Journal of Experimental Psychology 1972, Vol. 96, No. 1, 130-133

THE GENERALITY OF FITTS'S LAW LISANNE BAINBRIDGE 1 AND MOIRA SANDERS University of Reading, Reading, England J u P. M. Fitts's original equation describing the relation between amplitude, accuracy, and duration of hand movements, the duration depends on the logarithm of the ratio of amplitude and accuracy. Data presented by A. T. Welford suggest that the logarithmic transform gives additivity of amplitude and accuracy effects. This paper presents data from a variant of Fitts's dotting task, in which amplitude and duration of movements were controlled and accuracy measured. The results support the original ratio equation.

Fitts (1954) was the first to suggest an equation which related the amplitude, accuracy, and duration of hand movements. This equation was determined from empirical studies in which S moved his hand to and fro between two targets. The distance (.4) between, and width (W) of, the targets was varied and the time (MT) taken to make the movements was measured. The following equation fitted the data:

MT = a-\ogs(2A/W) + b, in which a and b are constants. From his equation, Fitts (1954) made inferences about the mechanisms of movement control in terms of information theory. Use of binary logarithms is not necessary to the analysis; however, the data would give a straight line if plotted using the above equation with logarithms to any base. Change of base, e.g., to log*, simply changes the constant a by a factor logz(2). The multiplier, 2, is not an empirically determined parameter but was added by Fitts to ensure that the ratio was always positive; it has the effect of adding a constant. The essential aspects of Fitts's equation are that the movement time is a function of the ratio of the amplitude and accuracy of the movement, whatever their absolute sizes, and that this function is logarithmic. Later researchers have studied the same type of task and suggested refinements of this equation. In Fitts's original equation, 1

Requests for reprints should be sent to Lisanne Bainbridge, Department of Psychology, University of Reading, Building 3, Earley Gate, Whiteknights, Reading, RG 6 2AL, England.

the A and W values were those defined by E and drawn as the targets. Since 5 may not be working according to the values thus defined, E should calculate the A/W ratios from S's A and W values, A' and W, in which A' is the measured distance between the midpoints of the two hit distributions and W is the measured width of the scatter of S's hit points about a given target. For the present purpose, the most important refinements of Fitts's equation using A' and W were given by Welford (1968, pp. 147 & 156): MT = k-log(A'/W + .5)

[1]

MT = k-\og(A'/(W - c) + .5). [2] In Welford's study, a single line could be drawn through the data on a plot of movement time against Equation 1, but a better fit to a line was obtained by plotting against Equation 2, with c — 3 mm. In a study by Welford (1969) in which the parameters were empirically determined, average values obtained were k = .116 and c = 3.16 mm., for Ss aged 20-60 yr. Welford (1968, p. 157) also presented a different analysis of the data, which brings into question the basic ratio relation, suggesting instead that the absolute sizes of A and W do affect the movement time. In Welford's plot of movement time against Equation 1, the points for one target width and different movement amplitudes did fall on a straight line. A set of such lines was obtained, one for each target width used. These lines were parallel. This means that a change in target width added a small constant to the movement time, and this constant was 130

131

THE GENERALITY OF FITTS'S LAW

unaffected by the particular amplitude being tested. Parallel lines were also obtained if points for any one amplitude at different target widths were connected. Thus the effects of accuracy and amplitude could be analyzed separately by taking their logarithms. Their effects are then additive and their interaction does not have to be considered. The equation describing this relation is given by Welford (1968): MT = a-logA' + 6-log(l/W")

[3]

From Welford's data the ratio b:a, for 5s aged 20-60 yr., was 1.75: 1; i.e., in determining movement time accuracy was weighted 1.75 times more heavily than amplitude. If a = b, Equation 3 simplifies to MT = a[\ogA' + log(l/PF')] = a>log(A'/W), the ratio as found in previous studies. This suggests that Fitts's finding, that movement time is determined by the ratio of amplitude and accuracy rather than their absolute values, may have been a happy accident of using a task in which amplitude and accuracy are equally weighted, rather than being a basic determinant of movement control. Welford's (1969) analysis did not resolve which of these two formulations, the ratio or the additive, best fitted the data. The data plotted against Equation 2 with c = 3 fitted a straight line. The data plotted against Equation 3, with a = .104 and b = .177, fitted a straight line just as well. If one of these equations is a general statement of the relation between amplitude, accuracy, and duration, then it should apply in other tasks. This paper presents a study of a simple inversion of Fitts's original task. Amplitude and duration of movement were controlled, and accuracy measured. Brown and Slater-Hammel (1949) varied the amplitude of movements made to a single line target, and found that duration of movement increased with amplitude, at an essentially constant accuracy of within less than 2 mm. from

TABLE 1 SUBJECTS' MOVEMENT ACCURACY (W) AND AMPLITUDE (A') FOR EACH CONDITION MT (in see.)

Item

.3

W W

A (in in.) 4

A'

.4 .6

A'

II"

A'

.99 4.06 .68 3.99 .34 3.99

g

16

1.75 7.77

3.13

14.66 2.0

1.09 8.01 .41 8.01

15.6 j

.79

15.86

Note.—Movement acc'uvaey and amplitude are measured in inches.

the target line. Woodworth (1899) studied movements of constant duration made over amplitudes of 50, 100, ISO, and 200 mm. The error scores were slightly less than would be predicted from Fitts's equation. In the experiment described here both amplitude and duration of movement were varied systematically. Analysis of the results should show whether the same type of relation generalizes to the description of this task, and if so should help to discriminate between the variant equations suggested for the original task. METHOD AND DESIGN The S dotted between lines drawn on paper. Each target was marked by a single line, so a target sheet contained a pair of parallel lines. The distance between the lines was the amplitude as denned by E. Three amplitudes were used, 4" (102 mm.), 8" (204 mm.), and 16" (408 mm.). Movement time was set by making 5 move in time with the beats of a metronome. Three metronome speeds were used, 100, 150, and 200 movements per minute, or .6, .4 and .3 sec. per movement. Each trial lasted for 1 min. The 5 was asked to be as accurate as possible in hitting the target line, but the importance of keeping up with the metronome was stressed. The S was told that accuracy should suffer rather than timing if the beats were too fast. The three amplitudes and three speeds gave nine experimental conditions. These nine conditions were presented in pseudorandom order to S. They were then presented again in the reverse order. Each S was given a different random order. Ten 5s, all volunteer university students, were used. The accuracy of hits around each target was found by the simple technique used by Welford (1969). W' was taken as the distance between extreme shots, measured along the axis of the movements, ignoring wild deviations. To measure S's amplitude, A', a

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LISANNE BAINBRIDGE AND MOIRA SANDERS

For comparison with earlier equations we wish to find MT in terms of W and A'. Taking logarithms of Equation 4 and solving for MT we obtain :

MT = .1279 logt(A'/(W - .176)) + .0215. • 6sec. W'=.O39A'+.I48 ?6 8 12 f

o 4

A' inches FIG. 1. Accuracy as a function of amplitude, with movement time as a parameter.

[5]

Equation 5 is similar in form to Equation 2. The target width W was the dependent variable in this experiment. A version of Welford's (1968) additive type of equation, 3, in terms of W would be: logW" =

' - MT).

The graph of \og^W against (\og*A'-MT) is shown in Fig. 2, which also shows the line was drawn down the linear center of the dis- equations for the three lines of constant tribution of hits for each target, perpendicular to the axis of movement. The A' in a given trial was A'. The differences in slope of these lines For the measured as the distance between these lines drawn were statistically significant. for the two targets. slopes at 4" and 8" A, F (1, 2) = 40.36; and for 8" and 16", F (1, 2) = 88.48, both significant at p < .05 ; for the slopes at 4" RESULTS and 16", F (1, 2) = 162.5, p < .01. The There were 20 readings for each condi- slopes and intercepts of these lines both tion; the means of these were calculated, changed consistently with A', and both in inches, and are given in Table 1. these functions were best fitted (by regresFigure 1 shows the accuracy achieved sion) by logarithmic equations. These with different movement amplitudes at the equations substituted in the general linear three levels of movement time. Inaccuracy equation give : (width of target) was linearly related to amplitude. This relation interacted with log, W"= (2.71 log*l'-.47)(log*4'-M:r) + (19.43- 13.61 Iog24'). [6] MT, as the slope of the function varied with Mr. The equations for the three lines, This equation cannot be solved for MT in obtained from least-squares regression cal- a way in which logW and log.4' are inculations, are given in Fig. 1. The lines dependent additive components. had a common intercept, of average value .176. The differences in slope were statistically significant. For the slopes at .6and .4-sec. MT, F (1, 2) = 40.34; and for .4- and .3-sec. MT, F (1,2) = 44.95—these were significant at p < .05; for the slopes at .6 and .3 sec., F (1, 2) = 221.05, p < .01. The relation of slope of line to MT could not be fitted by a linear, logarithmic, or power equation, but could be fitted by an exponential equation. In this equation, 2 is used as a base, as is common in these studies. Substituting this equation for -2J slope in the general equation for these straight lines gives:

W = (1.015 X 2-i

' + .176. [4]

FIG. 2. Log2W as a function of (logjX' - MT), with mean amplitude A' as a parameter.

THE GENERALITY OF FITTS'S LAW

DISCUSSION We wish to test whether the data from this experiment is best fitted by an equation such as Equation 2, in which A' and W interact as first suggested by Fitts (1954), or whether the logarithmic transform has the effect that A' and W are independent and additive as suggested by Welford (1968) in Equation 3. Two analyses of the data have been presented. Analysis of the untransformed data gave an equation, 5, which is very similar to Equation 2. In Equation 5, the constant .1279 is similar to Welford's k value of .116. Welford (1969) obtained a c calue of 3.16 mm. The constant .176 in Equation 5 can be taken as a measure in inches as used in this task and is equivalent to 4.47 mm. The discrepancy between these c values may be because c is a parameter which differs in the two tasks. Alternatively, Welford used a fixed added constant of .5 in his calculations, which may have influenced the c values he found. The data from this analysis therefore provides good support for an equation in which the effects of amplitude and accuracy interact, as suggested by Fitts. Support for an equation like Equation 3 would be obtained from the data after log transform. If the log transform imposed additivity of A' and W, the lines in Fig. 2 should be parallel; an increase in log;!' should not affect the slope of the line relating logW and MT. As shown in this article, the slopes of these lines are not the same, and they

133

vary consistently with A'. Also, it should be possible to solve for MT the one equation derived for all the lines in such a way that logW and log^4' are independent additive components as in Equation 3, and this cannot be done for Equation 6. The data from this experiment therefore does not fit an independent additive equation like Equation 3, the type suggested by Welford (1968). The results of an experiment on the amplitude, duration, and accuracy of movement, using a variant of the original Fitts dotting task, therefore further support Fitts's analysis that it is the ratio of amplitude and accuracy which determine the duration of a movement, and show that this relation describes performance in an inverse type of task. REFERENCES BROWN, J. S., & SLATER-HAMMEL, A. T. Discrete movements in the horizontal plane as a function of their length and duration. Journal of Experimental Psychology, 1949, 39, 84-95. FITTS, P. M. The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 1954, 47, 381-391. WELFORD, A. T. Fundamentals of skill. London: Methuen, 1968. WELFORD, A. T. Speed and accuracy of movement and their changes with age. Acta Psychologica, 1969, 30, 3-15. WOODWORTH, R. S. The accuracy of voluntary movements. Psychological Review, 1899, 3(2, Whole No. 13). (Received February 8, 1972)