My first thought was that it's too simplistic to look at the results in terms of ranks of ERA, WHIP etc. I'm not claiming that the Pirates bullpen has been really good, but it seems to me that bullpen performance should be affected by the starting pitching. Most bullpens consist of a closer, one or two solid setup guys, then some crappy middle relievers. If a team's starting pitching consistently fails to pitch deep into games, the crappy middle relief guys pitch more innings, and the bullpen ERA goes up. Roy Halladay can consistently pitch 7 innings and turn the game over to Antonio Bastardo and Ryan Madson. But when James McDonald consistently pitches 4.2 innings, guys like Dan McCutchen, Tony Watson, and Joe Beimel have to pitch more innings. It would be great if he could go 7 then turn the game over to Veras and Hanrahan.
My hypothesis is that bullpen ERA is associated with the number of innings a bullpen pitches. Luckily, this data is available. I've gone to Fangraphs and pulled bullpen data for the 2006-2010 seasons for all MLB teams (n=150). Due to bullpen turnover from year to year, I'm going to assume that a team's bullpen is independent from one year to the next. It's not totally true, but it's not an unreasonable assumption here. Let's take a look at a simple plot between bullpen ERA and IP.
Hmm...look at that. It looks like there's absolutely no relationship between bullpen ERA and innings pitched. Let's look at the simple regression model:
This shows that innings pitched is not a significant predictor of ERA (p=0.80). Well, there you go. Maybe I'm wrong, and you can judge a bullpen independently of starting pitching. Apologies to the Smize!