Ever since HDH was coined (link), one component of success has been the clean inning theory – that relievers do better coming in the game at the start of an inning.
To analyze the clean inning theory, I tabulated relief pitchers for just the inning in which they entered the game - start of inning versus coming in during the middle of an inning.
The comparison is done on the number of batters faced and how many of those batters reached base (RB) by any means – the lower the reached-base percent, the better.
Royals clearly made an organizational decision to bring in relief pitchers at the start of an inning, increasing each of the last five years.
Similarly, the Royals cut down on bringing relievers in during the middle of an inning, decreasing each of the last four years (by-product of more start of innings).
For the clean inning theory to be valid, the reached-base percent needs to be lower for pitchers starting an inning.
For the five-year averages, the theory holds for the league and both the Royals and Tigers.
What is interesting to note, is that when the Royals do bring in a pitcher during the middle of an inning, the reached-base percent has dropped significantly over the past four years – to the point that the RB% is essentially the same as a clean inning in 2015 (29.2% to 28.7%).
Another aspect of the clean inning theory is how often the pitcher retires the side in order (i.e. 1-2-3).
The five-year league average is 32.2%, with the Royals easily exceeding the average, at 37.6% in 2015 (that is three out of every eight innings).