that knowledge of what month it is (X) gives no information about expected temperature (Y). In general, any departure from a linear relationship degrades the correlation coefficient.
The first defense against nonlinear relationships is to transform one or both variables so that the relation between them is linear. Taking the logarithm of one or both is by far the most common transformation; taking reciprocals is another. Taking the logarithm of both variables is equivalent to fitting the relationship Y=bXm rather than the usual Y=b+mX. Our earlier plotting hint to try to obtain a linear relationship had two purposes. First, linear regression and correlation coefficients assume linearity. Second, linear trends are somewhat easier for the eye to discern.
A second approach is to use a nonparametric statistic called the rank correlation coefficient. This technique does not require a linear correlation. It does require a relationship in which increase in one variable is accompanied by increase or decrease in the other variable. Thus the technique is inappropriate for the Anchorage temperature variations of Figure 10b. It would work fine for the world population data of Figure 13, because population is always increasing but at a nonlinear rate. To determine the rank correlation coefficient, the steps are:
- assign a rank to each Xi of from 1 to N, according to increasing size of Xi;
- rank the Yi in the same way;
- subtract each Xi rank from its paired Yi rank; we will call this difference in rankings di;
- determine the rank correlation coefficient r, from r = 1 - [6(Σdi2)]/[N(N2-1)].
The rank correlation coefficient r is much like the linear correlation coefficient R, in that both have values of -1 for perfect inverse correlation, 0 for no correlation, and +1 for perfect positive correlation. Furthermore, Table 7 above can be used to determine the significance of r in the same way as for R.
For example, the world population data of Figure 13 obviously show a close relationship of population to time. These data give a (linear) correlation coefficient of R=0.536, which is not significant according to Table 7. Two data transforms do yield correlations significant at the 99% confidence level: an exponential fit of the form Y=b+10mx (although this curve fit underestimates current population by more than 50%), and a polynomial fit of the form Y=b+m1X+m2X2 (although it predicts that world population was much less than zero for 30-1680 A.D.!). In contrast, the rank correlation coefficient is r=1.0, which is significant at far more than the 99% confidence level.
Nonlinearities are common; the examples that we have just seen are a small subset. No statistical algorithm could cope with or even detect the profusion of nonlinear relationships. Thus I have emphasized the need to make crossplots and turn the problem of initial pattern recognition over to the eye.
Nonlinearities can be more sudden and less predictable than any of those shown within the previous examples. Everyone knows this phenomenon as ‘the straw that broke the camel's back’; the scientific jargon is ‘extreme sensitivity to initial conditions’. Chaos, a recent physics paradigm, now is finding such nonlinearities in a wide variety of scientific fields, particularly anywhere that turbulent motion occurs. The meteorologist originators of chaos refer to the ‘Butterfly Effect’: today’s flapping of an Amazon butterfly’s wings can affect future U.S. weather. Gleick [1987] gives a remarkably readable overview of chaos and its associated nonlinearities.
