Hi-

Both “<” and “<<” are mathematical symbols for comparing quantities. The “<” is the more commonly used “less than”, which has a fairly unambiguous or strict meaning: “A<B” is true if the size of A is less than that of B.

The “<<” symbol is “much less than”, which is a slightly vaguer means for comparing quantities. That is, “A<<B” denotes that A is not *just* technically less than B, but that it is *much* less than it-- different fields of math/physics might have different standards or general practices for defining how much “much” is. Typically in physics (=the right way to think about anything, as all mathematicians would agree), it typically means that A is a couple orders of magnitude smaller than B (an order of magnitude is roughly a factor of ten, so that means probably A<B/100) and usually has the practical consequence that if A and B were additively combined, their result could be approximated just by B; this is used often to simplify equations. In math, when one performs series expansions, one will typically use similar relations/argumentation to describe where the series was chopped off and hence the level of approximation.

So, in summary:

A) True: 10 < 11

B) True: 0.01 < 2500

C) False: 10 << 11

D) True: 0.01 << 2500

… where the exact case of where something transitions to being not just “<” but also “<<” something else is not exactly defined (so “C” and “D” there are a biiit fuzzily defined). Quantitatively, “<<” is typically use if there is a factor of 100 or more separating quantaties; notionally, it is used if the combination of the quantities can be well approximated (for the current calculation) as just the larger quantity, or the small one is “ignorably” small.

Sooooo, in the GUI: since the application of p-values is typically in hypothesis testing where we ask the question “is this null hypothesis rejected” and the answer of yes/no depends on significance being related to a pre-determined value, the p-values in the GUI are described with some different thresholds. The cases of p<0.01 are probably those where the cluster p-value is say, 0.005, 0.009, 0.0036452, etc., while those of p<<0.01 are more likely to be 0.0001, 0.000005, 0.00000094857, etc.

That is, in answering the question, “is the p-value of this quantity less than my threshold”, using “<” denotes “yes, it appears to be so,” while using “<<” denotes “heck yeah–it’s not even close!”

In terms of the cluster table, it makes sense that the largest clusters have the “<<0.01” designation, because they will likely be veeeeery unlikely to occur by chance (which is what a tiny p-value signifies).

–pt