How Can Adding a Movement Improve Target Acquisition Efficacy?

How can Adding a Movement Improve Target Acquisition Efficacy? Supplementary Material – Calculating Movement Amplitudes used in the Experiment

Alexander R. Payne1*, Beryl Plimmer1, Andrew McDaid1, Andrew Luxton-Reilly1, T. Claire Davies2

1University of Auckland, Auckland, New Zealand

[email protected], {b.plimmer, andrew.mcdaid, a.luxton- reilly}@auckland.ac.nz

2Queen’s University, Kingston, Canada

[email protected]

1. Controlling for Randomised Cursor Start Location

In order to make a fair comparison, we divide the screen into a number of equally sized targets and assume they are all equally likely to be selected. We also assume that the mouse is equally likely to start from any screen position. To control for the cursor starting from any position, we calculate an average movement time for each target position based on the integral of movement times across all positions. To do this, Fitts’s Law [1]and the Shannon formulation of ID [2] are used:

Once an average movement time is calculated for an individual target position, an amplitude of an average movement can be calculated. Notably, this amplitude does not equal the average amplitude of movements and it varies with target location. It is the amplitude for the average movement time. Importantly, our task set-up involves 1- D movements (Fig. 1). This simplifies the calculation of the amplitude of an average movement.

Fig. 1. A screen divided into equally sized 1D targets. SW = screen width, W = target width, i = th target number, n = number of targets on screen, Ai = movement amplitude for the i target. We split the screen up into n number of vertical targets. We will assume that there is an even probability of the cursor starting anywhere on the screen. E.g. it is not more likely to start on the left of the screen than the right. To demonstrate how movement amplitudes are calculated, Error: Reference source not found shows the amplitudes if the cursor was starting from the left edge of the screen. We are also assuming equal target sizes with equal likelihood of selection for each of them. Considering one target, we can express the average movement time for that target as the integral of movement times across possible amplitudes, A, divided by the range of those amplitudes (i.e. the screen width, SW).

Fig. 1. Movement time as a function of distance from that target. The integral divided by the range (i.e. the screen width) will equal an Ave MTA.

The function that defines the distance from the mouse start location to a target centre is discontinuous. Therefore, the integral needs to be split up into one for each direction, left and right.

If we define, A = 0 as the centre of the target, then to the left of the target, the distance goes from 0 to L and to the right of the target the distance goes from 0 to R where:

Therefore we can perform a definite integral to help evaluate the average movement time to specific target location from a randomised start location.

Therefore:

The above version of MTA is significant. It says that if we know a target width, its position on the screen, and the width of the screen, then we can defined the average movement time it will take to select that particular target. Now, we can substitute in the values of L and R.

Now, if we rearrange this in terms of the number of options, n instead of the target width (W), we can see that the screen width (SW) becomes irrelevant.

Therefore: By using this Ave MTA value and rearranging the Shannon formulation of ID, we can obtain an amplitude of an average movement.

This formulation contains the individual specific values a and b. However, once

Ave MTA is substituted in, the a and b values cancel out.

When calculating Ave MTA, the distance of the target from the centre of the screen is important. Therefore, our test scenario controls for this by using a target at every distance from the centre.

Ave MTA was used to set the movement amplitudes in the experiment.

2. References

3. 1. Fitts P. M.: The information capacity of the human motor system in controlling the amplitude of movement. J.Exp.Psychol. 47, 381 (1954). 4. 2. MacKenzie I. S.: Fitts' law as a research and design tool in human-computer interaction. Hum.-Comput.Interact. 7, 91-139 (1992).