Cashman was on ESPN Radio this afternoon, and in reference to Sabathia, he mentioned that if you look deeper into CC's numbers using more advanced stats you will see that he has been unlucky and that his numbers will normalize as the year goes on. Cashman at the very least understands FIP, so he likely understands other advanced metrics as well, which Girardi likely does as well based on the way he manages.
AA and Gibbons look at ERA, and don't give two shits about defense, that much is for sure.