  Gene set collections are a common ground to study the enrichment of genes for
specific phenotypic traits. Gene set enrichment analysis aims to identify genes
that are over-represented in gene sets collections and might be associated with
a specific phenotypic trait. However, as this involves a massive number of
hypothesis testing, it is often questionable whether a pre-processing step to
reduce gene sets collections' sizes is helpful. Moreover, the often highly
overlapping gene sets and the consequent low interpretability of gene sets'
collections demand for a reduction of the included gene sets. Inspired by this
bioinformatics context, we propose a method to rank sets within a family of
sets based on the distribution of the singletons and their size. We obtain
sets' importance scores by computing Shapley values without incurring into the
usual exponential number of evaluations of the value function. Moreover, we
address the challenge of including a redundancy awareness in the rankings
obtained where, in our case, sets are redundant if they show prominent
intersections. We finally evaluate our approach for gene sets collections; the
rankings obtained show low redundancy and high coverage of the genes. The
unsupervised nature of the proposed ranking does not allow for an evident
increase in the number of significant gene sets for specific phenotypic traits
when reducing the size of the collections. However, we believe that the
rankings proposed are of use in bioinformatics to increase interpretability of
the gene sets collections and a step forward to include redundancy into Shapley
values computations.
