Steering Law

Question 6 of Bruce Tognazzini's A quiz designed to give you Fitts [1] talks about the bottleneck in navigating a hierarchical menu system, which is the narrow horizontal channel you pass through to get from one level of the menu to the next. This is illustrated in figure 1.

Tog presents this as a Fitts's Law problem, and he's almost right: however, strictly speaking Fitts's Law predicts the time taken to point to a target, while in a hierarchical menu the problem is following a particular path or trajectory without deviating from it.

The law that defines human performance in path following is called the Steering Law, and is very closely related to Fitts's Law. It appears to have been discovered independently on at least three separate occasions, most recently by Johnny Accot and Shumin Zhai in 1997 [2] who gave it its most flexible mathematical formulation. This formulation is illustrated in figure 2.

As with Fitts's Law, the steering law is dependent on two constants, a (representing roughly the start and stop overhead) and b (representing pointing efficiency). The path is parameterised by s (vector distance along the path) and has a width of W*(*s) at point s. The movement time along the path (MT) is the integral of the width of the path along its length.

For the simple case of a straight tunnel of constant width, the law simplifies to

MT = a + b ⋅ (A/W)

where A is the tunnel length and W is its width. This gives us the intuitively correct answer that we can move faster along a wider (or shorter) path without going "off the edge".

Hick's Law

Hick's Law, also known as the Hick-Hyman Law, predicts the amount of time it will take a person to select between a number of alternatives. It has a similar form to Fitts's law:

ST = a + b ⋅ log₂ (n)

where ST is the selection time, a and b are constants, and n is the number of available choices. It is a very general law and holds for almost any circumstance where a person must respond to an easily-distinguishable stimulus by selecting one of an arbitrary set of choices (e.g., pressing one of several keys corresponding to a presented colour).

While it appears at first glance that applying Hick's Law to menu design would be a straightforward task, is actually quite subtle [3]. If the user doesn't remember all the menu items, or their order, a linear scan may be necessary---in which case selection time will be based more on the position of the desired item in the menu. However, if the menu can be divided up into logical groupings, Hick's Law can be made to apply.

Power Law of Practice

"Practice makes perfect" is a great slogan, but how perfect, and how fast do we get there? The Power Law of Practice gives an answer to those questions, at least for human performance improvement on consistent tasks involving skilled behaviour leading to motor memory. The form of the Power Law is

t_n = t₁ ⋅ n^-a

where t_n is the time to complete the n:sup:th trial, t₁ is the time to complete the first trial, and a is an empirically determined constant that varies with the task but is normally in the range of 0.4.

Weber's Law of Just Noticeable Differences

Weber's Law describes the human ability to immediately perceive the difference in intensity between two successively presented stimuli; for instance, to determine whether two successive weights are the same or different, or two successive lights are the same or different brightness. The form of Weber's Law is:

deltaI / I > k to cause a just noticeable difference

Here, I represents the intensity of the initial stimulus, deltaI represents the change in intensity, and k is a proportionality constant which varies with the type of stimulus, but which is generally in the range of 0.16.

What this means is that a difference in intensity of one part in six or more is immediately noticeable, where a smaller change is not.

This does not mean that human beings can only perceive in increments of one part in six: it speaks only to immediate (non-studied) perception, for successive presentation. Given time to study differences we can do significantly better, and if the stimuli are presented simultaneously (e.g., side by side) then we can perform extremely fine discrimination.

The human visual system

The human visual system is extremely subtle and complex. This is demonstrated by the many visual illusions we are subject to. For some excellent examples, see:

• Akioshi Kitaoka's illusion pages [5], as well as his comprehensive list of links;
• Edward H. Adelson's Checkershadow illusion [4]; and
• The Dragon Illusion [6].

— Greg Phillips

References

[1]	Bruce Tognazzini, A quiz designed to give you Fitts. Ask Tog, February 1999.

[2]	Johnny Accot and Shumin Zhai (1997). Beyond Fitts' law: models for trajectory-based HCI tasks. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. CHI '97. ACM Press, New York, NY, 295--302.

[3]	Landauer, T. K. and Nachbar, D. W. 1985. Selection from alphabetic and numeric menu trees using a touch screen: breadth, depth, and width. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. CHI '85. ACM Press, New York, NY, 73-78.

[4]	Edward H. Adelson. Checkershadow illusion

[5]	Akioshi Kitaoka. Akioshi's Illusion Pages

[6]	Grand Illusions. Dragon Illusion.