Measuring Nonlinear Relations

What do you do if a correlation is strong but clearly nonlinear (as concluded from examining scatterplots)?

Unfortunately, there is no simple answer to this question, because there is no easy-to-use equivalent of Pearson r that is capable of handling nonlinear relations. If the curve is monotonous (continuously decreasing or increasing) you could try to transform one or both of the variables to remove the curvilinearity and then recalculate the correlation.

For example, a typical transformation used in such cases is the logarithmic function which will squeeze together the values at one end of the range.

• To explore this possibility you could first try to switch the scale from linear to logarithmic in the scatterplot. For more information see the Axis: Scaling dialog.

• Another option available if the relation is monotonous is to try a nonparametric correlation (such as Spearman R. For more information, see Nonparametric Statistics, which is sensitive only to the ordinal arrangement of values. Thus, by definition, it ignores monotonous curvilinearity. However, nonparametric correlations are generally less sensitive and sometimes this method will not produce any gains.

Unfortunately, the two most precise methods are not easy to use and require a good deal of experimentation with the data. Therefore you could try these two methods:

• Try to identify the specific function that best describes the curve. A convenient facility to explore various fits is provided by the interactive Custom Function option in graphs. After a function has been found, you can test its "goodness-of-fit" to your data using options available in the Nonlinear Estimation module.

• Experiment with dividing one of the variables into a number of segments (for example, 4 or 5) of an equal width, treat this new variable as a grouping variable, and run an analysis of variance on the data (recode the variable).