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

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.

💡 Research Summary

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The paper investigates how to best compute the effective target width (Wₑ) in two‑dimensional (2D) Fitts’ law experiments, a crucial step for normalising the speed‑accuracy trade‑off and obtaining a fair throughput (TP) metric. ISO 9241‑411 recommends calculating Wₑ from the univariate standard deviation (σₓ) of selection endpoints projected onto the task axis. A later proposal by Wobbrock et al. suggested using a bivariate standard deviation (σₓᵧ) that also accounts for variability orthogonal to the task axis. However, the bivariate approach had only been evaluated under a single, neutral speed‑accuracy instruction, leaving its effectiveness for bias normalisation untested.

To fill this gap, the authors conducted a large‑scale online experiment with 346 crowdworkers. Participants performed a classic ISO‑style multi‑directional pointing task: nine circular targets arranged on a layout circle were selected in order, requiring movements in many directions. Crucially, three explicit speed‑accuracy bias instructions were used: (1) “fast but inaccurate”, (2) neutral (“as quickly and accurately as possible”), and (3) “slow but accurate”. For each trial the movement time (MT), error rate (ER), and final cursor coordinates were recorded.

The data were processed in two parallel pipelines. In the first, endpoints were rotated so that the task axis points right, then the univariate standard deviation σₓ (spread along the task axis) was computed. In the second, the same rotation was applied but the Euclidean distance from the target centre was used, yielding the bivariate standard deviation σₓᵧ. Effective target widths were obtained as Wₑ = 4.133 · σ (Equation 4), and effective indices of difficulty IDₑ = log₂(Aₑ/Wₑ + 1) were calculated. Throughput was then derived as the mean of IDₑ/MT across all amplitude‑width conditions (Equation 5).

The authors compared the two methods on three criteria:

1. Model Fit (R²) in Mixed‑Bias Condition – When MT data from all three bias instructions were pooled (the “mixed” condition) and regressed against IDₑ, the σₓ‑based model achieved an average R² of ≈ 0.86, substantially higher than the σₓᵧ‑based model (≈ 0.78). This indicates that σₓ better captures the systematic variation in MT caused by the bias instructions, leading to a tighter Fitts’ law relationship.

2. Throughput Stability Across Biases – Using σₓ, the three bias‑specific TP values differed by only about 0.05 bps (5.68 – 5.73 bps), i.e., less than 1 % variation. With σₓᵧ the spread widened to roughly 0.20–0.25 bps (≈ 3–4 % variation). Hence σₓ normalises the speed‑accuracy trade‑off more effectively, producing a more reliable single TP figure for a device or technique.

3. Robustness to Small Sample Sizes – The authors repeatedly sampled random subsets of participants (10, 20, 30 users) and recomputed the analyses 1,000 times per subset size. In >85 % of the resamples, σₓ outperformed σₓᵧ on both R² and TP stability, confirming that the advantage holds even for typical HCI studies with limited participants.

Additional exploratory analyses examined whether using nominal amplitude (A) versus effective amplitude (Aₑ), or redefining the task axis (e.g., using the target centre versus the goal centre) would affect the outcomes. Neither manipulation produced statistically significant changes in model fit or TP, suggesting that the choice of σ (univariate vs. bivariate) is the dominant factor.

A literature survey revealed that many recent 2D Fitts’ law papers have adopted the bivariate σₓᵧ, often without testing multiple bias conditions. Only a handful of studies directly compared σₓ and σₓᵧ, and those that did either used neutral bias only or employed different target geometries (e.g., rectangular targets). The present work therefore provides the first comprehensive empirical evidence, across three bias levels and a large participant pool, that the original ISO‑recommended univariate σₓ remains the superior choice for bias normalisation.

Implications:

• Researchers evaluating input devices, interaction techniques, or user groups should continue to compute Wₑ using σₓ, especially when speed‑accuracy instructions vary.
• The univariate approach yields more stable throughput values, facilitating fair comparisons across conditions.
• Even when only a small number of participants can be recruited, σₓ is likely to give more reliable results than σₓᵧ.

In summary, the study re‑affirms the ISO 9241‑411 recommendation, demonstrates its practical superiority over the newer bivariate proposal, and offers concrete guidance for future HCI performance evaluations involving 2D pointing tasks.