  The interactions between microbial taxa in microbiome data has been under
great research interest in the science community. In particular, several
methods such as SPIEC-EASI, gCoda, and CD-trace have been proposed to model the
conditional dependency between microbial taxa, in order to eliminate the
detection of spurious correlations. However, all those methods are built upon
the central log-ratio (CLR) transformation, which results in a degenerate
covariance matrix and thus an undefined inverse covariance matrix as the
estimation of the underlying network. Jiang et al. (2021) and Tian et al.
(2022) proposed bias-corrected graphical lasso and compositional graphical
lasso based on the additive log-ratio (ALR) transformation, which first selects
a reference taxon and then computes the log ratios of the abundances of all the
other taxa with respect to that of the reference. One concern of the ALR
transformation would be the invariance of the estimated network with respect to
the choice of reference. In this paper, we first establish the
reference-invariance property of a subnetwork of interest based on the ALR
transformed data. Then, we propose a reference-invariant version of the
compositional graphical lasso by modifying the penalty in its objective
function, penalizing only the invariant subnetwork. We validate the
reference-invariance property of the proposed method under a variety of
simulation scenarios as well as through the application to an oceanic
microbiome data set.
