I once found a book where one historian proposed a mathematical model for estimating the "cultural influence" of the dominant civilization within an empire upon the territory of a conquered civilization. Tried to test it on the roman empire, I believe. He considered as important factors the time duration of the occupation, distance, relative "civilization level", social cohesion within the conquered people, and the fraction of the population which were immigrants coming from among the conquerors.
It was something like t (years), d (in months of travel for the average commercial or private trip), c (development level, he used 6 grades depending on use of key technologies: hunter gatherer societies, agriculture, metal working, writing, industry with fossil fuels, modern telecommunications), s (five levels for social cohesion, from very weak to very strong), ef (number of immigrants from the occupying people/number of natives).
I could dig up the book and see the formula and weights he came up with, but obviously this is a rather dubious exercise. Anyway, I do thing that, while some general formula cannot be used, those 6 factors are well chosen. More, it takes a favorable situation for an occupier on all six in order to entirely assimilate a different, conquered people: centuries of occupation, ease of access, large relative numbers, technological superiority on key "civilizational game-changing" technologies, lack of strong institutions among the occupied people which can provide a rallying point for resistance, and large numbers of agents (traders, soldiers, settlers, bureaucrats) of the new power.
Your irish example... did the irish really remain distinct, or did they kind of re-invented Ireland in the 19th century? Anyway, the technological gap wasn't very big. And they did had a string distinct institution which provided a rallying point, the catholic church - the english did succeed in keeping those portions of Ireland where this specific institution was weaker and the immigrants were concentrated.