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Abstract
The time T needed to reach a target of width W located at distance D varies as a logarithmic or power function of the quotient of D/W. The received, strong version of Fitts’ law requires isochrony—the invariance of T across different D/W conditions with the same quotient—but there is room for a weaker, yet nontrivial version of Fitts’ law where scale influences T without interacting with the crucial quotient. The data of Fitts’s historic experiments are submitted to Cartesian/polar analysis. While tapping beautifully illustrates the strong, isochronous version of Fitts’ law, contrary to a widespread belief Fitts’s other two experiments were not just half-failed corroborations: The disc-transfer data eloquently illustrate the weak version of Fitts’ law, and the pin-transfer data flatly violate the law. Surprisingly, however, the pin-transfer data are remarkably simple in the alternative Cartesian description system, factors D and W exerting separate, additive effects on T.
PUBLIC SIGNIFICANCE STATEMENT. Sixty years ago Fitts published the first experimental demonstration of the quantitative rule famously known today as Fitts’ law. This paper reports a complete reanalysis of the numerical data that Fitts tabulated in detail in his article, revealing patterns of remarkable coherence that had been so far unsuspected due to undetected ambiguities concerning the dimensions of Fitts’ law. One particular intriguing discovery is that there exist two different versions of Fitts’ law, both eloquently illustrated by Fitts’s own data.
Notes
1 Google Scholar (December 2, 2018) counts 7,748 citations of Fitts’s (Citation1954) article.
2 There has been controversy on the necessity of a zero-intercept in Fitts’ law (e.g., Soukoreff & MacKenzie, Citation2004; Zhai, Citation2004). If possibly useful for comparison purposes, the intercept is uninterpretable in and of itself because the difficulty continuum has an arbitrary zero (Guiard, Olafsdottir, & Perrault, Citation2011). For example, Fitts’s ID = log2 (2D/W) zeroes out at the limit where W becomes so large as to equal 2D—in no way the point where task difficulty zeroes out, but rather the point where the target interval begins to include the start point, meaning that the movement task disappears.
3 With the intercept treated as a second free parameter, to match Equation 1a, the improvement is very small for the disc-transfer data (r² = .844 instead of .799) and for the pin-transfer data (r² = .890 instead of .888).
4 In the two transfer tasks W was simply the difference between hole diameter and pin diameter.
5 These authors were discussing reaction time specifically but their objection is quite general and obviously applies to the study of movement time.
6 In principle the question tackled in the Fitts paradigm—the trade-off of speed and accuracy in the execution of aimed movement—is tractable using not only just the popular time-minimization paradigm of Fitts (Citation1954), but also the spread-minimization paradigm of Schmidt et al. (Citation1979) and the dual-minimization paradigm of Guiard, Olafsdottir, and Perrault (Citation2011).
7 For his tapping experiment Fitts used a three-factor design, crossing not just target distance D and target tolerance W, but also stylus weight (1 lb. vs. 1 oz.), an auxiliary factor that turned out to exert virtually no effect on performance and to be involved in no interaction. Below we will leave aside stylus weight, the data being given on average over the two stylus weights.
8 Below the contrast will be between incomplete one-DoF and complete two-DoF descriptions of Fitts' law data, but that count refers to the number of factors involved in the experiment and/or the number of independent variables shown on the right-hand side of mathematical equations. However, recall that T, by definition the dependent measure of the Fitts paradigm, constitutes a further independent dimension. Thus, taking T into account, we will be contrasting incomplete two-dimensional descriptions of Fitts' law, of the form y = f (x), versus complete three-dimensional descriptions of the law, of the form y = f (x, z).
9 Many Fitts' law students (e.g., Meyer et al., Citation1988; Sheridan, Citation1979) have expressed curiosity about the respective contributions of D and W to the effect of ID. In fact this is a logically intractable question because this triad of independent variables has only two DoF—once you know two of them, you known the third.
10 The bad reputation of truisms in science is undeserved. Indeed no merit is attached to their formulation because they consist of trivially true propositions. But truisms are true propositions and they have, in comparison with sophisticated or far-fetched propositions, the advantage of offering safe foundations for theory building, as recognized after all in axiomatic mathematics since Euclid.
11 The general notion of a scale optimum is useful in a vast variety of contexts. In management, for example, Kuemmerle (Citation1998) gathered evidence of an inverted U-shape relationship between the performance and the size of research and development laboratories.
12 One simple reason why Fitts' law experimenters are in fact likely to go for about optimal ranges of S—even without conscious awareness of that variable—is that for any task a non-optimal scale level will induce an easily detectable effect of discomfort and fatigue.
13 In the search for the best possible description our default technique here is nonlinear regression, a robust technique that obviously accommodates the special case of linearity. Most of the best fits to be reported below were obtained from the online site http://www.xuru.org/rt/NLR.asp, which evaluates more than 100 mathematically “interesting” functions and ranks them according to goodness of fit.
14 Fitts duly reported the 16 corresponding values of error rate. That other dependent variable, although reasonably low on average (1.8%), correlates with T (r = .82), casting a doubt on the validity of T (Crossman, Citation1956; Soukoreff & MacKenzie, Citation2004). However, to take error rates into account—e.g., by trying to adjust the T—would take us too far off the focus of this paper and so we will leave them aside, as Fitts (Citation1954) did himself.
15 Taking into account the minute (roughly linear) effect of S on both the scaling coefficient a and the exponent b of the power function of Equation 5, a rather complicated two-DoF model obtains: log T = log (0.15 + 0.0014 S) + (0.4217 ‒0.0027 S) log F, and the improvement of the fit is quite negligible, the r² progressing only from .988 to .992.
16 One of the most influential Fitts' law studies in the HCI domain is MacKenzie (Citation1991). In his Exp. 1 the author ran a comparative experimental evaluation of three popular input devices (the mouse, the digitizing tablet, and the trackball) for two aimed-movement tasks (pointing vs. dragging). Fortunately, for one of his six conditions (mouse pointing), MacKenzie reported his data in detailed tabular form (Table 9, p. 87), making it possible to visualize the influence of scale on performance by plotting mean T as a function of S, with F (or the ID) as a parameter, as in Figures 4, 7, and 11 above. As could be expected, the pattern is nicely U-shaped with quite some converging evidence that the optimum was located at about 10 cm—a fact of potential relevance in the context under consideration. The effect of scale was small in size and there was no F * S interaction, and so the outcome was essentially the same as that obtained by Fitts in his tapping experiment. Unfortunately, the author did not inquire into the possibility of comparing his three input devices in terms of the performance vs. scale function, and for lack of the appropriate numerical tables there is no way to check.
17 Taking the viewpoint of practitioners of Fitts' law in HCI and holding tight to the one-DoF polar understanding of the Fitts paradigm, Soukoreff and MacKenzie (Citation2004, p. 768) have proposed a conventional criterion, justified by past experience, for deciding whether or not an experiment has corroborated Fitts' law. A normal fit, they suggested, is one with typically r ≥ .9, meaning r² = .81. Using this criterion all three experiments of Fitts (Citation1954) are corroborations of Fitts' law (see Figure 1).
18 Revisiting the data of Abrams, Meyer, and Kornblum (Citation1989), Guiard (in preparation) has found that the coefficient of variation of the saccade to visual targets ranges from about 10% up to about 25%. These values should be compared for example with the 0%–12% range found by Guiard et al. (Citation2011) in a hand-movement experiment especially designed to encourage their participants to explore their full spectrum of speed/accuracy strategies.