Luck

What effect does increasing the luck on, say, militia or infantry have?

I believe it increases chance of a critical hit.

Yes, I believe it increases the chance of a critical hit. For example, lucky ground secondary defense increases critical hit chance from 0% to 2% for militia.

I'll give you a better answer.
Luck upgrades, give more CRIT points. To be specific, +2 CRIT points.
Each CRIT point, gives you a 1% chance of doing a critical hit.
A critical hit, does Roll + maxDmg