This is Part II of a series that examines the Texas Rangers 2009 revenue outlook in a rough version of the framework laid out by Vince Gennaro in his fantastic book Diamond Dollars. Check out the Offline Reading list for other great reads.

In Part I, Texas Rangers Win-Curve Part I: Wins vs. Attendance, I walked through a model for predicting 2009 home attendance based on the team's on-field success as measured by wins.

Part II aims to add another piece to the puzzle by determining a team's chances of making the playoffs for a given number of wins.

WHAT EVERYONE KNOWS

Two types of teams make to the playoffs: 3 division champions and 1 wild card team.

The more games a team wins, the better its chances are for making it into the playoffs by either method.

In reality, for a given number of wins, a team will either make it to the playoffs or not. There are only two outcomes: 'yes' and 'no'.

MODELING THE DATA

Because there are only two outcomes, the data can be modeled with a logistics curve. The curve is created by a generalized binomial regression. Basically, using an independent variable (wins), it determines the probability that the dependent variable (team makes the post-season) is true.

I gathered 11 years of historical data for the American League in its current alignment - since Tampa Bay's inaugural season in 1998.

I ran regressions for each division and for the American League as a whole.

THE RESULTS

One hypothesis that I was eager to test was that for teams in smaller divisions, like the 4-team AL West, the odds of winning the division (and therefore the odds of making the playoffs) are greater than for teams in a 6-team division like the NL Central.

I tested this hypothesis by comparing the curves for each of the three divisions against the American League curve. Essentially, all 4 curves are the same but shifted either to the left or to the right.

The AL West curve, surprisingly, is shifted right, meaning it is harder to make the playoffs in the AL West than in the AL as a whole. The AL Central curve is shifted left, and the AL East curve showed a right shift approximately equal to the shift in the AL West curve.

At 92 wins, an AL team has had a 66.96% chance to make the playoffs. The AL West, AL Central, and AL East have had 62.69%, 78.24%, and 61.94% chances, respectively, at the 92-win level.

Since it has been easier to make the playoffs in the 5-team AL Central than it has been in the 4-team AL West, the hypothesis does not hold up. The difference between the AL West and AL East was barely noticeable.

TEXAS RANGERS POST-SEASON PROBABILITY

The two curves that apply to the Rangers, the AL curve and the AL West curve, are fairly similar. At 80 wins, the AL curve shows a 0.35% chance, and the AL West curve shows a 0.24% chance.

In what appears like it could be a weak division in 2009, 85 wins might be enough to get the Rangers in. Historically, though, 85 wins has resulted in only a 4.70% chance on the AL curve and an even smaller 3.58% chance on the AL West curve.

If the Rangers make the improbable jump from 79 wins to 95 wins, the AL curve gives them a 90.88% chance of making the post-season, while the AL West curve gives them an 89.59% chance.

Based on the 2009 outlook, if any AL West team can get to 95 wins, it should win the division handily. One team reaching that level would have a fair amount of shock value by itself, but if two teams hit the 95-win mark, it would be absolutely stunning.

APPLYING THE POST-SEASON EFFECT

When a team makes the post-season, the fan response typically includes increases in season ticket sales, television ratings, and merchandise sales. This post-season effect has a tangible benefit on team revenue for current and future seasons.

According to Gennaro's model, a net present value (NPV) is calculated for the post-season effect. For each win, the NPV is multiplied by the post-season probability for that win total, and the resulting value is added to that point on the win-curve.

In Part III, the dual focus will be on turning attendance figures into attendance dollars and assigning a value to the post-season effect.

Several days ago, it was reported by several local media outlets that Brandon McCarthy choked down 7,000 calories a day this off-season to add about 25 pounds to his slender frame. This weight gain has a lot of fans excited about the positive impact it could have on his 2009 season.

The expectation is that McCarthy will be stronger and more durable going into 2009 than he has ever been before, but while he may be stronger, he may not necessarily be any more durable.

At 7,000 calories a day, there is virtually no chance that all 25 pounds are added muscle, and even if it is all muscle, that isn't necessarily a good thing.

Since coming to the Rangers before the 2007 season, McCarthy has battled a stress fracture in his right scapula, a strained flexor tendon in his flexor-pronator mass, and a strained flexor tendon in his right middle finger.

GROUND ZERO

McCarthy's injury trouble started with a stress fracture in his right scapula in July 2007. A stress fracture occurs when a bone "flexes" repeatedly - an action obviously not meant for bones. The injury indicates that the bone is being bent, stretched, or pulled by some unnatural movement, but it is not out of the question that natural movement can cause it as well.

Following his 2007 shoulder injury, McCarthy spent the off-season working hard to get stronger for 2008. Word had it that he was up about 15 to 20 pounds. Without a doubt, his focus was on staving off another shoulder injury.

The nature of a stress fracture indicates (but does not guarantee) that a mechanical flaw is responsible. The other two injuries are indicative of a lack of physical fitness.

When throwing a pitch, the arm and hand have to be strong enough to overcome the inertia created by the rest of the body. If the inertia is stronger than the bones and soft tissue, the unfit tissues tend to break down.

THE WEAK LINKS

Strains occur when muscles attempt to handle larger loads than what they are capable of handling.

McCarthy's body got stronger, and thanks to both his increased mass and increased strength, he was creating more intense loads for his elbow, wrist, hand, and fingers. His forearm was not ready for the increased load, and his forearm flexor tendon suffered a very serious strain.

Pitching with a compromised flexor tendon puts the ulnar collateral ligament at serious risk for strains and ruptures, and it's usually pretty painful.

After spending several months strengthening and conditioning his forearm, McCarthy finally returned to the mound. With a stronger forearm, the next injury occurred at the next weak link in the kinetic chain, the flexor tendon of his right middle finger.

DOING THE MATH

If you've followed what I've said so far, you can see how it's important for a pitcher to be strong from his toes to his finger tips.

A pitcher's arm must be strong enough and conditioned to handle the loads generated by the rest of his body, otherwise, even someone with anatomically perfect mechanics can suffer muscle strains, fatigue, tendinitis, or worse.

Richard Durrett of the Dallas Morning News wrote the following on January 20, 2008 during the Rangers pitching mini-camp:

McCarthy said he's worked hard to strengthen not only the area that has proved bothersome, but also the areas around that.

To me, this is far more encouraging than reading that he gained 25 pounds. Perhaps now, his arm is ready to take advantage of the leverage his 6' 7" frame is capable of generating. If it is, McCarthy is primed for a break out season.

It has become somewhat fashionable to blame Mark Connor for Eric Hurley's shoulder problems - a torn rotator cuff and frayed labrum. Without a full blown analysis of Hurley's arm action, though, it is also hard to say with certainty that his mechanics were responsible.

Given the nature of these injuries, though, Hurley's mechanics and hamstring are far more likely than Connor's teachings to be the cause of Hurley's shoulder injuries.

ANATOMY

Like any soft tissue in the body, rotator cuff muscles and tendons are torn gradually over time as stress creates micro-tears that build up and compound. There are exceptions, of course, but most of them involve severe external trauma like violent collisions and power lifting.

In pitching, the rotator cuff contracts most powerfully during the deceleration phase as it tries to keep the humerus from twisting and flying out of socket.  When the arm moves across the body, the head of the humerus becomes an obstacle to this contraction.  This forces the muscles to contract "around a corner" which adds more tension to the muscle than it can create on its own.

A frayed labrum is an early stage SLAP (superior labrum from anterior to posterior) lesion. Later stage SLAP lesions are commonly referred to simply as "torn labrums". The lesions are caused by the compressive force and friction created when the long head of the biceps brachii contracts and pulls directly on the glenoid labrum in an unnatural manner.

Certain arm actions, most notably transverse hyperabduction of the shoulder (scap-loading), can position the head of the humerus as an obstacle to the contraction of the biceps creating extra tension on the labrum where the long head of the biceps attaches.

Since part of the long head of the biceps merges with the labrum, SLAP lesions can sometimes be misdiagnosed as biceps tendinitis.  This is was the reported initial diagnosis for Hurley's shoulder injury on July 30, just three days after his final start of 2008.  On August 1, the Rangers reported that it was, in fact, shoulder soreness.

ATTACK OF THE HAMSTRING

Hurley was cruising along fairly well before he injured the hamstring of his left leg - his landing leg.

The hamstring of the landing leg experiences an eccentric contraction as the upper body moves forward over the waist. A negative change in the muscle's flexibility can decrease the amount of trunk flexion and/or shoulder rotation that occurs during a pitch. Since the body is less engaged in the deceleration of the arm, the shoulder handles more of the load than it would with normal hamstring flexibility.

Limited trunk flexion or shoulder rotation can cause the throwing shoulder's forward movement to stop early, even though the arm tries to continue moving toward the plate. The force of this action slings the arm across the body and moves the head of the humerus into the path of the muscular contraction as described above.

Hamstrings are notoriously slow-healing muscles, and flexibility can be compromised for a long period even after the muscle is fully functional.

Hamstring injuries will not always lead to shoulder injuries, but they represent a huge risk factor for someone already dealing with a weakened shoulder.

WAS HURLEY'S SHOULDER ALREADY WEAKENED?

The answer to this question simply has to be, "Yes."

In 234 minor league innings over the last two seasons, Hurley worked primarily with Rick Adair, Terry Clark, and Andy Hawkins - one of whom has one of the greatest mustaches in baseball.  All three coaches are extremely well regarded; none of them is Mark Connor.

Mark Connor was Hurley's primary pitching coach for about 32 innings, all in 2009. (Hurley had a 7.1-inning rehab start in Frisco near the end of that 32-inning span.) 32 innings is simply not enough to tear the healthy rotator cuff of a professional pitcher - someone whose rotator cuff should be exceptionally strong and well conditioned.

Barring severe external trauma, his shoulder must have been compromised before reaching the Majors and long before he hurt his hamstring.

SUMMARY

Hurley likely began damaging his shoulder well before his injuries became apparent. One can argue about the inevitability of a major tear, but excluding an external traumatic event, Hurley's mechanics are the most likely cause of the injuries.

When his hamstring started giving him trouble, his body compensated for that injury, effectively placing more (too much) stress on a rotator cuff that was, in all likelihood, already damaged.

Rotator cuffs simply don't tear suddenly enough to blame Connor for the injury.

Why Hurley was allowed to start that last game (and remain in it for as long as he was) is a different matter entirely.

Fans respond to winning, but different fan bases respond differently. Fan response is most easily measured by a team's revenue stream, the largest factor of which is home attendance - essentially a measurement of demand. It follows that if one can understand the relationship between wins and attendance, then one can reasonably predict revenue at different win levels.

When plotted on a wins versus revenue graph, the function that predicts these points is called a win-curve. (NOTE: The win-curve is actually disjoint. Since modern-era Major League Baseball games do not end in ties, there are no fractional wins. The line itself serves strictly as an illustration because win totals will always be integers.)

The concepts discussed here are elaborated on in far greater detail in Vince Gennaro's Diamond Dollars - a book that I highly recommend if you find that this article sparks your interest (check the Offline Reading list). Mr. Gennaro developed win-curves for all 30 teams as a part of his research into these relationships. Using 37 years of historical data, I attempted to build my own for the Texas Rangers.

This is Part I: Wins vs. Attendance.

PREPARING THE DATA

Each team's win-curve is different. The trick to building an accurate win-curve is identifying trends in the relevant market. In Mr. Gennaro's model, he used a 50-50 weighted average of the previous year's wins and the current year's wins, and he compared that to a per-game average of the current year's home attendance.

He accounted for a "new franchise halo", overall industry growth, new stadium effects, and work stoppages. Due to the small sample size, these effects were identified and generalized for Major League Baseball as a whole, rather than for individual teams.

Based on the assumption that each market behaves differently, it is unreasonable to assume these effects will be the same from market to market, but they do represent a reasonable approximation.

In my attempts to recreate Mr. Gennaro's work, I struggled to capture these effects. Without doing similar studies for the other 29 teams, I tried to find other ways to account for these other attendance factors.

Wins

As Gennaro surely did, I played with several different values for my wins variable. Each was represented as a winning percentage to help adjust for seasons of different lengths. Here is the list of my various definitions for wins:

• Gennaro's average of previous wins and current wins
• Separate variables for previous wins and current wins
• Average wins of the three most recent seasons
• Separate variables for the wins of the three most recent seasons

For each definition of wins, I tried different weights. In nearly every model, the only significant variable from the group was current wins, though the previous wins variable was significant in a few. In the final model, I chose current winning percentage as my wins variable.

"New Franchise Halo"

This effect was minimal at best with the Rangers. In their first two years, 1972 and 1973, the Rangers averaged fewer than 4,000 fans per home game. Early on, the Rangers weren't very good, but after several years in Arlington, attendance climbed to well above 15,000 fans per game. This is counter to the typical new franchise halo, where a team sees an early boom that tapers off after a few seasons.

The Metroplex area has only had one Major League Baseball franchise, so there is only one from which to develop a model, leaving too few samples to effectively quantify the "new franchise halo" for Dallas/Fort Worth.

Industry Growth

To measure industry growth, a simple counting variable was added - a value of 0 in 1972, up to 36 in 2008. Effectively, this functioned the same as a "Years in Area" variable. In the final model, this variable is not significant - p = 0.255 (approx.) - but its inclusion in the model resulted in a smaller standard error and better R-square and Adjusted R-square values than when it was excluded.

New Stadium Effect / Work Stoppages

The Rangers are a unique franchise in this aspect. In 1994, the year they opened the only new stadium in their history, the MLBPA went on strike and the World Series was canceled. The strike continued into 1995, dramatically reducing and effectively negating the new stadium effect that was experienced prior to the strike. As a result, in every regression that was run, 1994 was a positive outlier and 1995 was a negative outlier.

Until 2008, these two seasons were the only outliers in every single model tested.

In 1996, the Rangers began the winningest period in the franchise's history, and in each of the models I ran, the results suggested that this winning period was responsible for the high attendance experienced during that period rather than the new stadium. The coincidence is somewhat striking, and in actuality, it was probably a combination of the two that resulted in the high attendance averages.

I added a value for stadium age (in years) to try and capture the new stadium effect, but even after figuring in Arlington Stadium's prior existence as Turnpike Stadium (making it 7 years old in 1972), the variable failed to be significant.

Other Variables

I tried several other variables to help build a better statistical model. Though none turned out to be significant, the following variables were included at some point during testing:

• Years since playoff appearance
• Made playoffs (1 or 0)

I also tried to include TMR's Fan Cost Index, but I was only able to find data for the 18 most recent of the 37 seasons. The lack of sample data for this variable resulted in its exclusion from the tested models.

THE FINAL MODEL

When narrowing my model down to the most relevant sample data, I greatly simplified my thinking. Instead of trying to identify individual factors that affect attendance, I realized that, historically, all of this information already existed as a single variable: attendance. It is definitely not the perfect solution, but last season's attendance is the most significant indicator of attendance for the current season.

Based on the models I ran, none was able to predict the huge drop off experienced in 2008. Something that I think might be responsible is the price of gasoline and the increased reliance on gas-guzzling vehicles. Not only did families have less disposable income to spend on baseball games, but the trip to the games became more expensive. Without an effective public transportation solution as an alternative means to get to the ballpark, the mostly commuter fan-base spent their money elsewhere.

After including the previous year's attendance in my models, its significance was immediately apparent, but two other variables remained viable: current winning percentage and growth factor (the counting variable discussed above). Logically, this makes sense.

With this three-variable model, three seasons stood out as dramatic outliers: 1994 (new stadium), 1995 (strike), and 2008 (transportation cost).

It is statistically questionable to eliminate outliers, but in this case, I think it makes sense. I understand that this brings the study as a whole into question, but I'm going to run with it anyway.

In 2009, the Rangers will not be moving into a new stadium; there is no labor conflict on the horizon; and for now, the price of gasoline has returned to a reasonable level.

Because another spike in gasoline prices is possible, 2008 was not removed from the data set. 1972, 1981, and 1995 were all removed because of work stoppages, and 1994 was removed because of the new stadium. (The years that followed still used accurate previous year attendance, so the effects of these events were carried forward to future years even though the immediate effects were not a part of the model.)

The final model included data from 1973 through 2008, skipping over 1981, 1994, and 1995. The dependent variable, of course, was current season average home attendance. The independent variables were the previous season's average home attendance, the current season's winning percentage, and the growth factor described above.

CONCLUSIONS

The model says that for 2009, Texas Rangers home attendance can be estimated within 2,646 attendees per game using a chosen win level, the 2008 attendance per game (24,021), and the growth factor (37).

The graph below represents the relationship between wins and attendance for 2009.

The yellow dot marks the 2008 average home attendance, and the red dot marks last season's win total. According to the model, the Rangers should see an increase in attendance over last season for as few as 69 wins.

I think the transportation cost will be a huge factor going into the 2009 season, since the most viable method of getting to the ballpark is to drive.

RELEVANT STATISTICS NOTES

For the final model, the R-square value is 0.904, the adjusted R-square is 0.894, with a standard error of 2,646.

The independent variables have the following p-values: previous attendance average < 0.000002, winning percentage < 0.0008, and growth factor < 0.255.

As discussed above, the removal of the growth factor variable resulted in smaller R-square values and a higher standard error, so it was left in the final model.

IN PART II

In Part II, I will tackle the topic of post-season probability at different win levels. Combined with this article, it will be possible to start turning these numbers into dollars.

[Updated January 2020 to reflect evolution of thought in the 11+ years since the post was originally published.]

In my first post -- Biomechanics: Ulnar Collateral Ligament -- I tried to focus on UCL injuries and how to prevent them. In one of my conclusions, I suggested delaying internal rotation until after arm extension. This post aims to analyze that concept in greater detail.

"Delayed internal rotation" is the term I use to describe the arm action in which internal rotation does not occur until after the elbow is mostly extended. In some instances, depending on the path of the elbow in space, this delay allows the triceps brachii to actively contribute to the throw. It remains to be seen whether this is beneficial to velocity or health when compared to intertial forearm acceleration which is believed to be more common.

In either case, given sufficient external rotation, delaying internal rotation allows elbow extension to occur in the plane established by the path of the humerus -- a sign of system efficiency, two segments working together -- and in the direction of the throw -- a sign of segment efficiency.

ARM ACTION - EFFICIENT SEQUENCING

As the humerus is accelerated, it generally moves in an arc. This arc establishes a plane of motion along with momentum (and therefore intertia) of the forearm and ball.

The distal end of the humerus (near the elbow) reaches peak forward velocity shortly after the humerus is perpendicular to the line between second base and home plate. At this point, from a laid back forearm position, two actions continue the acceleration of the forearm: elbow extension and internal rotation. Sequencing matters.

When internal rotation occurs prior to elbow extension, whether intentional or not, the forearm moves from the laid back position into a more upright position, changing the plane in which elbow extension occurs.

As internal rotation pulls the forearm further out of a laid back position, the further toward the target the humerus must be for elbow extension to be efficient. While that increases segment efficiency, it decreases system efficiency as the forward velocity of the humerus is sacrificed to achieve this.

Additionally, elbow extension after internal rotation may be a factor for valgus extension overload syndrome which can lead to a number of pathologies in the elbow including ulnar collateral ligament tears.

When the elbow extends prior to internal rotation, the extension accelerates the forearm more directly toward the target and in concert with the forearm's momentum established by the path of the humerus. In this sequence, elbow extension can maximally contribute to pitch velocity and is a strong link in the kinetic chain.

As elbow extension decelerates, pronation, wrist flexion, and internal rotation work together to powerfully finish the pitch directly toward home plate.

A HALL OF FAME EXAMPLE

Take a look at Nolan Ryan's arm action in the following 4-frame image.

In the first frame, you can clearly see that external rotation has taken place with the forearm trailing the elbow.

In the second frame, Ryan has finished accelerating his elbow, and elbow extension has begun.  His forearm is still trailing his elbow in a laid back position.

In the last two frames frames, Ryan's elbow extension is decelerating and nearly complete, internal rotation has begun, rotating his forearm toward the plate. As the pitch is released, pronation occurs, and internal rotation continues through the deceleration phase.

DR. MIKE MARSHALL AND ELBOW PATH

Dr. Marshall has something to say about elbow paths that have a large lateral component. From an email he sent me:

When, after 'traditional' baseball pitchers take the baseball laterally behind their body, they drive their pitching arm back to the pitching arm side of their body, they generate forces toward the pitching arm side of their body that 'slings' their pitching forearm laterally away from their body.

This can be fairly easy to see with the naked eye by looking for a lateral component to the elbow's path. For example, if a left-handed pitcher brings his arm toward third base (or a right-handed pitcher brings his arm toward first base), he must drive his elbow laterally to the other side of his body before he can accelerate toward the plate.

Much of this article assumes a traditional delivery in which the humerus moves in an arc as the shoulders rotate. Depending on exactly how the pitcher moves, this arc can be pronounced or nearly non-existent. A pronounced arc is the result of a greater lateral component in the arm path and creates more centripetal force on the forearm.

In order to control the slinging action, the brachialis contracts eccentrically. This not only opposes passive (inertial) elbow extension -- called "forearm flyout" -- it also prevents the triceps from maximally contracting.

Dr. Marshall believes those two effects to be inefficient and injurious because he believes:

• the triceps can generate elbow extension force that exceeds the centripetal force of a traditional delivery, and
• the lack of an effectual triceps contraction prevents the brachialis from properly decelerating elbow extension.

Frankly, I'm not sure either belief stands up to scrutiny, but I'm also far less qualified than Dr. Marshall in such matters. (It's also entirely possible that I've misunderstood or misstated his beliefs. You can investigate this for yourself at his website.)

IN A FEW PARAGRAPHS

A cursory analysis of primary arm action movements suggests that arm actions are generally more efficient when internal rotation is delayed until after arm extension. This means that less energy is wasted on movement that doesn't directly contribute to pitch velocity.

My conclusion: delayed internal rotation has positive performance implications.

This information, coupled with my previous conclusions regarding UCL health, leads me to believe that there are both performance and health benefits to delayed internal rotation.