residual standard deviation is 131 ms, and the first and third quartiles of the standardized within-participant residuals are -0.56 and 0.48, respectively. The large between-participant standard deviation for the intercept and small between-participant standard deviation for the slopes of the narrow paths with 45° corners implies that outliers across the IDS are associated with specific participants. The between-participant standard deviations of the slopes are larger than their standard errors implying that some participants consistently negotiate corners faster than other participants. None of the 95% confidence intervals for the slopes of paths with corners contain their respective slopes for the straight paths. We conclude that the effect of the corner is significant. Term (units)
Width Angle Estim. Std. Btw. Part. (px) Value Error Std. Dev. 45 82 46 64 20 90 243 43 0.05 Intercept 135 198 44 0.37 (ms) 45 22 45 50 80 90 27 44 0.03 135 112 43 0.06 45 58 8 0.006 20 90 72 10 21 Slope 135 53 10 19 (ms/bit) 45 67 8 5 80 90 120 12 32 135 70 10 19 Residual Std. Dev.: 131 ms Deg. of Free.: 381
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The failure rate for negotiating paths with corners is dependent on the path width, F1, 10 = 5.6, p = 0.04, but less significantly on the movement amplitude, F1, 10 = 3.7, p = 0.08, or corner angle F1, 10 = 4.2, p = 0.07. The rate of failure for paths with corners is greater than for the straight paths, 12% and 6% for path widths 20 px and 80 px. The trend of the failure rate with respect to corner angle is 4% for paths with 45° corners, and 11% for paths with 90° and 135° corner angles. Apparently 45° corners are easier to negotiate than 90° corners or even 135° corners.
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Table 2: Paths with corners’ estimated values, standard errors, and between-participant standard deviations.
The relative difference of the intercepts between the two path widths decrease by ≈70%, ≈90%, and ≈40% for corner angles 45°, 90°, and 135°; the narrow paths have larger intercepts. While the relative difference of the slopes between the two path widths increase by ≈10%, ≈40%, and ≈20% for corner angles 45°, 90°, and 135°; the narrow paths have smaller slopes. Also, apparent from the models in Figure 5(b), the MT for short wide corner paths is nearly the same as the MT for the short wide straight paths. We believe that the similarity of MT, and decrease in intercept for the wider paths are manifestations of “cutting off the corner,” which allows participants to negotiate wide corners with shorter movement amplitude than straight paths. The slope increases because “cutting off the corner” has less effect on the longer paths. The paths with corner angles 45°, and 90° benefit more from “cutting off the corner” than paths with 135° corners.
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Figure 5: MT vs. IDS for paths’ corner angles and widths, (a) narrow paths, (b) wide paths; 90° and 135° corner angle data are shifted right. Bold solid, dashed and dotted lines are models; model from the straight paths are lower solid thin lines.
The corner model diverges from the straight path models for both widths. The slope of the narrow paths with corners has increased by ≈40% (averaged over the three corner angles) from the straight narrow path slopes, while the average wide path slopes increased slightly more, ≈45%. There is little variation of the slope increase due to corner
Figure 5 shows the collected MT against IDS for both path widths and corner angles. To enhance visibility the 90° and 135° corner angle data are progressively shifted to the right. Also shown in Figure 5 is the fitted model. The model includes the path width, corner angle and ID factors; separate slopes
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slopes for the mean MT increase nearly linearly from 33 ms/bit for path width 20 px to 65 ms/bit for the 120-px wide path. 800
angle, but the paths with 90° corners have larger increases for both the narrow and wide paths, ≈50% and ≈65%, respectively. The increase of slopes suggests that the IDS for paths with corners have additional factors determining their difficulties.
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The comparison of straight and corner paths in Experiment 1 suggests that the difficulty of negotiating a corner is not solely dependent on IDS = A/W. Because the paths with 90° corners have the largest slope increase, we presume studying paths with 90° corners will best illuminate the effect of a corner on the MT. Also 90° cursor manipulations are very common in user interfaces, for example negotiating a pull-down menu hierarchy. Experiment 2 focuses on 90° corners and attempts to reveal the additional difficulties of steering through corners by studying the MT over a greater range of IDS, and path widths.
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Figure 6: Straight path factored mean MT for width and IDS. Design
The error rate for the corner paths is dependent on both the path width and movement amplitude, F1, 35 = 25, p < 0.0001, F6, 35 = 14, p < 0.0006, respectively. The percent failure negotiating paths are 6%, 4%, 4% and 1% for path widths 20 px, 40 px, 60 px, and 80 px respectively. No errors were performed on the 100-px and 120px wide paths. Except for the 20-px wide paths the failures were for larger IDS, IDS ≥ 8.
Fourteen experienced computer users volunteered to participate in this experiment. The average age was 23, and all participants were right handed. Six different path widths were studied, W = 20 px, 40 px, 60 px, 80 px, 100 px, 120 px, for both straight paths and paths with 90° corners. A fully crossed design of path widths with path length is not possible because of screen limitation at the wider paths. Table 3 lists all tested path dimensions for straight paths and paths with 90° corners. The straight paths were tested only in the horizontal direction, towards the right, and the paths with 90° corners started downward and then toward the right. The participants negotiated the 90 paths twice in a single seating. W (px) 20 IDS 1.25* 2.5 4 6 8 10 12 16
25 50 80 120 160 200 240 320
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60 75 150 240 360 480 600 720 960
80 100 200 320 480 640 800 960 1280
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Width Estim. (px) Value 20 32 40 45 Intercept 60 57 (ms) 80 72 100 62 120 53 20 32 40 34 Slope 60 40 (ms/bit) 80 45 100 48 120 64 Residual Std. Dev.: 145 ms
120 300 480 720 960#
Std. Btw. Part. Error Std. Dev. 19 2 19 2 31 91 26 53 26 2 50 3 5 16 4 11 5 15 5 15 7 16 14 29 Deg. Free.: 1028
Table 4: Straight path parameters; estimated values, standard errors, and between-participant standard deviations.
Figure 7 shows the collected data with the mixed-effects analysis models; the primary purpose of the figure is to show the distribution of the individual MT with the fitted models. Table 4 lists the estimated values for the coefficients, standard error, and between-participant standard deviations. The residual standard deviation is 145 ms, and the first and third quartiles of the standardized within-participant residuals are -0.39 and 0.28, respectively. The large between-participant standard deviations of the intercepts for paths 20-px and 60-px wide suggest that the outliers in the mid IDS range are associated with specific participants. The between-participant standard deviations for the slopes are significantly larger than their standard error, and imply a significant variation in the speed of
Table 3: Straight and corner path lengths. Lengths and widths are in pixels. *tested only for straight paths, #tested only corners. Straight Paths Results
The effects of both path width and IDS are significant, F6,1028 = 50, p < 0.0001, and F6,1028 = 59, p < 0.0001, respectively. The factored mean MT for IDS and path width are shown in Figure 6. Except for the 60-px wide path, the mean MT have regular increments with IDS. The linear regressions of the mean MT against IDS for each path width have high correlation, r2 > 0.98, except for path width 60 px which has r2 = 0.96, as implied by the regular vertical increments of the mean MT with IDS in Figure 6. The
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corners follows the same trend with width as the failure rate for straight paths, and the failure rate is twice the failure rate for straight paths. This implies that the paths with corners are significantly more difficult to negotiate.
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participants negotiating the paths. Only the 95% confidence intervals for the 120-px wide paths contain the results from Accot and Zhai [3].
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Figure 9: 90° path data with fitted model, bold lines. Lower light weight lines are the straight path models.
Figure 7: Straight path data with fitted mixed-effects model 90° Corner Paths Results
Figure 9 shows the collected data with the fitted models from the mixed-effects analysis (bold lines). Also shown for comparison are the fitted models for the corresponding width straight paths (lower and shallower thin lines). Table 5 lists the estimated values for the coefficients, standard errors, and between-participant standard deviations. The residual standard deviation is 239 ms, and the first and third quartiles of the standardized within-participant residuals are -0.40 and 0.30, respectively. The large between-participant standard deviations of the intercepts for the 20-px and 40px wide paths imply that outliers in the mid IDS range are associated with specific participants. The betweenparticipant standard deviation for the slopes is the same magnitude as for the straight paths.
ID=10 ID=12 ID=16
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Width Estim. (px) Value 20 205 40 185 Intercept 60 67 (ms) 80 87 100 82 120 30 20 85 40 75 Slope 60 94 (ms/bit) 80 98 100 105 120 118 Residual Std. Dev.: 238 ms
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The effects of both path width and IDS are significant, F6, 1001 = 181, p < 0.0001, and F6, 1136 = 173, p < 0.0001, respectively. The factored mean MT for IDS and path width are shown in Figure 8. Except for the 20-px wide paths, the increments of MT with IDS are regular. The linear regressions of the mean MT on IDS have high correlations, r2 = 0.99, for all path widths, as implied by the regular increments of MT with IDS in Figure 8. Unlike the straight paths the fitted regression line for 20-px wide path has a larger slope than the 40-px wide path, 85 ms/bit and 75 ms/bit respectively. The slopes of the fitted lines for larger width paths increase with width to 120 ms/bit for the 120-px wide path.
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Figure 8: 90° paths’ factored mean MT for IDS and path width
The failure rate for negotiating paths with 90° corners is dependent on the path widths, F1, 34 = 181, p < 0.0001, but is not dependent on movement amplitude, F1, 34 = 0.02, p = 0.87. The percent failure negotiating paths with corners are 12%, 8%, 3%, 2%, 1%, and 1% for path widths 20 px, 40 px, 60 px, 80 px, 100 px and 120 px respectively. The failure rate for paths with
Std. Btw. Part. Error Std. Dev. 59 165 41 57 38 0.7 38 0.7 45 1 62 1 8 27 5 13 6 17 6 18 8 21 13 21 Deg. Free.: 1001
Table 5: 90° corner paths’ parameter; estimated value, standard error, and between-participant standard deviations.
Only the 95% confidence interval of the slope of the 120-px wide paths with corners, (93 ms/bit, 142 ms/bit), contains the slope for the corresponding width straight paths. The
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can explain an increase in steering time for narrow paths (“stop and go” mode) due to deceleration and acceleration at the corner. Deceleration and acceleration are confined to the corners’ proximity, increasing the model’s intercept. But we have also observed increases in the slopes for corners at the higher IDS of Experiment 2. This implies that the effect of the corner extends far into the legs, and modeling steering through a corner as steering two straight paths is not adequate. Model Derivation
Negotiating a path with a corner requires continuous awareness of the corner; so that changing the cursor’s direction is synchronized at the corner. This synchronization is fundamentally an aiming or acquisition task, in other words a Fitts task. We propose that steering through the corner requires the participant to simultaneously perform two coupled tasks: steering along a straight path and synchronization at the corner. The local form of the steering law [4] for a path width W is: vS = W/τS,
vF = s/τF,
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(6)
where vF is the Fitts velocity, s is the distance to the target, and τF is the participant’s Fitts time constant. The law implies that participants move the cursor slower as it approaches the target. The two laws impose simultaneous constraints on the actual velocity, v, meaning v ≤ vF, vS. How the actual velocity is constrained is unknown, but a simple formula is to add the constraints in parallel,
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where vS is the steering velocity, and τS is the participant’s steering time constant. The law restricts the steering velocity by the lateral constraints of the path. A local form for Fitts’ law has many variations [10], one such law is:
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difference between the results from the straight and corner models in this experiment qualitatively agree with the difference found in Experiment 1. The slopes for paths with corners increase approximately 50 ms/bits from the estimated slopes for the straight paths, and the intercept differences decrease with path width. Figure 10 quantitatively shows the difference in slopes and intercepts between paths with corners and the straight paths. The line in Figure 10(a) is a linear regression of the slope differences, r2 = 0.23. The 0.07 ms/px slope is not significant, F1, 4 = 1.28, p = 0.3, implying that there is no path width dependency on the slope difference for straight paths and paths with corners. The trend for the slope difference in experiment 1 between straight paths and paths with 90° corners is much larger, 0.65 ms/px. This discrepancy in the trend of the slope difference between the two experiments may be due to the difference in the range of the measured IDS for the 80-px wide paths, (2.5 bits, 7 bits) for experiment 1 and (2.5 bits, 16 bits) for experiment 2. The effect of the corner would decrease with the length of the legs of the corner. The intercept difference should be more invariant to the range of observed IDS. The intercept difference, Figure 10(b), generally decreases with path width. The linear regression for the intercept difference, r2 = 0.80, has a slope of -1.9 ms/px. The slope is significant, F1, 4 = 15.8, p = 0.016. The 95% confidence interval for the slope, (3.2 ms, -0.58 ms), contains the trend from experiment 1, -2.8 ms/px.
(7)
Expression (7) can be by derived from a cognitive theory for divided attention, resource theory [17, 18]. This is not the first study to use resource theory to model the performance of cursor manipulation as a dual task; Gopher and Navon [8, 13] illustrated the fundamental principles of resource theory in a study for two dimensional tracking. When two tasks are structurally similar, they interfere with each other by demanding resources from the same processor with limited capacity. The performance of a single task, performed alone, is typically data limited, meaning limited by the data transfer rate. The performance of a steering task through a constant width straight path should be proportional to the average velocity, v =A/MT, and can be normalized by the steering velocity, Equation (5),
Figure 10: Corner and straight paths coefficient difference, (a) slopes’ differences, (b) intercepts’ differences. CORNER STEERING MODEL
In summary, we have observed that steering though a corner increases the slope in the fitted model, and there is minimal effect due to the angle of the corner. Surprisingly, paths with 45° corners are negotiated faster than 90° and 135° corners. We will ignore the effect of the corner angle because the existence of a corner of any angle has a profound effect on the steering time. The ≈50 ms/bit increase in the slopes for paths with 90° corners at all widths implies that the effect of the corner extends far into the legs of the corners. The effect of “cutting off the corner” is manifested by the smaller intercepts for the models of wider paths. “Cutting off the corner” reduces the time to steer the proximity of the corner by shortening the path length and reducing deceleration and acceleration. Modeling steering through corners as steering two straight paths
pS = v/vS.
(8)
Similarly as Accot and Zhai [4], we will associate the instantaneous velocity with the average velocity. We recognize perfect performance, pS = 1, as the data limited performance. Norman and Bobrow [14] propose using a
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performance-resource function (PRF) to describe the relationship between performance and resource allocation. The only requirement on the PRF is that it should be nondecreasing. A horizontal asymptote in the PRF implies that the task is data limited after sufficient resource has been allocated to the task. We propose that there is no data limited region in the PRF for a steering task and that the PRF is linear, in other words, pS = rS, where rS is the fraction of resource allocated to the steering task.
words the time to negotiate a path with a corner can be modeled as a the sum of a steering and Fitts terms, MTC = a + bIDS + cIDF.
where a, b, and c are experimentally determined regression coefficients. Model - Experiment Comparison
The Fitts index of difficulty, IDF = log2(A/2W +1), in the model for corner steering, Equation (13), is nonlinear in A/W =IDS, the steering index of difficulty. This suggests that a regression analysis on the MT for paths with corners using the model in Equation (13) can possibly discriminate the Fitts term from the steering term. The regression analysis found that the IDF dependency is not significant, F6, 995 = 0.3, p = 0.94. The lack of significance does not invalidate Equation (13) as a corner steering model; it can be due to residual error in the data obscuring the nonlinearity. Even in the presence of residual data error, the linear part of the Fitts term can contribute to the increase in the slopes for the regressions. In other words, the fitted constant for the slope of MT with respect to IDS, b, is equal to the sum of a factor due to steering without the corner and a factor from the linear part of the Fitts term. Because we observe different IDS ranges for the path widths, the linear fraction of the Fitts term, IDFSlope, should be estimated for each path width as the slope from a linear regression on the set of points
We define the instantaneous Fitts performance as the fraction of progress to the target, p F = ∆s v F ∆t = v v F ,
(9)
where ∆s is the progress to the target in time ∆t, and vF ∆t is the progress that perfect performance would achieve. Again we assume that the PRF is linear without a data limited region, pF = rF, where rF is the fraction of resource allocated to the Fitts task. Several Fitts studies suggest that the PRF does not have a data limited asymptote. The inclusion of even a minimal additional demand on the participants’ resource, such as dragging rather than clicking [12] and selecting the target during the aiming process [9], degrades the Fitts performance. When the two tasks are performed simultaneously, they can deplete the processor’s resource and the performance becomes resource-limited. The performance for negotiating paths with corners degrades because the Fitts task associated with the synchronization at the corner demands resource, rF, leaving only rS = 1 – rF resource for the steering task. The steering and Fitts performance must satisfy, pS = rS = 1 – rF = 1 – pF.
{(IDi, log2(IDi /2 + 1)},
(10)
b F = (bC − bS ) ID FSlope ,
IDC = ∫ ds v( s) C
,
(15)
where bC and bS are the slopes from the linear regressions for paths with corners and straight paths, respectively. We should also correct the slopes for the omitted 1/ln(2) factor in the steering index of difficulty for constant width paths. For the second experiment, the procedure above generates 6 estimates for bF. According to our steering corner model, the Fitts term contribution to the slope should be dependent only on A/W and independent of the width. We can consider the 6 estimates, bF, as representing a random sample of estimates for the fitted slope of a hypothetical Fitts experiment. The mean of the six bF, 227 ms/bit, differs from the fitted slope found in the literature [1, 11], 160 ms/bit, t5 = 4.9, p = 0.004. The conditions of this hypothetical Fitts experiment are significantly different from the Fitts experiments in the literature. Also some tasks are not accounted for in our model, for example while negotiating the approach leg to the corner the participants must glance at different locations; this adds additional demands on the resource [17]. Because changing direction and initiating steering along the exit leg of the corner may require centering the cursor better than clicking on a button, the effective Fitts target size of the corner may be smaller. Also changing the cursor movement direction in itself may be an additional task. Acceleration in the exit leg
Equation (7) can be integrated to get the form of the index of difficulty to negotiate a path with a corner, IDC,
C
(14)
where IDi = Ai /Wi are the IDS observed for that path width. Then the Fitts term contribution to the slope, bF, at each path width can be estimated from the experiment by
Equation (10) is the same formula for the actual velocity as Equation (7). We have shown that adding simultaneous constraints in parallel is consistent with interference between two tasks with linear PRF. The addition of data limited regions in the PRF will change the relationship between the performances, and Equation (10) will not be the same as Equation (7). But small variations from linearity in the PRFs will not substantially change the equation for the actual velocity.
= ∫ (τ S W + τ F s )ds
(13)
(11)
= τ S ( A W ) + τ F ln( A W )
where the Fitts term is integrated only along the leg approaching the corner and the steering term is integrated along both legs. We identify the first term as the steering index of difficulty, IDS, for the path without a corner and identify the second term as the Fitts index of difficulty, IDF, for a target width, W, and initial separation, A/2. In other
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CONCLUSIONS
increases movement time, but most of the effect should be to increase the intercept of the regression. All these unaccounted effects would tend to yield a larger bF.
The steering law is potentially a useful quantitative law in human-computer interaction research and design, especially for modeling gestures and cursor manipulations. But the law should be evaluated for each path shape. We have found that the slope of the fitted model is significantly larger for paths with corners compared to straight paths. Surprisingly, the slope is less for paths with 45° and 135° corners than paths with 90° corners. The paths with 45° corners have the lowest MT. The implication to user interface designers is that gestures with corners should be designed with acute angles rather than 90° angles. Because participants can negotiate wide corners by “cutting off the corner,” the wide paths with corners at small IDS have smaller movement times than corresponding straight paths. This implies that gestures with wide corners or large tolerances are significantly easier to articulate. Interface designers should expect that users will round off corners while gesturing or negotiating menu hierarchies, and should allow for large tolerances at the corners. Many user interfaces on Windows and Mac platforms allow the user to round off the corner while transitioning between menu levels. Because the difference in the slopes of the fitted models between straight paths and paths with corners persists to large IDS, up to IDS ≈ 16, we conclude that the effect of the corner is more than deceleration and acceleration at the corner. The user interface designer should be aware of the additional difficulty due to the corner, even in long gestures. Modeling steering through the corner as two tasks, steering through a straight path and a Fitts task associated with the corner, furnishes a simple formula with familiar terms for the difficulty of negotiating paths with corners,
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The six bF estimates have standard deviation 34 ms/bit, which is not that large because the bF statistics is a difference statistics of bC and bS. The Q-Q plot for the bF samples, Figure 11(a), has small deviations for the extreme quantiles from the line through the first and third quartiles implying that the bF distribution is close to normal. The residual plot, Figure 11(b), identifies that the first quantile is the estimate for the 120-px wide paths. This estimate is suspect because the paths are clearly in the region of “cutting off the corner,” wide paths at small IDS.
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Figure 11: Diagnostic plots for estimated Fitts slopes
We can construct a semi-empirical model for corner steering by adding the estimated Fitts contribution for the slope to the straight path slopes. Figure 12 compares the semi-empirical model for the mean of the calculated bF, 227 ms/bit, and the value found in the literature, 160 ms/bit, to the slopes for the corner data. Note that the slopes have been reduced by ln(2) to correct for the omitted factor in IDS. The model deviates significantly from the data only at 40-px wide paths and 120px wide paths. The 120-px wide path datum is suspect because it is clearly in the “cutting off the corner” mode, and the 40-px wide path with a 90° corner datum appears to be abnormally low. Comparison of the models for the two bF shows that the model is sensitive to bF.
IDC ~ IDS + IDF.
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The plus symbol in Relation (16) should not be taken literally as the sum of the two indices of difficulty; Relation (16) only expresses that the total index of difficulty is directly proportional to both the Fitts and steering indices of difficulty. A statistical model should use two factors, but because the nonlinearity of IDS in IDF can be obscured by the error in the data, the effect of IDF is to increase the slope of the regression with IDS. Our results show that the two effects are always present in paths with equal width legs from 20-px to 120-px, and that steering through a corner with equal width legs cannot be approximated as solely a Fitts task even in wide paths or as solely a steering task even in narrow paths. A study of steering through corners with unequal width legs could determine if steering through corners with narrower exit legs can be approximated by a Fitts task. If our model is correct, the effect of the corner for paths with long approach legs, IDS > 16, becomes proportionally less, but verifying this result is confounded by other effects, such as directional dependency [6] and scaling dependency [2] in the steering index of difficulty.
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The power of Relation (16) is that it allows the HCI designer to apply results from Fitts experiments to cursor manipulations with corners. For example refining Fitts’ law for bivariate pointing, Accot and Zhai [1] found that increasing the aspect
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Figure 12: Comparison of dual task model and 90° corner data
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April 22-27, 2006 • Montréal, Québec, Canada
REFERENCES
ratio of the target, W/H, where W is the amplitude tolerance, H is the directional tolerance, decreases the MT to the target. In light of Accot and Zhai’s results and our understanding of the difficulties involved with corner steering, we can explain the apparent difficulty of negotiating menus that do not allow rounding off corners. Typical menu hierarchies, curve (a) in Figure 13, require the user to negotiate a wide path, large directional tolerance, H, to a corner with small amplitude tolerance, W. Because the Fitts task of the corner is amplitude dominate, the larger directional tolerance does not alleviate the Fitts task making negotiating the corner difficult, and effects the negotiation all along the leg. Increasing menu hierarchy performance requires increasing the width of the corner in both directions. When the exit leg is wider then the entrance leg, Accot and Zhai’s results suggest that the MT for negotiating menus can be reduced. Examples of menu hierarchies with wide exit legs are pull-down menus from the tool bar of applications, curve (b) in Figure 13. Negotiating to the pull-down menu is easier than the corresponding command selection in the pull-downed menu, curve (a) in Figure 13. The exit leg does not need to widen much. Accot and Zhai found that increasing W/H beyond 3 did not reduce the MT significantly.
1. Accot, J., and Zhai, S., Refining Law Models for Bivariate Pointing, In Proc.CHI2003 ACM Press (2003), 193-200. 2. Accot, J., and Zhai, S., Scale Effects in Steering Law Tasks, In Proc. CHI2001, ACM Press (2001), 1-8. 3. Accot, J. and Zhai, S., Performance Evaluations of Input Devices in Trajectory-based Tasks: An application of the Steering Law, In Proc. CHI99, ACM Press (1999), 466-472. 4. Accot, J., and Zhai, S., Beyond Fitts’ Law: Models for Trajectory-Based HCI Tasks, In Proc. CHI97, ACM Press (1997), 295-302. 5. Ahlström, D., Modeling and Improving Selection in Cascading Pull-down Menus Using Fitts’ Law, the Steering Law and Force Fields, In Proc. CHI2005, ACM Press (2005), 61-70. 6. Dennerlein, J.T., Martin, D.B. and Hasser, C., ForceFeedback Improves Performance for Steering and Combined Steering-Targeting Tasks, In Proc. CHI2000, ACM Press (2000), 423-429. 7. Fitts, P.M., The Information Capacity of the Human Motor System in Controlling the Amplitude of Movement, Quarterly Journal of Psychology, (1954) 47, 381-391. 8. Gopher, D., and Navon, D., How is Performance Limited: Testing the Notion of Central Capacity, Acta Psychologica (1980) 46, 161-180.
(b)
(a)
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Figure 13: Corner paths while negotiating a menu hierarchy
This paper is only a first study of steering through corners. The study of steering through paths with different corner leg widths would have a direct application to negotiating menu hierarchies. Studying the cursor velocity through the paths could add more validity to modeling the effect of the corner as a Fitts task, and could extract the acceleration along the corner’s exit leg. Extending the theory of steering through corners to longer legs would require comparison of straight paths in both the horizontal and vertical direction.
13. Navon, D., and Gopher, D., Task Difficulty, Resources, and Dual-task Performance, in Attention and Performance, edited by Nickersor, R.S., Erlbaum, NJ, 1980, 297-315. 14. Norman, D.A.and Bobrow, D.G., On Data-limited and Resource-limited Processes, Cognit. Psych., 7, 44-64, 1975. 15. Pinheiro, J.C. and Bates, D.M., Mixed-Effects Models in S and S-PLUS, Springer, New York, NY, 2000. 16. Venables, W.N. and Ripley, B.D., Modern Applied Statistics with S, Springer, New, NY, 2002.
ACKNOWLEDGMENTS
The author wishes to thank the many colleagues who contributed to this study, especially the department chairperson, Dr. Linda Ott, for providing resources for this study and input during the analysis, and the students who developed the test platform: Joseph Vaillancourt, Scott Gross, Jacob Champlin, and Robert Miller. Also the author thanks all the participants who volunteered for the tests.
17. Wickens, C.D. and Hollands, J.G., Engineering Psychology and Human Performance, third edition Prentice Hall, Upper Saddle River, New Jersey, 2000. 18. Wickens, C.D., and Lilu, Y., Codes and Modalities en Multiple Resources: A Success and A Qualification, Human Factors, 1988, 30, 5, 599-616.
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