What is a Zero-Difficulty Movement? A Scale of Measurement Issue in Fitts' Law Research
Résumé
Fitts' law is the well-known empirical relation which states that the time it takes to complete a simple aimed movement varies linearly with movement difficulty, the latter being quantified with an index of difficulty (ID) computed from the ratio of target distance D to target width W. The paper asks about the level of measurement involved in the two most popular versions of the ID, the original Fitts ID and the Shannon ID. Analyzing the way these numerical quantities map onto the concrete geometry of a Fitts task, we show that they lack a true physical zero, meaning that their measurement runs on a non-ratio equal-interval scale and that Fitts' paradigm has been under-constrained so far from the measurement viewpoint. A simple way to force the independent variable of Fitts' law to run on a ratio scale of measurement is to calculate the ID as a function of relative target tolerance (W/D), whose zero is physically anchored, rather than relative target distance (D/W), whose zero is a numerical abstraction. Likewise it is advantageous to base the measurement of movement accuracy on the coefficient of variation of amplitude, which has a physical zero, rather than its inverse, which does not. We illustrate the practicability of our new measures of task difficulty and movement precision with the data of Fitts' (1954) classic tapping experiment. The emerging patterns can be modeled with simple equations whose coefficients are interpretable. We highlight two implications we think of special relevance to HCI research. One is that, contrary to a widely-held belief, the erratic behavior of the y-intercept of Fitts' law reported in the literature should not be a concern because a y-intercept is essentially uninterpretable in the absence of a physically defined zero on the x axis. The other implication concerns the reciprocal protocol popular in HCI, and which an ISO standard recommends explicitly. Not only is the measurement of movement time and task difficulty less rigorous with the reciprocal than discrete protocol, but a different measure of difficulty is needed.
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