How can a non-linear relationship among two variables influence the correlation coefficient?

Question

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?

Answer 1

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.

Answer 2

I'm assuming you are talking about the Pearson Product Moment correlation coefficient, which assumes a linear relationship between two variables.



If the relationship is best represented by a curve, then the strength of the relationship will be underestimated. Often times, nonlinear relationships are plotted on log or log-log scales. This will often allow the relationship to be plotted as a straight line.



You can either transform the values with a function or you can look at the values with a specific range that is known to be linear.

Answer 3

Pearson's r is a measure of the *linear* relationship between two variables. In terms of a scatterplot, it measures the extent to which the points fall on a single straight line. If the relationship is non-linear, then the points don't fall on a single straight line, and so Pearson's r is low. Even if the relationship is a *strong* non-linear relationship, Pearson's r says, "Hey, these points are not falling on a single straight line, so the relationship is weak." In short, Pearson's r is closer to zero.