My open questions on MacKenzie's formulation of Fitts' law

The HCI (human computer interaction) community claims to be a scientific community. The main topic to support this claim is Fitts' law. As a consequence there is a vast number of publications (see:MacKenzie's bibliography) on the topic. However, this big number of publication seems contributing more to confusion than to clarification.

Fitts published his now famous paper "The Information Capacity of the Human Motor System in Controlling the Amplitude of Movement" in 1954 [1]. In my view there is nothing wrong with this paper and the given formula for the index of difficulty (ID) is correct. However, the HCI community uses at least three different formulas for Fitts' law (Fitts, Welford, MacKenzie).

It seems that MacKenzie's formula is the most popular among HCI scientists. This formula originates from MacKenzie's publication "A Note on the Information-Theoretic Basis for Fitts' Law" from 1989 [2].

Here MacKenzie writes:

Fitts' law was developed from an analogy with physical communication systems. In such systems, the amplitude of a transmitted signal is described as perturbed by noise that results in amplitude uncertainty. The effect is to limit the information capacity of a communications channel to some value less than its theoretical bandwidth. Shannon's Theorem 17 expresses the effective information capacity C (in bits × s^-1) of a communications channel of band B (in s^-1) as

C = B log₂( (P + N ) / N )

(4)

where P is the signal power and N is the noise power (Shannon & Weaver, 1949, pp. 100-103). It is the purpose of this note to suggest that Fitts' model contains an unnecessary deviation from Shannon's Theorem 17 and that a model based on an exact adaptation provides a better fit with empirical data. The variation of Fitts' law suggested by direct analogy with Shannon's Theorem 17 is

MT = a + b log₂( (A + W ) / W )

(5)

My questions are:

Question 1:
What is wrong with Fitts' formula so that we need another formula?

We should not introduce new formulas without urgent needs. A better accuracy for the mean value on the third digit is not an urgent need if the sample data vary in a range of +/- 30%.

Question 2:
Why is an analogy with Shannon's Theorem 17 legitimate?

Why Shannon's Theorem 17 and not on any other theorem? What is a direct analogy? What is an unnecessary deviation?

Question 3:
What is analog to what and why?

Why should bandwidth (measured in bits × s^-1) be analog to time (measured in s)? Why is power analog to the target diameter (and not to radius or better the square of radius)?

Question 4:
What is the meaning of a?

The parameter a is introduced without mentioning and there is nothing analog in Shannon's Theorem 17. Actually, the a is not mentioned in Fitts' original work either. Used together with Fitts' definition of ID the a is the reaction time starting a pointing task. In MacKenzie's formula the a has no easy interpretation.

MacKenzie goes on with:

Fitts recognized that his analogy was imperfect. The "2" was added (see Equation 1) to avoid a negative ID when A = W ; however, log₂(2A / W ) is zero when A = (W / 2) and negative when A < (W / 2).

Question 5:
Where did Fitts state that his analogy is imperfect?

Actually, the factor 2 is necessary to get the radius of the target from its diameter W (see also question 3) and therefore it was not added. The simple question "radius or diameter?" is the source of MacKenzie's confusion. If A < (W / 2), the distance A to the target center is less than the radius and the pointer is already inside the target. This means the entry into the target happened in the past and Fitts' forumula delivers a negative time. Without the "2" there would be negative IDs for points outside the target.

MacKenzie states that using the Welford formulation with the additive term +0.5 results in better correlation and with his term +1 the correlation is even better.

Question 6:
Why MacKenzie did not go on and tested the formula MT = a + b log₂( (A / W ) + 1.5) or MT = a + b log₂( (A / W ) + 2) ?

On my small data set these formulas provide an even better correlation. By the way, I am sceptical that it is possible to prove the correctness of formulas by correlation.

Up to this point MacKenzie's publication was a single opinion. However, with the publication "A comparison of input devices in element pointing and dragging tasks" by MacKenzie, Sellen, and Buxton in 1991 [3] MacKenzie's theory became popular in the HCI community.

In this paper it is written:

There is recent evidence that the following formulation is more theoretically sound and yields a better fit with empirical data (MacKenzie, 1989):
MT = a + b log2(A/W + l). (3)

Question 7:
Who else than the author showed this evidence? Who agrees that 'the formulation is more theoretically sound'? Does this mean that Fitts was wrong?

In science we need confirmation by competitive researchers. We should not confirm ourselves.

Question 8:
Why did nobody ask these questions within the last 20 years?

In the case that there had been people who asked critical questions, what happened to them?

Question 9:
Is there anyone in the HCI community who can answer my questions?

There are so many scientists in the HCI community who published papers on Fitts' law and should be able to answer my questions. Alternatively, I would be glad to get some support for my view.

My questions are open since April 2010 [4]. This page went online on August 2012.

[1] Fitts, P. M. The Information Capacity of the Human Motor System in Controlling the Amplitude of Movement. In Journal of Experimental Psychology (1954), 47, 381 – 391.

[2] MacKenzie, I. S. A Note on the Information-Theoretic Basis for Fitts' Law. Journal of Motor Behavior (1989), 21, 323 – 330.

[3] MacKenzie, I. S., Sellen, A., and Buxton, W. A. A comparison of input devices in element pointing and dragging tasks. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. CHI '91. ACM Press (1991), 161 – 166.

[4] Drewes, H. Only one Fitts' law formula please! In Proceedings of the international Conference Extended Abstracts on Human Factors in Computing Systems. CHI EA '10. ACM Press (2010), 2813 – 2822.