Inferences about correlations when there is heteroscedasticity

Let (Y-i, X-i), i = 1,..., n, be a random sample from some bivariate distri bution, and let rho be the (Pearson) population correlation between X and Y . The usual Student's t test of H-0 :rho = 0 is valid when X and Y are inde pendent, 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 rejecti ng H-0, this might be due to rho not equal 0, but perhaps the main reason f or rejecting is that there is heteroscedasticity. This note compares two he teroscedastic methods for testing H-0 and finds that in terms of Type I err ors, the nested bootstrap performed best in simulations when using rho. Whe n using one of two robust analogues of rho (Spearman's rho and the percenta ge bend correlation), little or no advantage was found, in terms of Type I error probabilities, when using a nested bootstrap versus the basic percent ile 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.