A note on the interpretation of the statistical analysis of the
  $M_{\bullet}-M_{G}\sigma^2$ scaling relation
  In the context of scaling relations between Supermassive Black Holes and
host-galaxy properties, we aim to enhance the comparison between $M_{\bullet} -
M_{G}\sigma^2$ and $M_{\bullet} - \sigma$ relations from a statistical point of
view. First, it is suggested to take into account the predictive accuracy of
the scaling relation, in addition to the classical measures of goodness of fit.
Here, prediction accuracy is fairly evaluated according to a leave-one-out
cross-validation strategy. Then, we spread more light on the analysis of
residuals from the fitted scaling relation, in order to provide more useful
information on the role played by the different variables in their correlation
with the black hole mass. The findings from six samples are discussed.
