  Partial correlations quantify linear association between two variables
adjusting for the influence of the remaining variables. They form the backbone
for graphical models and are readily obtained from the inverse of the
covariance matrix. For compositional data, the covariance structure is
specified from log ratios of variables, so unless we try to "open" the data via
a normalization, this implies changes in the definition and interpretation of
partial correlations. In the present work, we elucidate how results derived by
Aitchison (1986) lead to a natural definition of partial correlation that has a
number of advantages over current measures of association. For this, we show
that the residuals of log-ratios between a variable with a reference, when
adjusting for all remaining variables including the reference, are
reference-independent. Since the reference itself can be controlled for,
correlations between residuals are defined for the variables directly without
the necessity to recur to ratios except when specifying which variables are
partialled out. Thus, perhaps surprisingly, partial correlations do not have
the problems commonly found with measures of pairwise association on
compositional data. They are well-defined between two variables, are properly
scaled, and allow for negative association. By design, they are
subcompositionally incoherent, but they share this property with conventional
partial correlations (where results change when adjusting for the influence of
fewer variables). We discuss the equivalence with normalization-based
approaches whenever the normalizing variables are controlled for. We also
discuss the partial variances and correlations we obtain from a previously
studied data set of Roman glass cups.
