Inferential procedures based on the weighted Pearson correlation coefficient test statistic

In this note, we evaluated the type I error control of the commonly used t-test found in most statistical software packages for testing the hypothesis on $H_0:\rho =0$ vs. $H_1:\rho >0$ based on the sample weighted Pearson correlation coefficient. We found the type I error rate is severely inflated in general cases, even under bivariate normality. To address this issue, we derived the large sample variance of the weighted Pearson correlation. Based on this result, we proposed an asymptotic test and a set of studentized permutation tests. A comprehensive set of simulation studies with a range of sample sizes and a variety of underlying distributions were conducted. The studentized permutation test based on Fisher's Z statistic was shown to robustly control the type I error even in the small sample and non-normality settings. The method was demonstrated with an example data of country-level preterm birth rates.