Inferences about correlations when there is heteroscedasticity

Let (Yb Xi), i = 1, ..., n, be a random sample from some bivariate distribution, and let rho be the (Pearson) population correlation between X and Y. The usual Student's t test of H0: rho = 0 is valid when X and Y are independent, so in particular the conditional variance of Y, given X, does not vary with X. But when the conditional variance does vary with X, Student's t uses an incorrect estimate of the standard error. In effect, when rejecting H0, this might be due to rho not equal to 0, but perhaps the main reason for rejecting is that there is heteroscedasticity. This note compares two heteroscedastic methods for testing H0 and finds that in terms of Type I errors, the nested bootstrap performed best in simulations when using rho. When using one of two robust analogues of rho (Spearman's rho and the percentage bend correlation), little or no advantage was found, in terms of Type I error probabilities, when using a nested bootstrap versus the basic percentile method. As for power, generally an adjusted percentile bootstrap, used in conjunction with r, performed better than the nested bootstrap, even in situations where, for the null case, the estimated probability of a Type I error was lower when using the adjusted percentile method. As for computing a confidence interval when correlations are positive, situations are found where all methods perform in an unsatisfactory manner.