Much ink has been spilled discussing Pythagorean win expectation (calculating a team's expected win-loss record based on the total numbers of runs they scored and allowed, instead of looking at specific game results). It has been hypothesized (and seems to make sense) that improving a team's bullpen could improve their record more than one might expect based on those pitcher's individual WAR totals because bullpen arms can pitch in innings likely to have a larger impact on games. I figured that it might be interesting to see if having a better quality bullpen would allow a team to outperform its expected win total. I estimated Pythagorean wins as RS**1.83 / (RS**1.83 + RA**1.83) and used a sample of all team*seasons between 2009 and 2011 (a total of 90 team*seasons). For quality of bullpen innings, I divided each team's Fangraphs bullpen runs above replacement by the team's relief innings pitched.
I anticipated one problem with my method. It is often said that teams that rely less heavily on their bullpens are able to get better quality innings. This makes sense: the fewer innings you need, the more of those innings you should be relying on your better arms. Bizarrely, however, a simple test for correlation between relief innings and relief pitcher runs above replacement per inning does not reveal this trend (p = 0.27; see figure after jump).
Since it seems like there is no interaction between these two variables, we can test the effect of quality of relief innings against deviation from Pythagorean win expectation and find that the quality of bullpen innings does not seem to affect a team's deviation from Pythagorean win expectation (p = 0.47; see figure below).
Contrary to what I expected, quantity of relief innings do not seem to be negatively correlated with quality of relief innings. I suppose this what could be driving this unexpected lack of a trend is that managers tend to rely on better bullpens more. It also seems like a possible factor is that worse teams pitch the ninth inning less frequently (since they're more likely to be behind going into the bottom of the ninth on the road), leading to fewer bullpen innings for worse teams.
I was equally surprised to find that quality of bullpen innings was not correlated with deviation from run expectancy. My expectation here is that, because our estimate of quality of bullpen innings was based on fangraphs runs above replacement, it reflected quality as a function of FIP, instead of actual runs. If we attempted to estimate quality of bullpen innings based on (for example) baseball-reference runs above replacement, we might find a different result. Another limitation of this study is that I did not break down how bullpen innings were distributed. Teams with greater variability in the quality of their relief pitchers likely better leverage bullpen innings by using their best pitchers in the tightest spots but this study included all bullpen innings as the same. On the other hand, managers do not always seem to use their bullpens optimally -- just look how long it took John Farrell to start trusting Casey Janssen last year. What do you all think about the role of a team's bullpen in allowing them to win more often than we might expect based on total runs scored and runs allowed?
Thanks to Modest Mouse's "Polar Opposites" for today's title.