March 2024 Applied regression analysis of correlations for correlated data
Jie Hu, Yu Chen, Chenlei Leng, Cheng Yong Tang
Author Affiliations +
Ann. Appl. Stat. 18(1): 184-198 (March 2024). DOI: 10.1214/23-AOAS1785


Correlated data are ubiquitous in today’s data-driven society. While regression models for analyzing means and variances of responses of interest are relatively well developed, the development of these models for analyzing the correlations is largely confined to longitudinal data, a special form of sequentially correlated data. This paper proposes a new method for the analysis of correlations to fully exploit the use of covariates for general correlated data. In a renewed analysis of the classroom data, a highly unbalanced multilevel clustered data with within-class and within-school correlations, our method reveals informative insights on these structures not previously known. In another analysis of the malaria immune response data in Benin, a longitudinal study with time-dependent covariates where the exact times of the observations are not available, our approach again provides promising new results. At the heart of our approach is a new generalized z-transformation that converts correlation matrices, constrained to be positive definite, to vectors with unrestricted support and is order-invariant. These two properties enable us to develop regression analysis incorporating covariates for the modelling of correlations via the use of maximum likelihood.

Funding Statement

Chen’s work is supported in part by the National Natural Science Foundation of China (No. 71771203) and Natural Science Foundation of Anhui Province (No. 2208085MA05). Leng’s research is partially supported by EPSRC (EP/X009505/1). Tang acknowledges support from NIH grant R01GM140476 and NSF grant DMS-2210687.


The authors express their gratitude to the anonymous referees, Associate Editor, and Editor for their valuable and constructive comments, which greatly enhanced the quality of this paper.


Download Citation

Jie Hu. Yu Chen. Chenlei Leng. Cheng Yong Tang. "Applied regression analysis of correlations for correlated data." Ann. Appl. Stat. 18 (1) 184 - 198, March 2024.


Received: 1 September 2022; Revised: 1 May 2023; Published: March 2024
First available in Project Euclid: 31 January 2024

Digital Object Identifier: 10.1214/23-AOAS1785

Keywords: correlated data analysis , Correlation matrix , Correlogram , generalized z-transformation , regression modeling , testing correlation structures

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 1 • March 2024
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