  Priorities in multi-criteria decision-making (MCDM) convey the relevance
preference of one criterion over another, which is usually reflected by
imposing the non-negativity and unit-sum constraints. The processing of such
priorities is different than other unconstrained data, but this point is often
neglected by researchers, which results in fallacious statistical analysis.
This article studies three prevalent fallacies in group MCDM along with
solutions based on compositional data analysis to avoid misusing statistical
operations. First, we use a compositional approach to aggregate the priorities
of a group of DMs and show that the outcome of the compositional analysis is
identical to the normalized geometric mean, meaning that the arithmetic mean
should be avoided. Furthermore, a new aggregation method is developed, which is
a robust surrogate for the geometric mean. We also discuss the errors in
computing measures of dispersion, including standard deviation and distance
functions. Discussing the fallacies in computing the standard deviation, we
provide a probabilistic criteria ranking by developing proper Bayesian tests,
where we calculate the extent to which a criterion is more important than
another. Finally, we explain the errors in computing the distance between
priorities, and a clustering algorithm is specially tailored based on proper
distance metrics.
