Comparing Correlation Tests

The Pearson product-moment correlation is widely used statistic to explore the association between two variables. To test whether the population correlation is zero, traditional parametric procedure utilizing Fisher's z transformation can be applied. However, this method relies on the assumption of normality, which is often violated in real-world scenarios. Bootstrapping, a resampling technique, provides more accurate and reliable solutions when data are not ideally distributed or have small sample sizes. But the question of whether various bootstrapping testing methods possess equal efficacy and how to determine the best among them remains unanswered. More importantly, the fourth-order moment significantly impacts the distribution of correlation, but this subject is typically overlooked by most researchers. There are further inquiries to consider regarding these testing methods. However, there exists a lack of literature addressing these issues. This project aims to investigate and compare the performances of four correlation testing methods-the traditional parametric procedure using Fisher's z transformation, bivariate bootstrapping, univariate bootstrapping, and bootstrap hypothesis testing-through theoretical derivation and Monte Carlo simulations. We focus on the inference of correlation, including data generation methods, covariance matrix distributions, and deriving correlation distribution. Simulation studies were conducted by applying the four methods to datasets having either high or regular kurtosis, generated from normal or non-normal distributions. The mis-coverage rates across various scenarios were summarized and compared. Drawing insights from the simulation results, the project aims to offer conclusive observations and recommendations, aiding in the selection of the most appropriate method for specific scenarios.