Normalizing Speed-accuracy Biases in 2D Pointing Tasks with Better Calculation of Effective Target Widths

TL;DR

The paper investigates how to normalize speed–accuracy biases in 2D pointing tasks by comparing univariate ($\upsigma_x$) and bivariate ($\upsigma_{xy}$) standard deviations for the effective width $W_ ext{e}$ within an ISO-style Fitts' law framework. Using three bias instructions and a large crowdsourced dataset (n=342) across multiple amplitudes and widths, the authors show that univariate $\upsigma_x$ yields higher model-fit ($R^2$) in mixed bias conditions, while TT-based effective width models with effective amplitude ($A_e$) offer the strongest throughput stability. A Monte Carlo simulation confirms these findings hold for smaller samples, with bivariate models rarely outperforming univariate ones as $N$ grows. The study argues for adopting ID$_{xTT}$ (and ID$_{xTT}A_e$ for TP stability) and TT task-axis definitions as practical defaults, contributing to standardized, bias-resistant evaluation of pointing performance in HCI. The work clarifies prior inconsistencies with Wobbrock et al. (2011) and provides guidance for reproducible reporting of $W_e$ calculations, task-axis choices, and bias normalization in future 2D Fitts’ law studies.

Abstract

For evaluations of 2D target selection using Fitts' law, ISO 9241-411 recommends using the effective target width (W_e) calculated using the univariate standard deviation of selection coordinates. Related research proposed using a bivariate standard deviation; however, the proposal was only tested using a single speed-accuracy bias condition, thus the assessment was limited. We compared the univariate and bivariate techniques in a 2D Fitts' law experiment using three speed-accuracy biases and 346 crowdworkers. Calculating W_e using the univariate standard deviation yielded higher model correlations across all bias conditions and produced more stable throughput among the biases. The findings were also consistent in cases using randomly sampled subsets of the participant data. We recommend that future research should calculate W_e using the univariate standard deviation for fair performance evaluations. Also, we found trivial effects when using nominal or effective amplitude and using different perspectives of the task axis.

Figures (18)

• Figure 1: The effective-width method for 1D targets. Cross marks ("$\times$") denote endpoints. (a) In a 1D Fitts' law task where the $x$-axis represents the task axis, a participant aims for a target of width $W$ located at a distance $A$ using a cursor. (b) When the target is repeatedly selected, the spread of endpoints tends to form a normal distribution, and larger targets generally produce greater endpoint variability. (c) When participants are biased toward performing in a slow and accurate manner, the error rate is low but the movement time increases. Conversely, (d) when participants are biased toward fast and inaccurate movements, the error rate increases while the movement time decreases. By adjusting target width a posteriori, we can compare performance fairly using $\mathit{TP}$ regardless of accuracy variability.
• Figure 2: (a) ISO 9241-411 multi-directional pointing with circular targets. When there are nine targets, select them in order from 1 to 10. (b--c) Rotating each trial so that the task axis points to the right when computing endpoint distributions.
• Figure 3: Dispersion in the plane is identical, but we have (a) $\upsigma_\mathit{x}=1.83$ and (b) $\upsigma_\mathit{x}=0$ (arbitrary units such as cm or pixels).
• Figure 4: Two definitions of the task axis. Depending on whether the start point is the previous target center or the previous successful click position (red cross), TT and CT yield different $\upsigma_\mathit{x}$ values.
• Figure 5: Abstract images of the experimental task. (a) Before starting each $\textsf{bias}$ condition, the goal for the seven sequences including practice was described. (b) Participants clicked the targets in the specified order. The given $\textsf{bias}$ was always written at the top of the window. (c) After completing a sequence, the results and a message to take a break if needed were displayed.