Applied Regression Analysis of Correlations for Correlated Data

TL;DR

This paper introduces a novel regression method for analyzing correlations in general correlated data, utilizing a generalized z-transformation to model correlations with covariates, demonstrated through applications to educational and health data.

Contribution

It develops a new generalized z-transformation for correlation matrices and applies it to regression modeling of correlations in diverse correlated datasets.

Findings

Revealed new insights into within-class and within-school correlations in educational data.

Provided promising results in modeling malaria immune response data.

Enabled regression analysis of correlations with covariates in complex data structures.

Abstract

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.