Circular variables, which
indicate direction or cyclical time, can be of great interest to
biologists, geographers, and social scientists. The defining
characteristic of circular variables is that the beginning and end of
their scales meet. For example, compass direction is often defined with
true North at 0 degrees, but it is also at 360 degrees, the other end
of the scale. A direction of 5 degrees is much closer to 355 degrees
than it is to 40 degrees. Likewise, times that represent cycles, such
as times of day (best expressed on a 24 hour clock), day in a
reproductive cycle, or month of a year are also circular. January,
month 1 is closer to December, month 12, than it is to June, month 6.
Examples of circular variables
are abundant in biology, geography, and the social sciences. One
experiment I saw in consulting compared the distance and direction
flown by male moths in comparison to unmated and mated female moths
under different weather conditions. Another compared the direction and
distance traveled by beetles under two different conditions of
crowding. Other examples include measures of wind and water flow
direction to understand the movement of pollutants and the timing of
events within a cycle, such as the when the number of heart attacks
peaks within a week or how body temperature fluctuates over a day. Note
that time can be considered either circular or linear. Time is circular
when it measures part of a cycle, such as the timing of a daily event.
It is linear when it measures length of time, such as the number of
days since an event.
Most familiar statistics do not
work with circular variables because they assume that variables are
linear--the lowest value is farthest from the highest value. For
example, the average of 5 degrees, 60 degrees and 340 degrees (which
are all northerly directions) is 135 degrees--a southerly direction.
This is clearly an unreasonable average. Changing 340 degrees to 20
degrees (an equivalent value) changes the mean to 15 degrees, which is
more reasonable. But 5 degrees could also be changed to 365 degrees,
giving a mean of 255 degrees, also reasonable. Which is right?
Because classical statistical
analysis does not work for circular variables, an entire field of
circular statistics has been developed. In circular statistics, each
datum is defined by its length and its angle from a chosen point on the
circle. In the case of the moths, each moth�s
final location would be designated by the distance it traveled from the
release point and the angle in degrees from true north. The mean
location of all the moths can be found using the sine and cosine of the
angle then adjusting for the length. Because the sine of 0 degrees and
360 degrees is the same, this solves the original problem of ends of
the scale being near each other.
Circular statistics include
tests of uniform direction around the circle, confidence intervals,
tests for comparing two groups of directions, circular graphs,
correlations, and regression, among others. Although the theory behind
these statistics is not new, there have been no mainstream statistical
packages that could implement them until recently. Now, both Stata and
S-Plus have implemented comprehensive circular statistics modules
within the last year.
References:
Batschelet, E. (1981). Circular Statistics in Biology. Academic Press:
London.
Jammalamadaka, S.R. & Sengupta, A. (2001). Topics in Circular
Statistics. World Scientific, River Edge, N.J.
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