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NFL home fields used to be loud, intimidating places for visiting teams. Nowadays, you can count the stadiums on one hand that truly give the home team a distinct advantage. New Orleans and Seattle come to mind, with New England, Pittsburgh and Houston in the mix as well (even though the Giants have won road games against all of these teams, save New Orleans, since 2007).
Without an additional advantage like last licks in baseball or having the last line change in hockey, the only real in-game advantage to playing at home in the NFL is the crowd noise.
Why has this lone leg up been minimized in recent years? For starters, I think that many of the newer outdoor stadiums are built in a way that minimizes the noise level. The seats are farther away from the field and the stadiums are flatter with seats spread farther apart to improve sight lines. In addition, with ticket prices increasing and PSLs (Personal Seat License) now the norm in 14 NFL stadiums, the average rabid fan has been priced out or moved upstairs. Taking their place is a richer, more fair-weather fan that is not as invested in the game.