*The following article, written by Keith Woolner with Rany Jazayerli, appeared
in Baseball Prospectus 2001.*

**Table of Contents**

- Long-term injury risk
- Pitch counts and injuries
- Data to be studied
- Identifying Injured Pitchers
- Defining Comparable Pitchers
- Career PAP as a Predictor of Injury
- The Workload Stress Metric
- The Injury Likelihood Equation
- Statistical Significance of Results
- Other PAP formulae
- 2000 Workload Stress leaders
- Conclusions and Futures
- Acknowledgements

In the previous article, we derived a new PAP formula

(dubbed PAP^3) that reflects the typical short-term decline in pitcher

performance following a high pitch count outing. In this article, we will

investigate whether PAP^3 has any value in predicting which pitchers are subject

to injury, and if not, whether any PAP-style metric can be derived that does

have predictive value.

Before claiming any success for any measure in predicting injury, we must

fundamentally recognize that any PAP-style metric will be positively correlated

with raw pitch counts. Pitchers with high pitch count totals will tend to have

high PAP totals. If a PAP function provides no additional insight into which

pitchers will be injured that pitch count totals alone, there is no reason to

add the added complexity of a PAP system to our sabermetric arsenal. Only if a

PAP function provides injury information above and beyond what can be learned

from aggregate pitch counts should we consider it successful.

As with the previous study, I looked at starts for all pitchers between 1988-98

for which there was pitch count data in the Baseball Workshop/Total Sports

database. The approach I used was to identify starting pitchers who suffered

major injuries during that span, and compare them to comparable pitchers who did

not suffer a major injury. Pitcher injury data was taken from Neft & Cohen’s The

Sports Encyclopedia: Baseball 2000.

In the annual season summary section of TSE:BB 2000, team rosters are presented,

and a notation is made if a player was injured for more than 30 days. For the

purposes of this study, I selected pitchers who were starting pitchers in the

year they were injured, and whose recent history indicated a pattern of starting

pitching. Generally speaking, if a pitcher was a full time or near-full time

reliever in either of the two seasons prior to the injury, he was excluded from

consideration. Pitch counts from relief appearances were not included for any

pitcher, since relief outings are generally low in total pitch counts, and the

hypothesis under consideration is that it is high pitch counts that overextend

pitchers, and lead to injury risk.

Furthermore, only certain types of injuries were considered. A two-letter code

indicated the type of injury (if known). Since pitcher overwork would most often

be associated with arm injuries, the only injury categories included were

shoulder injury, elbow injury, arm injury, and sore arm. Any injured pitcher

with one of these codes was presumed to have injured his pitching arm (the

reference does not specify which arm). Note that this categorization considers

only the most serious arm injuries, namely those which held a pitcher out of

action for a month or more. Less serious injuries, including missed turns in the

rotation, and DL stays of less than 30 days, are ignored (and in fact, these

pitchers are considered "healthy" as they did not miss 30 or more days due to

injury during the season).

Since I wanted to consider pitchers for whom we had pitch count data for most of

their career, any pitcher under consideration who accumulated more than 100

innings in the majors prior to 1988 was excluded.

Note that minor league pitch counts are not widely available at present, and

while a more thorough treatment of the impact of career usage and pitch counts

on pitcher injury susceptibility would certainly include them, I restricted the

investigation to major league pitch counts only.

Finally, several pitchers appeared on the injured list multiple times during

their career. Physiologically, prior injury makes one prone to future injury. To

account for this, only the first season a given pitcher suffered a major injury

is included in our data.

Using these criteria, a total of 73 injured pitchers were identified.

In order to identify a set of similarly worked pitchers who had not been

injured, I found matches for each injured pitcher’s age and career pitch count

total. By doing so, I would have several pitchers with similar age and usage

profiles, but who had not been injured. More specifically, for each injured

pitcher, I found all pitcher whose careers through the same age had amassed

within 10% or the injured pitcher’s career pitch count total. That is, if a

25-year-old Jason Bere had about 7800 career pitches in 1995, I matched him with

any other 25-year-old pitcher who had between 7020 and 8580 career pitches.

Of course, a further restriction was that any matching pitcher was not one of

the 73 injured pitchers, even if they were injured at a different age than the

one they were being compared for. If a single pitcher-season matched more than

one injured pitcher, the duplicate entries were removed, so that no

pitcher-season was counted more than once. A total of 569 healthy comparable

seasons were identified, for an average of 7.8 healthy comparables per injured

pitcher.

Note that the term "comparable pitcher" refers only to the aggregate

number of pitches thrown in a pitcher’s starts, not necessarily in the results.

Two 27 year old pitchers with 5000 career pitches would be considered comparable

in terms of workload, even if one had a 3.00 ERA, and the other a 5.50 ERA. They

are comparable in the total amount of work performed (pitches thrown), not the

in value of the results.

**Career PAP as a Predictor of Injury**

Our initial hypothesis is that PAP^3 has predictive power beyond raw career

pitch count totals in assessing the likelihood of injury for major league

pitchers. To test this hypothesis, I plotted career PAP^3 vs. career pitch

counts for all the pitcher-seasons in the sample, which is shown in the chart

below:

*(Click for full-size image)*

Over the course of any pitcher’s career, he will invariably pick up PAP in some

fraction of his outings. By looking at the usage patterns of many pitchers over

the years, you can ascertain the "typical" amount of PAP a major

league pitcher would accumulate given their pitch counts. Linear regression is

one technique for mathematically determining what this typical PAP level is. The

best fitting linear regression equation is plotted in the chart above as the

solid line.

If pitchers with greater than usual PAP are more likely to be injured, we would

expect more of the large dots indicating injured pitchers to lie above the trend

line in the chart above. It’s difficult to tell from visual inspection whether

this is the case or not. We can, however, analyze to the data itself to see if

this is true. Looking at the percentage of each group of pitcher that lie above

the trendline, we discover that:

- 31% of all injured pitchers had above average career PAP totals for their career pitch counts.
- 9% of all healthy pitchers had above average career PAP totals for their career pitch counts.

This suggests that high PAP pitchers are more than three times as likely to be

injured as low PAP pitchers of who’ve thrown similar numbers of pitches. We have

our first piece of evidence that PAP provides predictive information beyond what

pitch counts alone can tell us.

As a side note, the careful reader will note that there are four data points

that exceed a career-to-date PAP total of 2,000,000. These four pitcher-seasons

are all from the same pitcher, and far exceed the workload amassed by any other

pitcher. This workhorse is, of course, Randy Johnson, whose career workload

looks like a mistake in the chart. Whatever the results of our analysis of PAP

and injuries, Johnson is almost certainly an extreme outlier, a remarkable

physical specimen for whom comparison to regular major league pitchers may not

apply.

Though we now have some indication that high PAP totals are a predictor of

injury risk, the results are somewhat buried in the statistics. They key element

of the findings above is that more PAP for any given number of pitches leads to

higher risk. This leads to the concept of using PAP/NP as a measure of how

intense or stressful a pitcher’s pitches have been. I’ll refer to PAP/NP as

"Workload Stress" or simply "Stress".

I determined career-to-date Stress factors for each pitcher in our sample, with

the intention of plotting Stress versus rate of injury. However, since each

pitcher in the sample has an injury value of either 0 (healthy) or 1 (injured),

a straightforward plot of points would not be particularly revealing.

What I did instead was sort the list of all pitchers by Stress factors, and

created a moving average or "sliding window" of 50 data points at a

time. That is, I took pitchers 1-50 as one data point, pitchers 2-51 for the 2nd

data point, 3-52 as the third, and so on, such that with every step I was adding

one pitcher with a high Stress factor, and dropping the one with the lowest

Stress. I averaged the Stress factors for every pitcher in the window, and

computed the percentage of pitchers in the sample who were injured. This creates

a sample within the sample, for which we can estimate the injury rate for

pitchers with Stress factors similar to the sample’s average Stress. The results

are below:

*(Click for full-size image)*

Here we see a more compelling representation of the relationship between PAP and

injuries. There’s a clear trend between Stress and the percentage of pitchers

who get injured. There’s a relatively constant increase between 0 and 50, with a

leveling off thereafter. Over a quarter of pitchers with career Stress factors

above 40 have suffered a major injury at some point during the time of the

study, compared with less than 15% of those with career Stress factors below 20.

Interestingly, there are indications of a decline as you approach and exceed a

Stress factor of 100 (the chart is truncated at Stress=100 due to lack of sample

size above this level). However, the injury rate is still well above that of any

Stress factor less than 40. Given the small number of pitchers in the upper ends

of the chart, it could be a sample size effect. If we assume, for the sake of

argument, that this decline is not simply random fluctuation, I would speculate

that this represents a survival effect of sorts. The pitchers who can sustain

that high a workload stress are those whose managers have pushed them harder and

harder until they get a reputation as a workhorse who can consistently shoulder

130 pitch count outings. It takes awhile for both the pitcher to develop to a

point where he can be effective in the late innings (and hence won’t be pulled

for a reliever). Also, a manager may be cautious with a new arm until he’s

comfortable enough with a pitcher to "know" how far he can go. Thus,

the pitchers who end up with the highest levels of stress are the quality arms

who’ve survived the weeding out process.

**The Injury Likelihood Equation**

The shape of the line on the chart, with a steeper slope at the beginning and

leveling off as you go higher, suggests a logarithmic curve. An example of such

a curve is shown below:

*(Click for full-size image)*

The formula for the trend line shown above is (LN() is the natural log function):

Prob(Injury) = 0.06 * LN(Stress)

Or, equivalently:

Prob(Injury) = 0.06 * LN(PAP/NP)

(Technical note: This equation holds for Stress factors greater than or equal to

1. The curve is equal to zero for Stress factors below 1).

What this chart suggests is that a pitcher’s career stress factor can help

predict the likelihood of that pitcher suffering a major arm injury at some

point during his career. For example, a pitcher who’s consistently around a

Workload Stress of 30 has a 20% chance of missing a month or more due to arm

injury at some point in his career.

**Statistical Significance of Results**

Having derived these apparently impressive results, it’s only prudent to ask

whether they are statistically significant or not. One commonly used statistical

test is called a Chi-squared test. Though the details of the test will be

omitted here, for our purposes, the Chi-squared test determines the likelihood

that the results we’ve seen could result from a random split of a uniform

population, given the sample sizes. In other words, Chi-squared will check the

possibility that the high and low PAP pitchers are actually equally likely to be

injured, and the observed differences are due to chance (this is what’s called

the "null hypothesis" — that PAP has no predictive value). If the

resulting probability from the Chi-squared test is too high (traditionally

around 5%), then we can’t reject the possibility that the null hypothesis is

true, meaning that the differences could be explained by chance rather than due

to any predictive power of PAP. Conversely, a very low probability result from

the Chi-squared test increases our confidence that the results are not due to

chance, and that separating pitchers based on PAP does provide information about

their relative injury risks.

Turning first to the career PAP totals, we noted that pitchers with above

average PAP totals given their career pitch counts were far more likely to have

been injured than pitchers with below average PAP totals. Computing a

Chi-squared probability for this sample indicates that the split has only a

0.000018% chance of having occurred by chance. This easily passes the criteria

for statistical significance.

Looking then at the Workload Stress factor versus (PAP/NP), I took a more

granular approach, dividing the sample space into quintiles by PAP/NP, and

computed the injury rates in each of the five groups. I then computed the

Chi-squared probability of this split occurring by chance. The result were

comparable to our previous findings — a relationship like the one observed has

a miniscule 0.0000028% chance of happening by chance. Again, the Stress Workload

factor clears the bar for statistical significance.

As with the short-term PAP results, I examined other possible PAP formulae to

see if the relationship to injury risk was noticeably stronger. Though I do not

present the charts here, I tested classic PAP, other polynomial versions of PAP

(e.g. PAP = (NP-100)^2), and varying the baselines (100 pitches, 90 pitches, 110

pitches, etc). There was no dominant winner among the various formulae. In

general, they resulted in similar predictions as PAP^3. Perhaps isn’t that

surprising, given that unlike single starts, usage patterns tend to even out

more over the course of a career. Furthermore, even with the results we have,

predicting injury is an inexact science, and Workload Stress factors are no

guarantee for either health or injury. Therefore, any reasonable metric that

gives extra weight to high pitch count outings should yield a risk factor that

is in the same ballpark as PAP^3 (pardon the pun). Given that we have a

preferred metric for short-term impact that does acceptably for long-term injury

risk as well, we will stick to simplicity, and use a single metric for both

purposes. The PAP^3 formula will be the basis for our Pitcher Abuse Point work

going forward.

Though career Workload Stress has been shown to, we can compute Stress factors

for individual pitching seasons (or groups of seasons) to assess whether a

pitcher is "on pace" for difficulties. The list below shows the

pitchers with the highest and lowest Workload Stress rates for the 2000 season

(minimum 10 games started):

PITCHER GS PAP NP STRESS Hernandez,Livan 33 422979 3825 110.6 Johnson,Randy 35 439098 4021 109.2 Schmidt,Jason 11 101865 1203 84.7 Helling,Rick 35 313875 3791 82.8 Villone,Ron 23 150263 2246 66.9 Leiter,Al 31 229252 3478 65.9 Clemens,Roger 32 218043 3433 63.5 Hitchcock,Sterling 11 70714 1127 62.7 Wolf,Randy 32 217292 3528 61.6 Martinez,Pedro 29 190327 3165 60.1 Elarton,Scott 30 188275 3139 60.0 Appier,Kevin 31 194467 3314 58.7 Davis,Doug 13 78320 1338 58.5 Miller,Wade 16 97914 1724 56.8 Suppan,Jeff 33 181089 3488 51.9 Mussina,Mike 34 183194 3657 50.1 ... Dreifort,Darren 32 4498 3114 1.4 Karl,Scott 13 1339 1037 1.3 Yan,Esteban 20 2262 1801 1.3 Garland,Jon 13 1407 1198 1.2 Romero,J.C. 11 1009 961 1.0 Glynn,Ryan 16 1512 1456 1.0 Halladay,Roy 13 1253 1208 1.0 Rose,Brian 24 1728 1862 0.9 Blair,Willie 17 1342 1545 0.9 Guzman,Geraldo 10 737 896 0.8 Perez,Carlos 22 1531 1921 0.8 Ohka,Tomo 12 793 1096 0.7 Rupe,Ryan 18 757 1553 0.5 Bergman,Sean 14 512 1152 0.4 Gooden,Dwight 14 343 1161 0.3 Fassero,Jeff 23 152 1883 0.1 Schourek,Pete 21 126 1731 0.1 Cornelius,Reid 21 126 1828 0.1 Irabu,Hideki 11 27 853 0.0 Halama,John 30 63 2607 0.0 Arroyo,Bronson 12 8 958 0.0 Johnson,Mike 13 8 981 0.0 Stottlemyre,Todd 18 1 1496 0.0 Eiland,Eiland 10 0 667 0.0

Injuries to a key pitcher can have a devastating effect on a team’s fortunes,

not to mention that they can shorten or hinder a pitcher’s career. With

escalating salaries, proper pitcher usage is increasingly important to

maximizing a team’s investment in its personnel. As a result, pitch counts are

in prominence, managers and pitching coaches are scrutinized more closely in how

they handle a staff, and player development systems in the minors are

increasingly aware of protecting young arms.

The research presented here has shown, in essence, that not all pitches are

created equal. It is the high pitch count outings that represent the greatest

risk for both short-term ineffectiveness, and long-term potential for injury.

The PAP^3 system represents the most comprehensive attempt to date to quantify

the impact of starting pitcher usage over both time horizons, allowing us to

estimate, based on empirical evidence, the tradeoffs of having a star pitcher

throw deep into a game.

However, before placing too much weight on these discoveries, some caveats

apply. The results of this study should not be considered final because many

active pitchers are included in the study. It will be several years before a

large sample of pitch counts for entire pitcher careers becomes available, and

such a resource is necessary before we can complete the analysis has been

started here.

It’s important to note that the Workload Stress factor is not a prediction of

injury risk for a specific season, but rather a risk of injury over several

years of pitching at that level. Also, PAP^3 may underestimate the relationship

between high pitch counts and injuries. This study considered only the most

major injuries, and did not look at minor injuries, missed turns in the

rotations, or shifts from starting to relief pitching. We also proceeded

assuming that the injury effect of high pitch counts would manifest itself in

arm problems. It’s possible that there would also be effects for other kinds of

non-arm injuries (especially back and leg injuries).

The research questions are far from resolved, and there are still many facets to

the problem that have yet to be fully addressed. For example, a pitcher’s age

may be of considerable importance when assessing the risks of specific pitch

count limits, but was not included in this study. Important data is still

missing from the study, such as minor league, spring training, and post-season

pitch counts. The interactions and spacing between pitcher outings may prove to

have a significant effect — does starting on 3 days rest vs. 4 days rest

substantially affect the risk of either injury or ineffectiveness? There may yet

be better estimates of injury risk as I did not conduct an exhaustive search for

all mathematical representations, favoring the simplicity of a single measure

like PAP^3. Biomechanical experts may help identify physical characteristics

that indicate which pitchers are more or less susceptible or have greater

endurance, allowing personalized PAP formulae for individual pitchers.

There is also the possibility that the relationship between pitch counts and

injury risk is not static over time. Improved training methods, changing usage

patterns and strategies, new medical technology and techniques, new diagnostics

and screening could all impact the negative effects of high pitch counts. Pitch

count data from 1950 may not be terribly informative about the effects on modern

pitchers. Similarly, twenty years from now, an entirely different PAP formula

may need to be developed to take into account the impact of a machine that

rejuvenates muscle tissue instantly that some scientist has yet to discover.

Clearly, we have not learned all we need to know about the effects of pitcher

usage.

For now, however, we can confidently say that PAP^3 yields information about

pitcher performance and durability not answered by pitch counts alone under

current playing conditions. Long pitch count outings noticeably decrease

expected short-term performance, and high stress workloads over time increase

the chances for serious injury. Any strategic analysis of pitcher usage will

have to consider the tradeoff between winning the current game and the long-term

cost. There are clearly times when you will want to ride a workhorse hard, such

as a key playoff game (though Al Leiter will attest that there are limits even

in the World Series). Finding the right balance between winning now and winning

tomorrow remains a interesting challenge, and today we have another tool in our

arsenal to assess a team’s sustainable pitching strategy.

I’d like to thanks Dr. Lutz Mueller of Lumina Decision Systems for his advice

and consultation on the design and statistical testing methods in this research.

*Keith Woolner is an author of Baseball Prospectus. You can contact him by
clicking here.*

#### Thank you for reading

This is a free article. If you enjoyed it, consider subscribing to Baseball Prospectus. Subscriptions support ongoing public baseball research and analysis in an increasingly proprietary environment.

Subscribe now