paper: 2D fitt's law

Extending Fitt’s Law to Two-Dimensional Tasks
I. Scott MacKenzie and William Buxton
Proceedings of CHI’92

This paper extends the famous Fitt’s Law for predicting human movement times to work accurately in two-dimensional scenarios, in particular rectangular targets. The main finding of the paper is that two models, one which models target width by projecting along the vector of approach and another which uses the minimum of the width or height achieved equal statistical fits, and showed a significant benefit over models which used (width+height), (width*height), and (width-as-horizontal-distance-only) models.

For those who don’t know, Fitt’s Law is an empirically validated law that describes the time it takes for a person to perform a physical movement, parameterized by the distance to the target and the size of the target. It’s formula is one-dimensional: it only considers movement along a straight line between the start and the target. The preferred formulation of the law is the Shannon formulation, so named because it mimics the underlying theorem from Information Theory --

MT = a + b log_2 (A/W + 1)

Where MT is the movement time, A is the target distance or amplitude, W is the target width, and a and b are constants empirically determined by linear regression. The log term is known as the Index of Difficulty (ID) of the task at hand and is in units of bits (note the typo in the paper).

The Shannon formulation is preferred for a number of reasons

• Provides the best statistical fit
• Mimics the underlying information theory
• Always provides a positive value for the ID

This paper then considers two-dimensional cases. Clearly you can cast the movement along a one-dimensional line between start and the center of the target, and the amplitude is the Euclidean distance between these points. But what to use as the width term? Historically, the horizontal width was just used, but this seems like an unintuitive choice in a number of situations, particularly when approaching the target from directly above of below. This paper studies five possibilities: Using the minimum of the width and distance (“smaller-of”), using the projected width along the angle of approach (“w-prime”), using the sum of the dimensions (“w+h”), using the product of the dimensions (“w*h”), and using the historical horizontal width (“status quo”).

The study varied amplitude and target dimensions crossed with 3 approach angles (0, 45, and 90 degrees). Twelve subjects were used, who performed 1170 trials each over four days of experiments. The results found the following ordering among the models in terms of model fit: smaller-of > w-prime > w+h > w*h > status quo. Notably, the smaller-of and w-prime cases were quite close – their differences were not statistically significant.

The w-prime case is theoretically attractive, as it cleanly retains the one-dimensionality of the model. The smaller-of model is attractive in practice as it doesn’t depend on the angle of approach, and so require one less parameter than w-prime. The w-prime model. However, doesn’t require that the targets be rectangular as the smaller-of model assumes. Finally, it should be noted that these results may be slightly inaccurate in the case of big targets, as the target point is assumed to be in the center of the target object. In many cases users may click on the edge, decreasing the amplitude.