A new tool for measuring pitcher stress

Gerrit Cole and Francisco Liriano's innings totals this season have them headed into territory they've scarcely charted. Cole is likely to throw over 200 innings this season after only accumulating 240.1 in his first two seasons combined. Liriano will likely approach his career season high of 191 innings.

The high number of innings pitched by the Pirates' top two starters has led reporters to ask Clint Hurdle and Neal Huntington about their workloads. Hurdle consistently says the situation is monitored internally and both pitchers are "in a good place." Sunday, Huntington basically said the same thing, while adding some limited insight into how the Pirates measure fatigue.

"We don't anchor ourselves to innings pitched," Huntington said. "We recognize there is a much better metric to measure than pure innings pitched. We're not going to get in the details of how we measure [or] what we measure. You can look at a six-inning, 100-pitch outing, and look at another six-inning, 100-pitch outing, and they may be completely different, in terms of the stress levels and the intensity. So there is a huge number of variables we work in when we are looking to identify when we should begin to limit where our young pitchers are going."

Both innings pitched and pitches thrown are blunt instruments to measure pitchers' wear and tear. Neither accounts for many of the host of factors that contribute to stress on a pitcher's arm.

"There is no right or wrong answer," Huntington said. "And that's where it is more art than science and we are trying to use science to refine our art."

Pitcher Stress Index

The recent conversations at PNC Park piqued my curiosity about how to sharpen the tools that we outsiders use to measure overall pitcher stress levels and workload. Baseball Prospectus already offers one such measure, called Pitcher Abuse Points (PAP). What I'm presenting is a variation on that metric called Pitcher Stress Index (PSI). The PSI does not try to replace PAP, merely adds another metric to the conversation.

The goal is to provide a simple way to compare the workloads of pitchers by factoring in the number of pitches thrown and the context in which they were thrown.

The intuition that PSI shares with PAP is that stress on a pitcher's arm increases as he throws more pitches. What it adds to PAP is that the context in which pitches are thrown might add stress. Throwing pitches in a tense situation is assumed to add more overall stress.

What I did

Like PAP, PSI factors in pitches thrown. However, where PAP assigns  "abuse points" to every pitch over 100 in a start, and assigns 0 for starts under that threshold,  PSI looks at total pitches thrown in a season and multiples it by the average leverage a pitcher faced. Then, it adds in the number of pitches thrown over what we'd expect from league average pitcher with the same number of starts. The formula is straightforward:

Total pitches * aLI + pitches over 96.5 per start. (aLI is the average leverage index that a pitcher faced and 96.5 is the league average number of pitches per start.)

EXAMPLE:

David Price leads the league in Total Stress with 3364 PSI. Gerrit Cole is 11th with 3180 and Matt Harvey 63rd with 2342.

Name	Pitches	aLI	Pitches over average	Equation	PSI
Price	3074	1.0	290	(3074*1.0)+290	3364
Cole	2816	1.084	128	(2816*1.084)+128	3180
Harvey	2533	0.91	37	(2533*0.91)+37	2342

Each pitcher can be compared to a league average pitcher by giving him the same number of starts:

Name	Starts	Pitches	aLI	Pitches over average	Equation	PSI	PSI - LG AVG PSI for X Starts
League Average	28	2702	0.976	0	(2702*0.976)+0	2637	0
Cole	28	2816	1.084	128	(2816*1.084)+128	3180	3180 - 2637 = 543

Cole has accumulated 543 points of PSI beyond what a league-average pitcher would accumulate in the same number of starts.

Now, a look at the league leaders and Pirates in cumulative PSI and PSI above average.

Leaderboards

Here are the pitchers with highest PSI.

Name	PSI	PSI - Lg Avg
David Price	3364	633
Zack Greinke	3321	590
Cole Hamels	3307	670
Dallas Keuchel	3291	560
Chris Archer	3249	423

Here are the pitchers who have accumulated the most PSI over average relative to their number of starts -- in other words, pitchers who are adding the most PSI per start over average.

Name	PSI	PSI over league average
Cole Hamels	3307	670
David Price	3364	633
Chris Sale	3244	607
Zack Greinke	3321	590
Dallas Keuchel	3291	560
Gerrit Cole	3180	543

Not surprisingly, the leaderboard is dominated by some of baseball's best pitchers. They are throwing a lot of pitches and/or working through stressful situations.

Here are the Pirates' totals and where they rank. I included all pitchers with over 15 starts, so the rankings are out of 108 total pitchers.

Name	PSI (RK)	PSI - Lg Avg. (RK)
Gerrit Cole	3180 (10)	543 (6)
Francisco Liriano	2765 (33)	128 (41)
Jeff Locke	2351 (61)	-286 (91)
A.J. Burnett	2182 (71)	109 (45)
Charlie Morton	1616 (93)	-267 (87)

The Pirates' top three starters (Cole, Liriano and Burnett) each have accumulated more PSI per start than the league-average starter. Cole and Liriano are the only two starters to crack the top half of the league in terms of total PSI.

Cole has not only thrown a lot of pitches (four more than the average starter per start), he has had the third highest leverage of any starter, 1.084.

Locke and Morton, however, have accumulated less PSI than a league-average pitcher in the same number of starts. That is, they endure less stress on average than a league-average pitcher.

Finally, here are some pitchers of interest:

Name	PSI (RK)	PSI - Lg. Avg. (RK)
Jake Arrieta	2996 (19)	264 (20)
Max Scherzer	2910 (23)	178 (31)
Madison Bumgarner	2884 (27)	152 (36)
Matt Harvey	2342 (63)	-106 (71)

Harvey had had less stress per start than a league-average pitcher. An average starter would have accumulated a total of 2448 PSI over the same number of starts as Harvey.

Conclusion

Innings pitched is a blunt instrument for measuring pitcher workloads. The goal of PSI is to sharpen that instrument somewhat.

The more complex models I came up for measuring workloads included variables like days of rest, weather, leverage levels for each pitch, and leverage levels for each pitch thrown over 90 in a game. But the more detailed the analysis became, the more time-consuming it was to arrive at a single workload number. It was also a monumental task to come up with a league average, which is vital for comparisons. Most importantly, many of the results either weren't all that different from the fairly straightforward methods I detail above, or I had to come up with arbitrary weights for things like the impact of a day off or weather.

In sum, the PSI is designed to be quickly and easily measure pitchers' workloads without relying simply on innings pitched. While it is still a blunt tool, it does attempt to capture some of the phenomenon of "stress," which is an important variable to consider when thinking about pitchers' wear and tear.

There are many ways to improve upon this model, and any suggestions are greatly appreciated. In particular, my method of giving equal weight to each pitch over average is inadequate. It is most likely the case that each pitch thrown over a certain threshold probably adds greater and greater magnitudes of stress, but I couldn't think of a non-arbitrary way of applying progressive weight to each pitch.