# Independent Variables in Fitts' Law Experiments

I. Scott MacKenzie
Dept. of Electrical Engineering and Computer Science
York University
mack@cse.yorku.ca

Last update: 29/3/2015

## Background

An important consideration for research in human-computer interaction is the design of experiments that involve testing with human participants.  Two fundamental features of such experiments are the independent variables and the dependent variables.1  There is at least one of each.  An independent variable is a circumstance of interest to the investigator, while a dependent variable is a measurable human behaviour potentially influenced by the circumstance.  For example, an investigation on computer pointing devices may ask whether speed or accuracy in selecting on-screen targets is influenced by the type of pointing device or the transfer function that relates device motion to cursor motion.  In the preceding sentence, two independent variables are identified: "device" and "transfer function".  Two behaviours, or dependent variables, are also identified: "speed" and "accuracy".

In designing an experiment, the circumstances are organized into factors (i.e., independent variables) and levels.  In the example, "device" and "transfer function" are factors.  Device might be tested over, say, three levels: "mouse", "trackball", and "joystick", while transfer function might be tested over, say, two levels: "linear" and "accelerated".  In this scenario, the result is a "3 × 2 design"; i.e., an experiment with six circumstances organized as two factors with three levels on the first and two on the second.  Circumstances are also called "test conditions".

The procedure for such an experiment would likely use a dozen or more participants with a series of trials administered to each participant under each test condition while the speed and accuracy of responses are measured.  Issues such as counterbalancing the order of conditions are also relevant, but are not considered here.

## What Makes a Variable "Independent" or "Dependent"?

The discussion above is simple enough, but let's dig a little deeper on the terminology.  Why are behaviours such as speed and accuracy called "dependent variables" while circumstances such as device or transfer function are called "independent variables?  What do they "depend on"?  What are they "independent of"?  This too is simple (but see below).  As expressed by Martin (2004), dependent variables are so-called because the behaviour they represent depends on the participant – the behaviour depends on what a participant does.  Speed and accuracy in selecting targets may differ among the test conditions, but for all conditions, they are also a direct result of participant behaviour.

Independent variables are so-called because the test conditions are independent of the participant – the test conditions are independent of anything a participant does.  In testing three devices, there is nothing a participant can do to influence the fact that they are using a particular device for a series of trials.  Perhaps the participant is given a trackball for one series of trials, and a mouse for another.  No behaviour the participant exhibits can influence this.

## Variables in Fitts' Law Experiments

The preceding might seem like a tutorial on experiment design, but there is an interesting an important twist in the case of Fitts' law.  In most respects, a Fitts' law experiment is like any other experiment seeking to measure human behaviour under circumstances of interest.  The dependent variables are typically speed, accuracy, and often throughput.  Numerous independent variables appear in Fitts' law experiments, such as device (e.g., Card, English, & Burr, 1978; Epps, 1986; MacKenzie, Sellen, & Buxton, 1991), transfer function (Jellinek & Card, 1990; MacKenzie & Riddersma, 1994), or system lag (MacKenzie & Ware, 1993).

However, there are two rather special independent variables in the vast majority of Fitts' law experiments: movement amplitude (A) and target width (W).  Typically the task involves tapping/selecting back and forth between two targets of width W separated by amplitude A.  Both A and W are organized as factors and are tested over several levels each.

Fitts developed his model by analogy with Shannon's theory of communications.  Accordingly, movement amplitude is analogous to the amplitude of a transmitted signal, and target width is analogous to Gaussian noise in the communications channel.  Since noise is superimposed on, and perturbs, the signal, a proper test involves measurements conducted over multiple transmissions of the signal.  In experimental terms, this means administering a sequence of trails for a given movement amplitude and target width.  Averaged over the sequence, the mean movement amplitude should be A and the end-point or selection coordinates, perturbed as they are, form a distribution with the central 96% of the selections falling within the target and 4% falling outside the target.1  Hence, a sequence of trials in the ideal case includes a 4% error rate.

Fitts sought to measure the information capacity of the human motor system under several circumstances of interest.  One was a stylus held in the hand and manoeuvred to tap targets arranged in a fashion similar to the figure above.  His experiment used a range of signals (movement amplitudes) and noise (target widths).  He reported that the information capacity of the human motor system was relatively constant over the tested range of signals and noise.

The scenario above is repeated in many dozens of Fitts' law experiments found in the literature.  But, there is a problem.  While the circumstances of interest are often exemplars in the field of research (e.g., "device" for research on computer input, or "system lag" for research on virtual reality), the additional circumstances of signal amplitude (viz. movement amplitude) and noise magnitude (viz. width of the distribution of selections) are always included as additional independent variables, if for no other reason to ensure that the measurements cover a representative range of movement amplitude (signal) and target width (noise) conditions. But, are movement amplitude and target width independent variables?  Are the movement amplitudes and the distribution of selections independent of participant behaviour?  Clearly, they are not.  It is entirely possible for a participant to behave in a manner such that Ae does not equal A and/or We does not equal W.  Here, Ae is the actual or "effective" movement amplitude measured over a sequence of trials and We is the measured "effective" width of the target – operationally, the central 96% of the distribution of selection coordinates.  Because the presented and measured values may differ depending on what a participant does, amplitude and width cannot be viewed as independent variables in the same manner as, for example, device or transfer function. That is, they are not independent of participant behaviour.

## Does it Matter?

Despite the possibility that A and W are not, in the strictest sense, independent variables, it is not clear that the conjectured dependence of these variables on participant behaviour actually occurs in practice or that it makes a difference in the outcome of an experiment.  One way to accommodate the "dependent" nature of these independent variables is to use Ae and We in building a Fitts' law model.  This possibility is explored further in Chapter 2 in Fitts' Law as a Performance Model in Human-Computer Interaction (MacKenzie, 1991).  A more recent discussion is given in Chapter 7 in Human-Computer Interaction: An Empiricial Research Perspective (MacKenzie, 2013).

## Acknowledgment

The motivation for this Research Note emerged from a lunchtime conversation with Ravin Balakrishnan and Darius Miniotas in May 2002.

## References

Card, S. K., English, W. K., & Burr, B. J. (1978). Evaluation of mouse, rate-controlled isometric joystick, step keys, and text keys for text selection on a CRT. Ergonomics, 21, 601-613.

Epps, B. W. (1986). Comparison of six cursor control devices based on Fitts' law models. Proceedings of the Human Factors Society 30th Annual Meeting, 327-331. Santa Monica, CA: Human Factors Society.

Jellinek, H. D., & Card, S. K. (1990). Powermice and user performance. Proceeding of the ACM Conference on Human Factors in Computing Systems - CHI '90, 213-220. New York: ACM.

MacKenzie, I. S. (1991). Fitts' law as a performance model in human-computer interaction. Doctoral Dissertation, University of Toronto (http://www.yorku.ca/mack/phd.html).

MacKenzie, I. S. (2013). Human-computer interaction: An empirical research perspective. Waltham, MA: Morgan Kaufmann.

MacKenzie, I. S., & Riddersma, S. (1994). Effects of output display and control-display gain on human performance in interactive systems. Behaviour & Information Technology, 13, 328-337.

MacKenzie, I. S., Sellen, A., & Buxton, W. (1991). A comparison of input devices in elemental pointing and dragging tasks. Proceedings of the ACM Conference on Human Factors in Computing Systems - CHI '91, 161-166. New York: ACM.

MacKenzie, I. S., & Ware, C. (1993). Lag as a determinant of human performance in interactive systems. Proceedings of the INTERCHI '93 Conference on Human Factors in Computing Systems, 488-493. New York: ACM.

Martin, D. W. (2004). Doing psychology experiments (6th ed.). Belmont, CA: Wadsworth.

## Footnotes

1. Synonymous terms are manipulated variable and response variable, respectively. An independent variables is also called a factor.