Computing Touch-Point Ambiguity on Mobile Touchscreens for Modeling Target Selection Times
CCalibration Methods of Touch-Point Ambiguity for Finger-Fitts Law
SHOTA YAMANAKA,
Yahoo Japan Corporation, Japan
HIROKI USUBA,
Meiji University, Japan
Finger-Fitts law (FFitts law) is a model to predict touch-pointing times that was modified from Fittsβ law. It considers the absolutetouch-point precision, or a finger tremor factor π a , to decrease the admissible target area and thus increase the task difficulty. Amongchoices such as running an independent task or performing parameter optimization, there is no consensus on the best methodologyto measure π a . By integrating the results of our 1D and 2D touch-pointing experiments and reanalyses of previous studiesβ data,we examined the advantages and disadvantages of each approach to compute π a , and we found that there is no optimal choice tomaximize the prediction accuracy of FFitts law.CCS Concepts: β’ Human-centered computing β HCI theory, concepts and models ; Pointing ; Empirical studies in HCI .Additional Key Words and Phrases: Fittsβ law, touchscreens, finger input, pointing, human performance modeling
For human motor performance modeling, researchers have sought to develop new models and modify existing modelsto improve their prediction accuracy (i.e., model fitness). The model that we focus on here, the Finger-Fitts law (a.k.a.,FFitts law) proposed by Bi et al. [3], is a modified version of Fittsβ law [9] for predicting operational times in targetpointing on touchscreens. FFitts law is based on the effective width method [7], which adjusts the target size (or width π ) from the nominal value drawn on the screen to an effective width that takes the actual touch-point distributionsinto account. Bi et al. modified this effective width method to deal with finger touch ambiguity and empirically showedthat FFitts law is superior to Fittsβ law in terms of model fitness [3]. As touchscreen devices have become common inour daily life, deriving a model with a high prediction accuracy will contribute directly to human-computer interaction(HCI), e.g., when designers create user interfaces for webpages and apps.As another type of contribution in performance modeling, standardizing a modelβs methodology is important forfuture researchers in terms of the replicability [16, 20, 23]. Unfortunately, while several research groups have examinedFFitts law [3, 14, 21, 24, 26, 27], there is no consensus on a standard methodology, which is an obstacle to future researchon finger-touch pointing. The methodology inconsistencies include the computation method for touch ambiguity(running an independent finger calibration task, computing the intercept for a linear regression of the target sizesand touch-point distributions, or performing parameter optimization from the regression expression of Fittsβ law); theinstruction for the finger calibration task (balancing the speed and accuracy or concentrating on the accuracy); and thetarget size for the calibration task (a 1-pixel target or the smallest size used in the main Fittsβ law task). There are twoother issues of inconsistency: in contrast to Bi et al.βs finding [3], the model fitness of FFitts law was also found to beinferior to that of Fittsβ law [24]; and FFitts law sometimes cannot be used because of a mathematical error that resultswhen the value inside a square root is negative [26, 27].Leaving these inconsistent methodologies and issues unsolved could be harmful to HCI studies such as evaluatingnovel pointing techniques and comparing different user groups. This point was previously mentioned by Soukoreff andMacKenzie in regard to Fittsβ law [20], and rethinking the finger-touch model (FFitts law) is a timely notion given therecent trend of widespread smartphone and tablet use. In this paper, taking a step toward a standard for measuringtouch-pointing performance, we explain the concept of FFitts law, survey the inconsistencies of its methodologies in the a r X i v : . [ c s . H C ] J a n amanaka and Usuba literature, and discuss the related problems. Then, we empirically examine how the inconsistent methodologies changethe results of FFitts law for both 1D and 2D target-pointing tasks, through eight sub-tasks in total. Our contributionsare twofold. β’ Surveying related work on FFitts law to explore inconsistencies in its methodologies (Section 3), and reanalyzingprevious FFitts law studies with modern methods (Section 7). These sections emphasize better understanding of theprinciple of FFitts law, the reasons why previous researchers have run different procedures for a single model, andthe relative advantages and disadvantages of each approach. For example, parameter optimization can always beapplied (i.e., to avoid the error due to a negative value inside the square root), but it induces the risk of overfittingthe data. β’ Conducting eight sub-tasks in total, including two main Fittsβ law tasks with 1D and 2D targets. The results showthat (1) using the baseline Fittsβ law model is the best and safest choice for predicting the MT under a single taskcondition; and (2) for comparing different input devices or user groups, using parameter optimization is the bestchoice for 2D circular targets, while using the conventional effective width method is the best choice for 1D targets.In conclusion, by integrating our results and reanalyses of previous studiesβ data, we found that there is no singlebest methodology for FFitts law. We stress the importance of conducting more user experiments under different taskconditions, such as different input devices or user groups, to achieve a better understanding of touch-pointing behaviorsand to derive more accurate models. According to Fittsβ law, the movement time MT to point to a target is linearly related to the index of difficulty, ID [9]: MT = π + π Β· ID , (1)where π and π are empirical constants. In the HCI field, the Shannon formulation is widely used for the ID value [16]: ID π = log ( π΄ / π + ) , (2)where π΄ is the distance to the target and π is its width. Here, π΄ and π are nominal values shown on the display.In typical pointing experiments, participants are instructed to βpoint to a target as rapidly and accurately as possible,βwhich emphasizes balancing the speed and accuracy [20]. However, it is common that some participants tend to showshort MT values and high error rates, while others show long MT values and low error rates [29]. To normalize suchbiases in comparing those participantsβ performance, or to compare the performance with different input devices (e.g.,finger vs. stylus), using Crossmanβs post-hoc correction for calculating the effective target width π π [7] is recommended[16, 20, 23]: π π = β πππ obs = . π obs , (3)where π obs is the standard deviation SD of the observed endpoints. The basis of this adjustment is that the spread ofhits then follows a normal distribution. By using this method, the π π is adjusted so that βΌ
96% of hits fall inside thetarget. The effective ID using the π π is defined as follows: ID π = log ( π΄ / π π + ) . (4) alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law While the theoretical justification of the effective width method has been questioned recently [10], several advantageshave been shown empirically (e.g., [17, 25, 29]). FFitts law is also based on this effective width method.
Bi, Li, and Zhai hypothesized that the observed spread of hits ( π obs ) includes the relative and absolute components; theformer component follows the speed-accuracy tradeoff rule, while the latter one solely depends on the finger touchprecision [3]. The tapped point is considered a random variable π following a normal distribution ( π βΌ π ( π, π ) ).Then, π is the sum of two independent random variables for the relative and absolute components, both of whichfollow normal distributions: π r βΌ π ( π r , π ) and π a βΌ π ( π a , π ) , respectively. Bi et al. named this βthe dual Gaussiandistribution hypothesis.β Although the relative spread of hits, π r , decreases as the movement speed and target widthdecrease, the absolute finger precision π a cannot be controlled via a userβs speed-accuracy priority. The means of bothcomponents ( π r and π a ) are assumed to be close to the target center: π r = π a = π r is what the effective width method models. Thus, from Equation 3, we have π π = β πππ r . (5)Then, because Bi et al. assumed that π is the sum of the independent random variables π r and π a , we obtain π = π + π . (6)From Equations 5 and 6, we obtain the effective width for FFitts law, π π : π π = βοΈ ππ ( π β π ) . (7)By applying Equation 7 to π in the Fittsβ law ID expression (Shannon formulation), we obtain the ID for finger touching: ID π = log (cid:16) π΄ / π π + (cid:17) . (8) π a Bi et al. also obtained π a via 1D and 2D fingercalibration tasks conducted independently from the Fittsβ law task [3]. In the 1D task, participants repeatedly tapped asclosely to a 2.4-mm-high horizontal bar target as possible, and the SD of the signed biases from the target was computedas π a . For the 2D condition, a 2.4-mm-diameter circle was used as the target, and the bivariate SD was taken as π a . Inboth tasks, the participants were instructed to tap the target as rapidly and accurately as possible. After the screen wastapped, the next target appeared after a 1-sec break. Because this task does not require a movement to a target from aspecific position, Bi et al. stated that the speed-accuracy tradeoff rule has a negligible effect on π a .Woodward et al. conducted FFitts law tasks with children and circular targets [24]. Overall, they followed theprocedure of Bi et al. For the calibration task, they used a target with π = . Luo and Vogel tested the applicability ofFFitts law to touch-based goal-crossing tasks [15]. They drew a 2-pixel line for the finger calibration task and instructedthe participants βnot to rush and focus on accuracy,β because βmeasuring π a is not about speed.β Hence, in contrastto Bi et al.βs instruction, Luo and Vogel removed the instruction of βoperating as rapidly as possible.β They reported amanaka and Usuba somewhat negative results: the data fit for the discrete crossing condition decreased from π = .
26 (conventionalFittsβ law) to 58 .
53 (FFitts law). After removing the data point with the highest ID n , the FFitts law fitness improved to π = .
3, but this was likely due to an arbitrary choice of data-point removal to increase π .Yamanaka tested Fittsβ and FFitts laws for touch-pointing tasks with unwanted target items (called distractors ) [26, 27].In the finger calibration tasks, 1-pixel targets were used (a bar for 1D and a crosshair for 2D). As in Luo and Vogelβs study,Yamanaka instructed the participants to βtap as close to the target as possibleβ and emphasized that the βparticipantswere instructed to concentrate on spatial precision and not on time.β He reported that FFitts law could not be used,because in some task conditions, in particular for the smallest target ( π = . π obs values were smaller than π a , in both the 1D and 2D tasks, resulting in a negative value inside the square root in FFitts law (Equation 7). This waspartially because the participants in Yamanakaβs studies had to pay attention to avoid the distractors, in addition toaccurately aiming for the target, but this mathematical error occurred even in no-distractor conditions. π obs and π . In Bi and Zhaiβs 2D touch-pointing task, at thebeginning of each trial, a circular target appeared on the screen, and the participants tapped it as rapidly and accuratelyas possible [4]. Bi and Zhai assumed that the endpoints when using a fine probe like a mouse cursor are proportionallyrelated to π (i.e., π r = constant Γ π ), thus giving π = πΌπ , (9)where πΌ is a constant. By substituting this π from Equation 9 into Equation 6, we obtain π = π + π = πΌπ + π . (10)Figure 1 shows this relationship. They used five circular target diameters ( π = 2, 4, 6, 8, and 10 mm), and their regressionexpression for π versus the corresponding π values on the (e.g.) y-axes gave π = . π + . π a was computed as β . = .
153 mm.
Recently, Ko et al. proposed to obtain the finger tremor factor used in FFitts law byparameter optimization [14] : MT = π + π Β· log (cid:169)(cid:173)(cid:173)(cid:171) π΄ βοΈ ππ ( π β π ) + (cid:170)(cid:174)(cid:174)(cid:172) = π + π Β· log (cid:169)(cid:173)(cid:173)(cid:171) π΄ βοΈ πππ β πππ + (cid:170)(cid:174)(cid:174)(cid:172) = π + π Β· log (cid:169)(cid:173)(cid:173)(cid:171) π΄ βοΈ π π β π + (cid:170)(cid:174)(cid:174)(cid:172) , (11)where π is a free parameter that represents finger tremors ( π = πππ ). They empirically confirmed that using thenominal π instead of π π gave a higher model fitness, which is consistent with previous studies on the effective widthmethod (e.g., [25, 29]).In fact, Equation 11 was proposed by Welford in 1968 (p. 156, l. 30 in [22]) with the β+0.5β version of Fittsβ law insteadof β+1β. His aim was the same: π represents hand tremors in stylus-tapping tasks. Also, he empirically confirmed thatthe following βno square root, no powerβ formulation was even better in terms of model fitness: MT = π + π Β· log (cid:18) π΄π π β π + (cid:19) . (12) Ko et al.βs model is for rectangular targets whose width and height are defined as π and π» , respectively. We invoke their method instead for circulartargets whose size is solely defined by π . 4 alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law a ππ o b s [ p i x e l s ] ππ [pixels ] Mouse:y = 0.0012 + 0.013xR = 0.98 b ππ o b s [ mm ] ππ [mm ] Touch:y = 1.07 + 0.011xR = 0.96 W=1px W=1mm c d a W=1px
Fig. 1. Overview of the dual Gaussian distribution hypothesis with hypothetical endpoint data (the βXβ marks). (a) In a mouse-pointingtask, if a user spends a sufficient time to point to an extremely small target, the observed endpoint variability π obs is close to zero;thus, (b) the intercept in the regression of the squares of π obs and π is also close to zero. (c) In contrast, for touch pointing, even if auser spends a long time, there is a remarkable variability when tapping a small target [11, 12]; thus, (d) the regression has a clearnonzero intercept. Because the π π and π a use standard deviations, it would be more mathematically sound to subtract after squaring themand then taking the square root; however, the dimensions of π π and π are the same (both in mm), and Equation 12is thus valid. This modelβs superiority with respect to the baseline (Equation 2) for small targets was confirmed byChapuis and Dragicevic [6]. There are two kinds of approaches: using the smallest π used in a Fittsβ law task (2.4-mm [3] or 4.8-mm target [24]) orthe minimum visible target (1 pixel [26, 27] or 2 pixels [15]). In pointing tasks with a fine probe, π obs is assumed tobe proportional to π when users can spend sufficient time. In this case, users can accurately point to a small targeteven if the width is quite narrow: e.g., π = π a . Hence, even if users can spend a long time, there is a slight bias from the intendedtarget position to the actual tapped position sensed by the system [3, 12]. The aim of a finger calibration task is tomeasure this lower bound of precision in the Fittsβ law paradigm. For this purpose, pointing to a 1-pixel target with theinstruction to operate as rapidly and accurately as possible is a straightforward method.There is, however, an issue related to using the smallest π in the main Fittsβ law task. The issue is that we mayobserve a mathematical error in the square root in Equation 7 ( π π = βοΈ ππ ( π β π ) ). For example, Woodward et alused a target with π = . π a was 1.590148 mm [24]. The smallest π obs measured in themain Fittsβ law task was 1.591275 mm for the ( π΄,π ) = ( , . ) mm condition; the difference was only 0.001127 mm.Because π a and π obs are variability values (i.e., standard deviations), it is possible to observe π a greater than π obs bychance. According to Equation 10 ( π = πΌπ + π ), using a π = π = . π = . π a should not be affectedby the speed-accuracy rule, so π a must be the π obs value for the π = π = π = amanaka and Usuba ππ o b s [ mm ] ππ [mm ]00 Random- A task:Intercept methodwith 5 W values:y = 1.05 + 0.019xR = 0.92 ππ a2 = 1.05 mm a ππ o b s [ mm ] ππ [mm ]00 Preset- A task (i.e.,Fittsβ law task):Intercept methodwith 4 A Γ W :y = 0.96 + 0.020xR = 0.83 ππ obs2 = 0.70 mm ππ a2 = 0.96 mm b ππ obs2 = 1.01 mm Fig. 2. A potential problem in using the intercept method to compute π a . The intercepts obtained from the (1) random- π΄ and (2)preset- π΄ tasks could be greater than some of the π values in the Fittsβ law task. There have been two instruction choices: balancing the speed and accuracy [3] or concentrating on accuracy [15, 26, 27].We assume that both instructions are valid for measuring π a . For the βrapid and accurateβ instruction by Bi et al. in Fittsβlaw tasks, as the π becomes smaller, participants have to be more careful to avoid missing the target, which causes themto spend a longer time. Therefore, even if the participants were instructed to tap the target βas rapidly (and accurately)as possible,β the operational time for a 1-pixel (or smallest- π ) target would be quite long, and the difference from theinstruction to βconcentrate on accuracyβ becomes almost negligible. Still, the effect of this instruction difference onFFitts law fitness has been neither discussed nor empirically compared. Hence, we empirically assess this difference inour data analyses. π a : Calibration Task, Intercept of Regression, or Parameter Optimization As discussed in Section 3.1, the use of a finger calibration task induces a risk of a mathematical error in the square-rootcalculation. Another method to obtain π a is to use the intercept of the regression expression for π vs. π (Equation 10, π = πΌπ + π ). Bi and Zhai [4] and Yamanaka and Usuba [28] conducted target-pointing tasks in which a new targetappeared at a random position (i.e., π΄ was not controlled by the researchers), with several π values; they then obtainedregression expressions. Yamanaka and Usuba also ran regressions for Fittsβ law tasks in which four π΄ values were presetto use this method.If we apply π a computed by this intercept method to FFitts law, it is possible to obtain π a greater than π obs , whichcauses the mathematical error. Figure 2 illustrates this problem with hypothetical data observed in a Fittsβ law task. Inthis case, for the π values computed from both the random- and preset- π΄ conditions, several π values at the lowest π condition in the main Fittsβ law task (Figure 2b) are smaller than the intercept.To avoid this issue, a possible choice is to use large target width values for the main Fittsβ law task. For example, ifwe had not used the narrowest π condition in Figure 2b, all the π obs values would be greater than π a . Using only wide π values also lowers the risk of the mathematical error in using the π a measured by a finger calibration task. Yet, thisapproach has a clear limitation: it prevents researchers from using a small target, and the threshold for the smallesttarget to avoid the error is unclear and would vary among participant groups. In addition, the effectiveness of FFitts lawis for small targets; when targets are large, FFitts law approximates the original effective width method without using π a [3].The state-of-the-art method to obtain the finger tremor factor is parameter optimization [14]. The methodβs drawbackis that it uses an additional free parameter π , which is adjusted to maximize π for the regression of MT vs. ID . Dependingon the measured MT values and the number of task conditions, introducing additional free parameters could lead to alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law overfitting. In contrast, using a π a value computed from a calibration task or the intercept method has no such problem,because π a is then independent of the MT values measured in a Fittsβ law task.Regarding the model fitness in terms of π , using parameter optimization would theoretically give the best fit amongthe candidates. Also, it does not require an independent finger calibration task and is thus less time-consuming forresearchers and participants. However, if other model-fit metrics that consider the model complexity show a worseresult due to the free parameter π , then using π a instead of π is recommended. To assess this issue, we also compare themodel fitness by using the Akaike Information Criterion and Bayesian Information Criterion in our data analyses. We conducted touch-pointing experiments with a smartphone, as shown in Figure 3a. The experiments were conductedon two separate days: Day 1 for 1D horizontal bar-shaped targets, and Day 2 for 2D circular targets. The procedures forthe two days were the same. Under both the 1D and 2D conditions, we conducted four sub-tasks. The main one was aFittsβ law task with 4 π΄ Γ π conditions, and the remaining three sub-tasks were used to compute π a values: one wasfor the intercept-based method with five π values and random π΄ values, and the other two were for finger calibrationtasks. The order of the four sub-tasks was balanced using a Latin square pattern among 12 participants for both days.Each participant took 40 to 50 min for the experiment on each day.For both the 1D and 2D conditions, our π a data computed by the intercept method for Fittsβ law and the random- π΄ tasks was reported before [28]. The data for the two finger calibration tasks is newly reported here. Because our novelcontribution in this paper is the evaluation of the model fitness for MT , we repeat the minimum necessary explanationof the experiments (e.g., the mean error rate) to make this paper self-contained, while taking care to avoid plagiarism.For example, we could have reported all the pairwise test results for the error rate, but that data would not relate tothis paperβs main contribution. Thus, we mainly report the MT and π obs results, and readers who are interested in thedetailed error rate results and error-rate prediction models are directed to [28]. The participants were instructed to tap as rapidlyand accurately as possible on a 1-pixel horizontal bar target or a 25-pixel-wide crosshair target in the 1D or 2D conditions,respectively. For the 2D condition, we emphasized that the intersection of the crosshair was the target to aim for. A1-sec break was enforced before the next target appeared so that the participants did not have to aim for the targets oneafter another extremely rapidly. Each participant repeated this procedure 50 times, which entailed five practice trialsfollowed by 45 data-collection trials. The signed biases of the tap point from the target were used to compute the SD (i.e., π a ) on the y-axis for the 1D case and the bivariate SD on the x- and y-axes for the 2D case. For this sub-task, only the instruction wasdifferent from the previously explained sub-task. That is, the participants were instructed to tap as closely as possible tothe target without paying attention to the operational time.
This was a discrete pointing task with preset π΄ and π values. For the 1D task, a 6-mm-wideblue start bar was displayed at the top of the screen, and a green target bar was at the bottom, as shown in Figure 3b.The midway point between the start bar and the target was at the screen center. The movement direction was alwaysdownwards. When participants tapped the start bar, it disappeared and a click sound played. Then, if they successfullytapped the target, a pleasant bell played, and then the next set of start and target bars appeared. If the tap point fell amanaka and Usuba Start circle AW TargetStart bar
A W
Target b ca
Fig. 3. Experimental environment: (a) a participant attempting a 1D Fittsβ law task, and the visual stimuli used in the (b) 1D and (c)2D Fittsβ law tasks. outside the target, they had to aim for the target again until they succeeded; the trial was not restarted from tappingthe start bar. The participants were instructed to tap the target as rapidly and accurately as possible. For the 2D task,circles were used instead of horizontal bars, and the start and target circlesβ positions were randomized while keeping adistance π΄ between them.This sub-task used a 4 Γ π΄ = 20, 30, 45, and 60 mm) and five target widths ( π = 2, 4, 6, 8, 10 mm). Each π΄ Γ π combinationentailed a single repetition of practice trials followed by 16 repetitions. The order of the π =
20 conditions wasrandomized. Thus, we recorded 4 π΄ Γ π Γ repetitions Γ participants = MT , the standard deviation of the endpoints ( π obs ), and the error rate. For the 1D case, a 6-mm-high start bar was initially displayed at arandom position. When the participants tapped it, the first target bar appeared at a random position, and then theysuccessively tapped new targets. If a target was missed, a beep sounded, and the participants re-aimed for the target. Asuccessful tap resulted in a bell sound. To reduce the negative effect of the screen edges, the random target position wasat least 11 mm away from the top and bottom edges [2]. For the 2D case, circular targets were used.This sub-task used a single-factor, within-subjects design with an independent variable of π : 2, 4, 6, 8, and 10 mm.The dependent variable was the observed touch-point distribution, π obs . First, the participants performed 20 trialsas practice, which included 4 repetitions of the 5 π values appearing in random order. In each session , the π valuesappeared 10 times in a random order. The participants were instructed to successively tap the target as rapidly andaccurately as possible in a session. They each completed four sessions as data-collection trials. In total, we recorded5 π Γ repetitions Γ sessions Γ participants = On Day 1, 12 university students participated in this study (2 female, 10 male; 20 to 25 years, π = . SD = . π = . SD = . βΌ US$ 45) in compensation for one day. Theparticipants were instructed to hold the smartphone in their non-dominant (left) hand and perform tapping operationswith their dominant (right) index finger, as shown in Figure 3a. They were instructed to sit on an office chair and not torest their hands or elbows on the table or their lap. alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law On both days, we used an iPhone XS Max (4 GB RAM; iOS 12; 1242 Γ Γ As in previous studies, data points for which the distance between the tap point and the target center was greater than15 mm were removed as outliers before we analyzed the ππ , π a , π obs , and error rate [4, 28]. Among the 540 trials (45 repetitions Γ
12 participants), we observed no outliers. Two participantsβ data did not pass thenormality test (Shapiro-Wilk test with alpha = . SD of the tap positions (i.e., π a ) for each participant rangedfrom 0.5448 to 1.325 mm, and the mean was 0.8837 mm. We again observed no outliers, while two participantsβ data did not pass the normality test. The π a values ranged from0.4569 to 1.296 mm among the participants, and the mean π a was 0.7362 mm. Among the 3840 trials, four data points were removed as outliers (0.10%). According to the experimenterβs observation,the outliers resulted mainly from participants accidentally touching the screen with the thumb or little finger. Two ormore taps were observed in 347 trials, and the mean error rate was thus ( / ) Γ = . π΄ Γ π Γ participants ) passed the normality test, or 90.8%.Throughout this paper, we use repeated-measures ANOVA with Bonferroniβs π -value adjustment method for pairwisecomparisons. Although our results showed that the dependent variables did not pass the Shapiro-Wilk test (alpha =0.05) in some cases, it is known that ANOVA is robust against violations of the normality test assumptions [8, 18]; thus,we consistently use repeated-measures ANOVA. For the πΉ statistic, the degrees of freedom for the main effects of π΄ and π , as well as their interactions, were corrected using the Greenhouse-Geisser method when Mauchlyβs sphericityassumption was violated (alpha = 0.05).For the endpoint variability π obs , we found significant main effects of π΄ ( πΉ , = . π < . π π = .
21) and π ( πΉ , = . π < . π π = . π΄ Γ π ( πΉ , = . π = . π π = . MT , we found significant main effects of π΄ ( πΉ , = . π < . π π = .
95) and π ( πΉ , = . π < . π π = . π΄ Γ π was significant ( πΉ , = . π < . π π = . π vs. π regression. The π a value was β . = . π value: the intercept π was thus greater than some π ,causing the mathematical error in FFitts law. amanaka and Usuba y = 0.0154x + 1.0123RΒ² = 0.935301234 0 20 40 60 80 100 ππ [mm ] ππ o b s [ mm ] y = 0.0191x + 0.9543RΒ² = 0.814101234 0 20 40 60 80 100 ππ [mm ] ππ o b s [ mm ] a Fittsβ law task with preset A b Random target position
Fig. 4. Regression results of π vs. π for the 1D conditions. The intercepts show the π values. We removed 13 outlier trials (0.54%). The Shapiro-Wilk test showed that the touch points followed a normal distributionunder 47 of the 60 conditions ( = π Γ participants ), or 78.3%. The value of π had a significant main effect on π obs ( πΉ , = . π < . π π = . π a was β . = .
006 mm,and some π values in the Fittsβ law task (Figure 4a) were smaller than 1.0123 mm, which caused the mathematicalerror when we applied the π a measured with this intercept method to Fittsβ law data. Before analyzing the FFitts law fitness, we found that we could only use the parameter optimization method. As listedin Table 1, among the π =
20 data points for fitting, any method using π a had one or more mathematical errors (due toa negative value inside the square root in π π = βοΈ ππ ( π β π ) ). This result shows the low robustness of FFitts lawwhen using π a , regardless of whether the π a value is directly measured by a finger calibration task or calculated by theintercept method.For model fitness comparison, we use both the absolute and adjusted π . The latter balances the number of coefficients.We also compare models through the Akaike Information Criterion ( π΄πΌπΆ ) [1]. This statistical method balances thenumber of free parameters and the fitness to identify a comparatively best model. As a brief guideline, (a) a model witha lower
π΄πΌπΆ value is a better one; (b) a model with
π΄πΌπΆ β€ ( π΄πΌπΆ minimum +
2) is probably comparable with better models;and (c) a model with
π΄πΌπΆ β₯ ( π΄πΌπΆ minimum +
10) should be rejected. We also use the Bayesian Information Criterion(
π΅πΌπΆ ) [13] for comparison. For this metric,
π΅πΌπΆ differences of 0β2 are not significant, of 2β6 are positive, and of 6β10are strong; differences greater than 10 are very strong [13]. Hence, a model with a higher π and adjusted π is better,while one with a lower AIC and
BIC is also better. The
AIC penalizes using additional free parameters the least, whilethe
BIC penalizes it the most.Table 2 lists the model fitness results. Overall, the baseline model of Fittsβ law showed the best model fitness in termsof the adjusted π , AIC , and
BIC . While Model π , it was due to the additional free parameter;thus, the adjusted π was slightly lower than that of Model AIC , Models
BIC , however, Models π π ( π or using π π . alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law Table 1. Measured data for the 1D tasks. The units are all in mm, except for the MT in ms. The four π a values were computed from thedata as follows. Calib (R&A): the finger calibration task with the βrapid and accurateβ instruction. Calib (Acc): the finger calibrationtask with the βconcentrate on accuracyβ instruction. Fitts: the intercept method for the Fittsβ law task. Random π΄ : the interceptmethod for the pointing task with a random target position. The π π values were calculated by Equation 7 ( π π = βοΈ ππ ( π β π ) ).The yellow cells noting β!errβ indicate that the π π value was not defined (because of a negative value in the square root). π΄
20 20 20 20 20 30 30 30 30 30 45 45 45 45 45 60 60 60 60 60 π MT
444 364 328 305 298 489 400 353 327 315 529 459 400 369 347 602 511 436 407 393 π a π obs π π !err 3.87 8.15 10.4 8.52 0.694 3.83 4.01 9.02 8.93 !err 3.07 5.32 9.2 11.1 1.35 4.14 8.01 9.41 12.5Calib (Acc) 0.7362 π π !err 4.36 8.39 10.6 8.75 2.14 4.33 4.49 9.25 9.15 0.728 3.68 5.69 9.42 11.3 2.43 4.61 8.26 9.63 12.7Fitts 0.9769 π π !err 3.47 7.96 10.2 8.34 !err 3.42 3.62 8.86 8.76 !err 2.55 5.04 9.04 11 !err 3.77 7.82 9.26 12.4Random π΄ π π !err 3.32 7.9 10.2 8.28 !err 3.27 3.48 8.8 8.7 !err 2.34 4.94 8.98 10.9 !err 3.64 7.76 9.2 12.4 Table 2. Model fitness results for the 1D tasks. The yellow cells indicate the best fit for each criterion.
Description ID formulation π adj. π AIC BIC π π π ( π΄ / π + ) ID π log ( π΄ / π π + ) π π , no sqrt) log (cid:16) π΄π π β π + (cid:17) π π , sqrt) log (cid:18) π΄ β π π β π + (cid:19) π , no sqrt) log (cid:16) π΄π β π + (cid:17) π , sqrt) log (cid:16) π΄ β π β π + (cid:17) y = 90.032x + 132.73RΒ² = 0.98130200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 108.52x + 112.36RΒ² = 0.91070200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 101.62x + 119.82RΒ² = 0.91330200400600800 0 2 4 6 M T [ m s ] ID [bits]y = 103.05x + 121.64RΒ² = 0.91410200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 88.581x + 134.95RΒ² = 0.98140200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 88.327x + 136.38RΒ² = 0.98150200400600800 0 2 4 6 M T [ m s ] ID [bits] a b cd
πΌπΌπΌπΌ = log + 1 e f
πΌπΌπΌπΌ = log ππ + 1 πΌπΌπΌπΌ = log ππ βππ + 1 πΌπΌπΌπΌ = log ππ2 βππ + 1 πΌπΌπΌπΌ = log + 1
πΌπΌπΌπΌ = log βππ + 1 Fig. 5. MT vs. ID regressions of Models Again, data points for which the distance between the tap point and the target center was longer than 15 mm wereremoved as outliers. Among the 540 trials for this sub-task, we observed no outliers. Two participantsβ data did not pass amanaka and Usuba y = 0.0299x + 1.7593RΒ² = 0.83580123456 0 20 40 60 80 100 y = 0.0213x + 1.6155RΒ² = 0.94750123456 0 20 40 60 80 100 ππ [mm ] ππ o b s [ mm ] ππ [mm ] ππ o b s [ mm ] a Fittsβ law task with preset A b Random target position
Fig. 6. Regression results of π y vs. π for 2D conditions. The intercepts show the π values. the normality test. The SD of the tap positions (i.e., π a ) for each participant ranged from 0.8717 to 2.148 mm, and themean was 1.372 mm. We again observed no outliers, while three participantsβ data did not pass the normality test. The π a values ranged from0.7107 to 1.752 mm among the participants, and the mean π a was 1.163 mm. Among the 3840 trials, nine outlier trials were removed (0.23%). The mean error rate was 17.91%. Under 184 (76.7%)conditions, the touch points followed a bivariate normal distribution.For the tap point distribution π obs , we found a significant main effect of π ( πΉ , = . π < . π π = . π΄ ( πΉ , = . π = . π π = . π΄ Γ π was also not significant ( πΉ , = . π = . π π = . MT , we found significant main effects of π΄ ( πΉ , = . π < . π π = .
94) and π ( πΉ . , . = . π < . π π = . π΄ Γ π was significant ( πΉ . , . = . π < . π π = . π vs. π regression. The π a value was β . = .
326 mm. The regression lineclearly passes above the four data points at the smallest π value, causing the mathematical error in FFitts law. We removed 33 outlier trials (1.375%). Under 41 (68.3%) conditions, the touch points followed a bivariate normaldistribution. The value of π had a significant main effect on π obs ( πΉ , = . π < . π π = . π a was β . = .
271 mm, and this π was greater than some π values in theFittsβ law task (Figure 6a), which caused the mathematical error. In contrast to the results for the 1D task, we can use the π a value obtained from the finger calibration task with theβconcentrate on accuracyβ instruction. As listed in Table 3, the π =
20 data points for fitting had no negative valuesinside the square root in FFitts law. Thus, in Table 4, we add Model π , AIC , and
BIC values. According to the
AIC , Models alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law Table 3. Measured data for the 2D tasks. The units are all in mm, except for the MT in ms. The four π a values were computed from thedata as follows. Calib (R&A): the finger calibration task with the βrapid and accurateβ instruction. Calib (Acc): the finger calibrationtask with the βconcentrate on accuracyβ instruction. Fitts: the intercept method for the Fittsβ law task. Random π΄ : the interceptmethod for the pointing task with a random target position. The π π values were calculated by Equation 7 ( π π = βοΈ ππ ( π β π ) ).The yellow cells noting β!errβ indicate that the π π value was not defined (because of a negative value in the square root). π΄
20 20 20 20 20 30 30 30 30 30 45 45 45 45 45 60 60 60 60 60 π MT
440 373 322 294 278 506 410 361 345 314 560 476 413 368 354 622 517 446 407 385 π a [mm] π obs π π !err 2.65 4.22 5.29 5.95 !err !err 4.36 6.18 5.32 !err 2.6 4 6.05 7.38 !err 2.45 4.95 6.67 7.67Calib (Acc) 1.16 π π π π !err 3.02 4.46 5.49 6.12 !err 0.386 4.6 6.35 5.51 0.758 2.98 4.26 6.22 7.52 !err 2.85 5.16 6.83 7.8Random π΄ π π Table 4. Model fitness results for the 2D tasks. The yellow cells indicate the best fit for each criterion.
Description ID formulation π adj. π AIC BIC π π π ( π΄ / π + ) ID π log ( π΄ / π π + ) π π , no sqrt) log (cid:16) π΄π π β π + (cid:17) π π , sqrt) log (cid:18) π΄ β π π β π + (cid:19) π , no sqrt) log (cid:16) π΄π β π + (cid:17) π , sqrt) log (cid:16) π΄ β π β π + (cid:17) π a ) log (cid:32) π΄ βοΈ ππ ( π β π ) + (cid:33) model fitness to that of Model BIC , Models π π ( π π factor(2 πππ = π π ), it showed significantly worse fits than those of Models π or using π π . Here, we reanalyze three sets of data reported in previous studies: Woodward et al.βs study using circular targets [24]and Bi et al.βs 1D and 2D targets [3]. The results are summarized in Table 5. We also examined using π a obtainedby the intercept method, but the mathematical error occurred for all the three data sets. Thus, we report the fits forModels π a was obtained from the fingercalibration task with the βconcentrate on accuracyβ instruction, while Bi et al. used the βrapid and accurateβ instructionand Woodward et al.βs instruction was unclear from their paper.For Woodward et al.βs data, the model fitness for the baseline (Model ID e (Model π partially improved the fit (adjusted π increased from 0.0253 to 0.122); these results are consistent amanaka and Usuba y = 99.57x + 109.72RΒ² = 0.99040200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 147.21x + 22.863RΒ² = 0.73170200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 107.98x -1.3991RΒ² = 0.940200400600800 0 2 4 6 M T [ m s ] ID [bits]y = 119.09x + 35.291RΒ² = 0.93410200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 99.075x + 110.48RΒ² = 0.99050200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 99.124x + 110.67RΒ² = 0.99050200400600800 0 2 4 6 M T [ m s ] ID [bits] y = 120.1x + 33.141RΒ² = 0.9340200400600800 0 2 4 6 M T [ m s ] ID [bits] a b cd
πΌπΌπΌπΌ = log +1 e f πΌπΌπΌπΌ = log ππ +1 πΌπΌπΌπΌ = log ππ βππ +1 πΌπΌπΌπΌ = log ππ2 βππ +1 πΌπΌπΌπΌ = log +1 πΌπΌπΌπΌ = log βππ +1 g πΌπΌπΌπΌ = log obs2 βππ a2 +1 Fig. 7. MT vs. ID regressions of Models π values are not exactly the same as in our analyses. Woodward et al. [24] Bi et al. [3], 1D Bi et al. [3], 2DDescription ID formulation π adj. π AIC BIC π adj. π AIC BIC π adj. π AIC BIC ( π΄ / π + ) ID π log ( π΄ / π π + ) π π , no sqrt) log (cid:16) π΄π π β π + (cid:17) π π , sqrt) log (cid:18) π΄ β π π β π + (cid:19) π , no sqrt) log (cid:16) π΄π β π + (cid:17) π , sqrt) log (cid:16) π΄ β π β π + (cid:17) π a ) log (cid:32) π΄ βοΈ ππ ( π β π ) + (cid:33) with ours. In contrast to our results, however, according to the AIC and
BIC , the model fitness was slightly degraded byusing Models π and π (Models π a value (Model π π -based candidates (Models AIC and
BIC differences were both less than 2 and thus insignificant; this point was notdiscussed by Woodward et al. [24].For Bi et al.βs 1D task results, Model
AIC and
BIC , which is a unique outcome among all theanalyses in this paper. The second best model using π π was AIC and
BIC differences from π value was determined as βΌ π ; thus, while the π values were the same asfor Model π , AIC , and
BIC were worse than the baseline because of the additional free parameter. ForBi et al.βs 2D task results, among all models, the best fit was shown by Model π π gave abetter fit than those using the nominal π .Through these reanalyses, we found the benefit of introducing an additional free parameter π regardless of whetherwe used the nominal π (to analyze a single task condition) or π π (to compare different conditions). However, thisconclusion did not always hold. For Woodward et al.β data, using π for π π slightly degraded the fitness (Model alibration Methods of Touch-Point Ambiguity for Finger-Fitts Law bit better than AIC and
BIC ), and for Bi et al.βs 1D data, using π a gave the best AIC and
BIC results.
The first inconsistency with the previous studies is that we sometimes could not use FFitts law with π a because ofthe mathematical error. For the 1D task, we could not use it for any derivations (finger calibrations with two differentinstructions; square root of the intercept in the regression of π vs. π for the Fittsβ law task and random- π΄ task; seeTable 1). For the 2D task, only the π a computed from the finger calibration task with the βconcentrate on accuracyβinstruction could be used (Table 3). This clearly shows a limitation of the conventional FFitts law: because it dependson both the π a and π obs values, we cannot often use this methodology.Even when we applied FFitts law with π a to the 2D results, the model fitness was significantly degraded as comparedwith the baseline model (Models π , we found a limited benefit. For the 2D task, while Model ID e )showed adjusted π = .
72, using π improved the fitness: Models π = .
93, and the
AIC and
BIC differences were significant (Table 4). Because comparing different user groups or devices requires using theeffective width method to normalize the speed-accuracy biases, this additional parameter for finger tremor helps toimprove the prediction accuracy. For the 1D condition, however, we observed the opposite finding. The adjusted π ofModel ID e ) was 0.91, and those of Models BIC comparison, the use of Model
AIC . For FFitts law and its alternatives (Models π was always superior for both the 1D and 2Dconditions. If we use the nominal π , however, the baseline (Model MT s under untested conditions for a single user group or a single device, the baseline model is recommended; thisis consistent with previous studies comparing the nominal and effective width methods [25, 29].When researchers try to compare several conditions with 2D circular targets, models using π π are required. Ratherthan the ID e (Model π are recommended: ID e model is still a better choice than using π .Among all the 1D and 2D conditions, we recommend not using π a , because it often causes the mathematical error inFFitts law. Use of the parameter optimization method is convenient for both researchers and participants. In addition, byavoiding the finger calibration task with a 1-pixel target, we can use and compare Fittsβ and FFitts laws by conductingonly Fittsβ law tasks with reasonably sized targets. This enables testing of the model fitness by using data measuredfrom (e.g.) a gamified task of tapping bubbles on the screen, as Woodward et al. did [24]. amanaka and Usuba Our conclusions are limited by the task conditions that we used. It is unclear whether our findings, e.g., on the best modeland on when a mathematical error occurred, would hold under other conditions, such as operating a smartphone with athumb in a one-handed posture and using much longer target distances. For the model-fitting results, we sometimes didnot observe a great difference in the
AIC and
BIC values. For example, we sought to state more clearly whether Model
AIC difference was 1.3 (no significant difference), while the
BIC differencewas 2.2 (positively different). This prevented us from concluding that using ID e is better, because the results could easilychange depending on the user group and the task parameters π΄ and π . Much more data is needed to understand thispoint, which will inform our future work.Another unresolved point is the timing of when to compute the model fitness. Following previous studies on FFittslaw [3, 15, 24], we examined the fit for 4 π΄ Γ π =
20 conditions. For the effective width method, however, Soukoreff andMacKenzie stated that the ID e values should be calculated for each task condition for each participant; the participantsβdata should then be averaged last in order to compute the throughput (i.e., a unified performance metric) [20]. By thatmethodology, we should have calculated Equation 7 ( π π = βοΈ ππ ( π β π ) ) for the 20 data points for each of the 12participants. This would have increased the chance to observe the mathematical error, because it would have requiredchecking for it 240 times. This notion indirectly supports that researchers should avoid using π a when they seek toapply FFitts law robustly. According to Olafsdottir et al., there are at least 20 approaches to compute the throughput,depending on the order of aggregating the data [19]. We did not get deeply involved in this point and simply followedthe previous FFitts law studies, yet it will be worth revisiting in the future. We have revisited FFitts law and the inconsistencies in its methodology. The parameter optimization method showedsome relative advantages in comparison with measuring the finger tremor factor π a , which often causes a negativevalue inside a square root in both our data and the data in previous studies. While we discussed the best and suboptimalmodels in consideration of the research goal, such as a device comparison or MT prediction under a single condition,our conclusions could change for different user groups and task conditions in future experiments. To better understandtouch-pointing performance and derive better models, we hope that researchers will report more data from touch-pointing experiments, even if the data shows that a novel model exhibits a lower fitness than the baseline or the datacannot be fitted because of mathematical errors. REFERENCES [1] Hirotugu Akaike. 1974. A new look at the statistical model identification.
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