Fitts Law: [Fitts and Posner, Human Performance, Wadsworth, 1967]

Our estimate for the time pointing the mouse, P, in GOMS Key Level Model is imprecise. The time required to locate the mouse is dependent on the target size, S, and the distance, D.  Intuitively the time is proportional to D/S, in other words it is easier to locate a mouse on a closer larger target. In order to get the units correct (remember I am a physicist) the measure of the size of the target is a target diameter.  Fitt user’s movement time, MT, to locate the mouse and discovered:

MT  = a +b lg(D/2S)

or
MT = a + b lg(D/S + 1)

where a and b are constants for the particular user and mouse.  Raskin suggests using a = 50 msec and b = 300 msec.  The logarithmic dependence might be surprising to us, but I hope by the end of this lecture you will believe it correct.  Naturally if the mouse is in the correct position, D = 0, the time is zero; therefore the reason for the 1 in the logarithmic argument.  Fitts explains why buttons are faster than selecting menus.  Typically the user perceives the target to be smaller than it actually is.  The law also explains the frustration I have locating a cursor in order delete a character; I must move the mouse a large distance and precisely locate it between characters.  (Another reason to use the vi editor.)

 

In order to improve our estimates for gui user input time we need drawings of the gui.  If we assume all buttons are the same size (of course they are not) and we move the mouse the same distance, then we can use the same time. The estimate for P = 1.1 sec assumes D/S = 9 (using Raskin's values for a and b); for 40 px target represtents moving approximately 400 px.
 

We can derive Fitts’ in one dimension by assuming that the selection of a target is performed by successive approximation.  Assume a single dimension with the pointer distance D from the center of a linear target width, W.  Also assume that the user moves to the target by successive approximation, with accuracy a.  Let Xi represent the distance from the center of the target for each successive try.  For example:

X0 = D

X1 = aX0 = aD

…

Xn = anD

The process stops when the pointer gets inside the target:

anD < W/2

Solve for n:

n = lg(W/2D)/lg a

Now we assume that the movement time, MT, is proportional to the number of tries, n, and a < 1.

MT ~ lg(W/D),

the form of Fitts law.  The law can also be derived modeling the user a feedback system and that is speed is proportional to distance from the target. 


MacKenzie and Soukoreff reevaluated Card, English, and Burr study of Fitts’ Law for text selection with word 1, 2, 4, 8, 16 cm away from the cursor and word length 1, 2, 4, 10 characters long.  Assuming characters 0.246 cm wide and 0.456 cm high we can calculate the index of difficult, ID, for the tasks:

        ID = lg(D/W +1)

The results:

 

MacKenzie revaluation of Card’s Fitts’ Experiment for text selection


Note that the reciporical of the b coefficients is called the Performance Index:
PI ~ 1/b
What does it meausre?

Steering Law:

Most is the material for this section is from Accot and Zahai CHI’97 conference paper, “Beyond Fitts’ Law: Models of Trajectory-Based HCI Tasks.

 

In a series of experiments they illustrate and propose a law for moving the computer cursor through constraints, steering law.  Steering law is derived as a generalization of Fitts’ Law. 

 

Experiment 1:

The first experiment is a repeat of Fitts’ Law experiment through two goal posts. 

 

 

Then Fitts Law would define the index of difficulty, ID, as:

 

ID = lg(A/W + 1)

 

and the movement time, MT, and the plot of MT verses ID should be a straight line.  The average result from ten subjects from a fully crossed experiments using A = 256, 512, and 1024 pixels, and W= 8, 16, and 32 pixels, each subject performed all tasks 10 times:

 

 

 

The correlation of the best fit regression line is r2 = 0.987, quite good but the data is highly averaged. 

 

The authors derive the steering law by proposing a sequence of Fitts’ goal post experiments.  So if the subject performs a sequence of N goal post experiments the index of difficulty, IDN, is:

 

IDN = N lg(A/NW + 1)

 

If the we let N go to infinity then:

 

            ID∞ = A/W ln 2

 

The second experiment tests this relation by requiring subject to constrain the cursor to a rectangle, and the subject moves the cursor from one side to the opposite side.  The results from 13 subjects and constraining rectangles with A = 250, 500, 750, and 1000 pixels, and W = 20, 30, 40, 50, 60, 70, 80, and 90 pixels.  Each subject performed two sets of 5 tasks of all the rectangles. 

 

The measure time is MT =  -188 + 78 ID,   r2 = 0. 98 and average error rate is 6.4%. 

 

The authors derive a steering law for a tapered constraint.

 

 

By integrating dx/W(x) across the path the steering law for the index of difficult is:

 

ID = A ln(W2/W1)/(W2 – W1)

 

The third experiment tested this law with ten subjects and W1 = 20, 30, 40, 50; W2 = 8 and A = 250, 500, 750, 1000.

 

 

With MT = -532 +92 ID with r2 = 0.978 and the average error rate is 18%. 

 

A fourth experiment test steering law a more elaborate constrain, a spiral, r = (θ+ w)3.  The index of difficult is now the integration of ds = f(w, θ) dθ, rather elaborate integral. 

 

 

 

Tested with fully crossed set of spirals with number of turns, n = 1, 2, 3, 4 and width factor, w = 10, 15, 20, 25, with each subject performing 2 sets of 10 trials.

 

 

 

The best fit line is MT = 115 + 169 ID with r2 = 0.97 and averaged error rate is 13.7%.

 

Note that the coefficient, 169, the slope of the best fit line is appreciably different from the slope for two previous experiments, 78 and 92.  This suggest that significant aspects of the experiment/task is missing.  In later papers the authors speak about domains of steering.  Curved steering and straight steering are in different domains. 

 

In this paper the author derive a local law, that the instantaneous velocity of the cursor is proportional to the velocity. 

           

            v(s) = w(s)/τ

 

The author also proposed that the instantaneous velocity might also depend radius of curvature and direction. 

 

Hick’s Law:

 

Our estimate for thinking time, M, is not precise.  We should consider the amount the information the user has to process and communicate.  Hick's law says given n choices that user can select with equal probability then the time to decide on the choice is:

 

Time = a + b lg(n + 1)

 

The form of the equation is the same as Fitt's law.  The coefficients, a and b, are dependent on the presentation of the choices and how well the user knows the effects of alternatives.  Certainly if the user always uses a single choice the time is much shorter.  Given that the user can always choose not to respond then the formula derives directly from information theory.  Consider the fastest way a user can make a choice.  We could make a decision tree and follow the decision tree to a leaf.  I find the implications of this law impressive, the law implies that users do not make choices by considering one alternative with each other, that would take exponential time. Users make choices from a global perspective.  (By the way I have a saying based on Hick's law; a genius makes a choice in less then logarithmic time.)  An implication of Hick's law is that it is better to present to the user a broad menu selection rather then a deep menu hierarchy.  Not only is a broad menu selection easy for the user to choose from it takes less keystrokes.  Naturally any single design style can be abused.  A Menu that has 100 selections would not appear on the screen and taxes the user's short term memory. My experience is the expert user can be presented with 20 items.  If the selections can be ordered appropriately then it is possible to present 100 selections that works most of the time.