Editing Fitts's law (section)

== Welford's model: innovations driven by predictive power ==
Not long after the original model was proposed, a 2-factor variation was proposed under the intuition that target distance and width have separate effects on movement time. Welford's model, proposed in 1968, separated the influence of target distance and width into separate terms, and provided improved predictive power:<ref name="Welford"/>

<math display="block">MT = a + b_1 \log_2 (D) + b_2 \log_2 (W)</math>

This model has an additional parameter, so its predictive accuracy cannot be directly compared with 1-factor forms of Fitts's law. However, a variation on Welford's model inspired by the Shannon formulation,

<math display="block">MT = a + b_1 \log_2 (D+W) + b_2 \log_2 (W) = a + b\log_2 \left(\frac{D+W}{W^k}\right)</math>

The additional parameter ''k'' allows the introduction of angles into the model. Now the users position can be accounted for. The influence of the angle can be weighted using the exponent. This addition was introduced by Kopper et al. in 2010.<ref name="Kopper2010">{{cite journal |title=A human motor behavior model for distal pointing tasks |journal=International Journal of Human–Computer Studies |last1=Kopper |first1=R. |last2=Bowman |first2=D. A. |last3=Silva |first3=M. G. |last4=MacMahan |first4=R. P. |volume=68 |issue=10 |pages=603–615 |doi=10.1016/j.ijhcs.2010.05.001 |year=2010 |s2cid=12530345 }}</ref>

The formula reduces to the Shannon form when ''k = 1''. Therefore, this model ''can'' be directly compared against the Shannon form of Fitts's law using the [[F-test]] of nested models.<ref name=Shoemaker_2012>{{cite journal	|last1=Shoemaker	|first1=Garth	|last2=Tsukitani	|first2=Takayuki	|last3=Kitamura	|first3=Yoshifumi	|last4=Booth	|first4=Kellogg	|title=Two-Part Models Capture the Impact of Gain on Pointing Performance	|journal=ACM Transactions on Computer-Human Interaction	|date=December 2012	|volume=19|issue=4|pages=1–34|doi=10.1145/2395131.2395135|s2cid=6201126	|doi-access=free	}}</ref> This comparison reveals that not only does the Shannon form of Welford's model better predict movement times, but it is also more robust when control-display gain (the ratio between e.g. hand movement and cursor movement) is varied. Consequently, although the Shannon model is slightly more complex and less intuitive, it is empirically the best model to use for virtual pointing tasks.