Detecting Nonlinear Associations, Plus Comments on Testing Hypotheses About the Correlation Coefficient

Let (Y i , X i ), i = 1, . . . , n, be a random sample from some p + 1 variate distribution where X i is a vector of length p. In the social sciences, the most common strategy for detecting an association between Y and the marginal distributions is to test the hypothesis that the corresponding correlations are zero using a standard Student’s t test. There are two practical problems with this strategy. First, for reasons described in the article, there are situations where the correlation between two random variables is zero, but Student’s t test is not even asymptotically correct. In fact, the probability of rejecting can approach one as the sample size gets large, even though the hypothesis of a zero correlation is true. Of course, one can also apply standard methods based on a linear regression model and the least squares estimator, but the same practical problems arise. Second, Student’s t test can miss nonlinear associations. This latter problem is the main motivation for this article. Results of a former study suggest an approach that avoids both of the difficulties just described. Based on simulations, it is found that the Cramér-von Mises form of the test statistic is generally better than the Kolmogorov-Smirnov form. Situations arise where this method has less power than Student’s t test, but this is due in part to t test’s use of an incorrect estimate of the standard error.