Abstract
When subjects were asked to estimate the degree of overlap of pairs of circles, their median estimates were close to the true values but showed marked variability. A tendency for the estimates to be less variable when squares rather than circles were displayed suggests that overlapping squares are a superior method of portraying correlation.
Silverstein (1978) describes a technique for portraying the relationship between two variables (x, y) by means of overlapping circles, where the circles represent the total variance of the two variables and the overlap represents r2, the proportion of xs variance associated with y and vice versa. Silverstein provides a table showing the distance between the centers of the two circles which is necessary in order to represent specified values of r2.
Although there have been experiments studying the accuracy with which subjects can judge the relative sizes of circles (Rule, 1968), there do not appear to have been any studies of subjects’ ability to estimate the amount of overlap of circles. Since the technique described by Silverstein is intended to communicate the size of r2 by a visual representation, it is important to establish the effectiveness of the display mode by discovering how accurately subjects perceive the amount of overlap shown.
Four displays were prepared. Each consisted of two circles (5 cm diameter), with the distance between the centers varying so as to give amounts of overlap representing r2 of 4, 16, 36, and 64%. Each display was drawn in the center of a 20-cm square card. The displays were shown in a separate random order to 27 school students aged 16 to 18 yr. Subjects were tested individually, and the cards were handed singly to each subject who was asked “What proportion of the area of the left-hand circle is overlapped by the right-hand one? Please express your estimate as a percentage.”
The tendency for responses to be numbers ending in 5 or 0 meant that the response distributions were bimodal. The median estimates (and semi-interquartile ranges) for each display were as follows: 5.2 (2.5); 19.6 (5.0); 32.6 (5.1); 65.0 (7.1). The medians were close to the true values of 4, 16, 36, and 64%, but the estimates demonstrated considerable variability.
A difficulty with this display format is the complexity of preparing the diagrams, to which Silverstein's paper bears witness. An alternative, easier procedure is to use squares rather than circles. If two squares of equal side length and standing on the same plane are used, the amount of overlap and hence the positioning of the two squares can be calculated much more readily than is the case with circles. The effectiveness of using overlapping squares was tested in similar fashion to the experiment described above, using a fresh sample of 21 subjects. The displays showed the same levels of overlap as the circles used previously had done.
The medians (and semi-interquartile ranges) of the responses to the squares were 5.4 (2.5); 15.1 (4.8); 30.3 (1.7); and 65.6 (3.2). The medians are similar to those for the circle displays (for no level of true value do the two medians differ significantly, according to Mann-Whitney tests). The semi-interquartile ranges for the larger amounts of overlap are notably lower for the squares than the circles, but the difference is only significant for the largest level of overlap. Responses were categorized according to whether they were within 2.5 of the median. For the true value of 64, χ2 = 13.89 (p < 0.002). On the grounds of simplicity of preparation and the tendency to lower variability in the responses, overlapping squares may be recommended in preference to overlapping circles as a method of portraying r2.