Proposição de testes bootstrap para igualdade de matrizes de correlação

Abstract

The comparison of correlation matrices plays a fundamental role across various fields of knowledge, including medical, agricultural, animal, and social sciences, as it enables the investigation of the dependence structure among variables in different populations. Linear correlation is widely used to assess relationships between data sets; however, simple inspection of pairwise coefficients is insufficient to statistically evaluate the equivalence of two or more correlation matrices. In such cases, traditional asymptotic tests often prove to be overly liberal, especially in small samples or under high-dimensional settings. As an alternative, the bootstrap method stands out as a robust and efficient approach, with the main advantage of not requiring prior knowledge of the underlying probability distribution. This study aims to propose tests for the equality of correlation matrices in multivariate populations that are not necessarily normally distributed, while handling outliers and high dimensionality through nonparametric bootstrap methods. To this end, we reviewed the main methods available in the literature for testing the homogeneity of I population correlation matrices and proposed bootstrap versions of Jennrich’s and Kullback’s tests, as well as the MPCBT (Maximum Pearson Correlation Bootstrap Test), MSCBT (Maximum Spearman Correlation Bootstrap Test), MSEDBT (Maximum Spearman Euclidean Distance Bootstrap Test), and MPEDBT (Maximum Pearson Euclidean Distance Bootstrap Test). The performance of these tests was assessed through Monte Carlo simulations, using type I error rate and power as performance measures, under scenarios of both low and high dimensionality and in the presence of outliers. Regarding the results, MSEDBT and MPEDBT yielded the best performances, achieving “very good” and “good” type I error rate control, respectively, particularly for larger sample sizes (n = 50 and 80). When outliers were present, MSEDBT and MSCBT were the most effective in controlling the type I error rate for larger samples, regardless of dimensionality. Taken together, these findings highlight the superior performance of MSEDBT across all scenarios and configurations, supporting its recommendation as a reliable tool for testing the equality of correlation matrices.