The general goodness-of-fit tests for correlated data

Analyzing correlated data by goodness-of-fit type tests is a critical statistical problem in many applications. A unified framework is provided through a general family of goodness-of-fit tests (GGOF) to address this problem. The GGOF family covers many classic and newly developed tests, such as the minimal p-value test, Simes test, the GATES, one-sided Kolmogorov-Smirnov type tests, one-sided phi-divergence tests, the generalized Higher Criticism, the generalized Berk-Jones, etc. It is shown that the omnibus test that automatically adapts among GGOF statistics for given data, i.e., the GGOF-O, is still contained in the GGOF family and is computationally efficient. For analytically controlling the type I error rate of any GGOF tests, exact calculation is deduced under the Gaussian model with positive equal correlations. Based on that, the effective correlation coefficient (ECC) algorithm is proposed to address arbitrary correlations. Simulations are used to explore how signal and correlation patterns jointly influence typical GGOF tests' statistical power. The GGOF-O is shown robustly powerful across various signal and correlation patterns. As demonstrated by a study of bone mineral density, the GGOF framework has good potential for detecting novel disease genes in genetic summary data analysis. Computational tools are available in the R package SetTest on the CRAN.