This is what it says on PANTHER with regards to the test:
The expression data analysis statistics now include a Bonferroni correction for multiple testing. The Bonferroni correction is important because we are performing many statistical tests (one for each pathway, or each ontology term) at the same time. This correction multiplies the single-test P-value by the number of independent tests to obtain an expected error rate.
For pathways, we now correct the reported P-values by multiplying by the number of associated pathways with two or more genes. Some proteins participate in multiple pathways, so the tests are not completely independent of each other and the Bonferroni correction is conservative. For ontology terms, the simple Bonferroni correction becomes extremely conservative because parent (more general) and child (more specific) terms are not independent at all: any gene or protein associated with a child term is also associated with the parent (and grandparent, etc.) terms as well.
To estimate the number of independent tests for an ontology, we count the number of classes with at least two genes in the reference list that are annotated directly to that class (i.e. not indirectly via an annotation to a more specific subclass).
So I have been submitting gene lists with LFCs to PANTHER. When I apply the correction, I get no results. However, when I run panther without it, I get lots of significant results. How important is it to the results? Is it too stringent?