  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.
