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How can a non-linear relationship among two variables influence the correlation coefficient?
How can a non-linear relationship among two variables influence the correlation coefficient? For example, ability to pay attention and the level of stimulation in the environment do not demonstrate a linear relationship. At some point, too much stimulation will limit the ability to focus and pay attention. What would happen to the size and the sign of the correlation coefficient if we correlated these two variables?
The correlation coefficient is determined by observing the behaviour over the entire set of responses -- it is an average measure. It is used to measure the linear dependence between two variables -- it is 1 or -1 if there is a truly linear dependence and some value inbetween otherwise. If there is a non-linear dependence, then the correlation coefficient is not very useful. As humans, we may see a very strong pattern of dependence between the variables, but the correlation coefficient may be zero. Visit
http://en.wikipedia.org/wiki/Correlation
On the top right side there is a picture of various data sets and their correlation coefficients -- you will see in the bottom row many strong patterns of dependence which give zero correlation coefficient.
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