A robust and fast two-sample test of equal correlations with an application to differential co-expression

A robust and fast two-sample test for equal Pearson correlation coefficients (PCCs) is important in solving many biological problems, including, for example, analysis of differential co-expression. However, few existing methods for this test can achieve robustness against deviation from normal distributions, accuracy under small sample sizes, and computational efficiency simultaneously. Here, we propose a new method for testing differential correlation using a saddlepoint approximation of the residual bootstrap (DICOSAR). To achieve robustness, accuracy, and efficiency, DICOSAR combines the ideas underlying the pooled residual bootstrap, the signed root of a likelihood ratio statistic, and a multivariate saddlepoint approximation. Through a comprehensive simulation study and a real data analysis of gene co-expression, we demonstrate that DICOSAR is accurate and robust in controlling the type I error rate for detecting differential correlation and provides a faster alternative to the bootstrap and permutation methods. We further show that DICOSAR can also be used for testing differential correlation matrices. These results suggest that DICOSAR provides an analytical approach to facilitate rapid testing for the equality of PCCs in large-scale analysis.