In winning 87 games last season, the Texas Rangers drew an average attendance that was nearly what my model predicted for that win level -- predicted attendance: 27,958 per game; actual attendance: 27,641 per game.

For this year's model, there have been no tweaks to the methodology. I have simply added last year's data to the model. For details on my wins-attendance model, click here. It is based on the model presented by Vince Gennaro in his book Diamond Dollars: The Economics of Winning in Baseball.

Here's this year's model of Attendance versus Wins:

At 2009's level of 87 wins -- represented by the red dot -- my model predicts the Rangers to crack the 30,000 mark for average attendance at 30,593 per game. The model also predicts the Rangers to maintain last year's attendance level with as few as 73 wins -- represented by the yellow dot.

Regression Notes

The standard error is down from last year's 2,646 attendees per game to 2,602. The R-square and Adjusted R-Square values are nearly identical.

The growth factor variable is slightly more significant than last season, but still seems more significant to the calculations than its relatively low t Stat value (1.326) suggests. Removing it from the regression results in smaller R-Square values and a larger standard error.

Playoff Chances

Using a logistics regression for the past 12 seasons (since the Tampa Bay Rays franchise came into existence), I took a look at the odds of making the playoffs for a given win level. This is based on historical probability rather than a super complex mathematic system. For a more in-depth explanation of this process, click here.

Josh Hamilton predicted that the Rangers would win 96 games. Historically, 96 wins gives an American League West team a 94.54% chance of making the playoffs (94.50% across the entire American League).

Team president Nolan Ryan predicted 92 wins. Those four wins dramatically change the team's playoff chances. 92-win AL West teams can expect to make the playoffs 62.77% of the time, while a 92-win team from any AL division can expect to make it 68.44% of the time.

Various projection systems predict the Rangers to win between 81 and 87 games. This represents quite a wide range of playoff chances -- AL West: < 0.50% to 8.39%; AL overall: 0.73% to 14.02%.

After about the half-way point in a season, the results from such a logistics regression become fairly meaningless for that season. At that point, the division and wild-card races are taking firm shape, and a daily look at the standings tells a much more complete story.

[Note: When properly applied during the off-season (or at the trade deadline), though, playoff probability added can be used to more accurately estimate a player's true dollar value to an organization. This was to be explained in Part III of my Texas Rangers win-curve series, but I stopped at Part II. I may take another crack at finishing that series this year.]

For a long time, I've been frustrated by spin movement (Magnus effect) charts because they don't genuinely show how much a pitch actually moves. These charts perfectly demonstrate how the spin of the ball changes its path, but they don't show how velocity adds a vertical element to the pitch's movement.

Take this chart for example. These are the pitches thrown by Texas Rangers LHP Derek Holland during September and October of last season.

Even though they are much slower pitches, Holland's change ups are located in the exact same place on the graph as his fastballs. If his fastball and change up start with the same trajectory, the change up will always cross the plate lower than the fastball. I wanted to capture this on a chart, so I put gravity back into the equation.

Using Gameday's physics data (initial position, initial velocity, acceleration), I calculated how long each pitch was in the air. Keep in mind, though, that PITCHf/x starts at 50 from the plate and ends just in front. The mapped data covers only about 48 1/2 feet.

With the flight time for each pitch, I calculated the drop caused by [sea-level] gravity. After converting this number from feet to inches, I added the vertical spin movement. Here's how it turned out:

Success. The change ups now appear below his fastballs. The chart reflects not only gravity's effect on a pitch, but it also helps separate pitches by velocity, making identification a little bit easier.

This chart does not replace virtualizations by any stretch of the imagination, but I think it does show how different two pitches can be from each other even when spin movement alone can't show it. Taking this a step further could lead to a "hitter's decision" chart that would represent how different the pitches look at a certain time or distance from the plate.

The gravity charts are now available for all pitchers in TexasLeaguers.com's PITCHf/x Database.

[[Update: On April 24, 2010, the Spin Movement w/Gravity charts were updated to reflect gravity's effect from y = 40 to y = 1.417. This change was made based on the information that can be found at Alan Nathan's PITCHf/x site: MLB Extended Gameday Pitch Logs: A Tutorial]]

If you're a betting man, you should know that the odds are good that this won't be my last article featuring the mechanics and health of the Texas Rangers starting pitcher Brandon McCarthy.

As my favorite subject, his mechanics have spent a lot of time on my computer monitor playing forward and backward, in slow motion, and in still shots. As a result, I have a small tendency to see a little bit of McCarthy in just about every pitcher. Every once in a while I run into a pitcher whose mechanics have a lot in common with him.

Meet University of Texas at Dallas junior Marvin Prestridge.

In light of recent mechanical changes, Prestridge doesn't look much like McCarthy does these days [Edit: this may not actually be true since I haven't seen high-speed video of McCarthy's new mechanics], but when I pulled up the video I shot of McCarthy last spring, the similarities were striking. The angles aren't quite the same, so you may have to use a little imagination in places.

They don't look too similar at the top of their leg kicks, but they appear to have a similar degree of reverse rotation (turning their backs to the plate). McCarthy is more compact, and Prestridge lifts his knee much higher.

At hand-break, their mechanics are starting to run together. McCarthy sits a little lower on his back leg. Prestridge breaks his hands much closer to his body.

Before foot plant, this is the frame where their elbows stop moving upward and backward (toward 1B), and their arms begin external rotation. You can clearly see McCarthy's inverted W and that Prestridge's arm is below shoulder level with an extended elbow. Both pitchers have their arms well behind their shoulders.

I much prefer Prestridge's method of picking up the baseball to McCarthy's method from last spring. As a part of the changes he has made to his mechanics over the past 9 months or so, McCarthy's current pick-up features a full arm swing that positions his pitching arm much like Prestridge's arm.

By the time they hit foot plant, there's only one evident difference between the two: Prestridge is pulling his glove arm back toward second base. McCarthy's glove arm is essentially dead weight, while Prestridge's arm helps create additional rotational force through his shoulders.

Again, the only difference is the glove arm action and position, though it appears that Prestridge has a greater degree of trunk tilt toward 1B.

At this point, the pitchers are literally inches away from letting go of the baseball. Prestridge is able to reach a little more toward vertical, thanks to his 1B-side trunk tilt.

After release, the pitching arm continues internal rotation while the body tries to keep the arm from flying out of socket. This frame attempts to capture the moment where internal rotation stops.

What's clear in this frame is that McCarthy's arm continued to fly forward, winding up closer to his head than to his chest. Prestridge's arm, on the other hand, is still essentially at shoulder level. This is the most significant difference between the two deliveries.

With McCarthy's arm positioned like this, the head of his humerus is placed in an anatomically questionable position while his rotator cuff applies extreme compressive force at the glenohumeral joint, driving the humerus awkwardly into the scapula.

Prestridge's arm is in a more natural position at this point, and as a result, I do not view his mechanics as risky despite their on-the-surface similarity to McCarthy's old, problematic mechanics.

[Edit: For reference, here's a link to the video I shot of McCarthy at spring training in 2009.]

Back in January, I stumbled my way through a brief study of the relationship between Texas Rangers wins and attendance. The end result was the following graph. The yellow dot on the graph marks the 2008 attendance level, and the red dot marks the 2008 win level.

The Texas Rangers won 87 games in 2009, and the 2009 attendance numbers for Major League Baseball were compiled by Maury Brown in early October.

The model I prepared says that 87 wins should be worth an average attendance of 27,958. According to the data gathered and prepared for the Brown article, the average attendance of Texas Rangers home games in 2009 was 27,641. A difference of only 317 attendees per game translates to an overshot of only +1.15%.

As much as I would like to pat myself on the back for this, I have to acknowledge the extreme amount of luck involved with the startling accuracy of my prediction.

My model came with a sizable standard error attached to it: 2,646 attendees per game. You don't need to be a statistician to recognize how large that is or the uncertainty that it projects. I addressed this briefly in the comments of the original article:

The line in the graph marks the raw estimate based on the information provided by the model. At any given point on the line, the standard error says that the attendance level could be 2,646 higher or lower than the line.

With the reason for the 2008 drop off in question, it is probably unreasonable to expect that attendance will simply rebound to the 2006 or 2007 level. For this reason, I expect that actual attendance will fall somewhere below the line but within 2,646 attendees per game.

The luck of this season will definitely narrow the standard error of the 2010 model. Look for the 2010 model some time in February as the new season approaches.

If you haven't read the original article (or if you're into economics and data modeling) and you have 10-15 minutes to kill, I suggest giving it a read: Texas Rangers Win-Curve Part I: Wins vs Attendance.

Kenny Rogers (2001). Hank Blalock (2007). John Rheinecker (2008). Matt Harrison (2009). Jarrod Saltalamacchia (2009). These are the five major leaguers from the Texas Rangers who have been diagnosed with thoracic outlet syndrome in recent history.

Aaron Cook (2004). Kip Wells (2006). Jeremy Bonderman (2008). Noah Lowry (2009). These are the four major leaguers from all other teams who have been diagnosed with thoracic outlet syndrome in recent history. [Note: There may be more, but there aren't many. This is all I could find.]

Texas Rangers 5, Everyone Else 4. The Texas Rangers also had a minor leaguer diagnosed with thoracic outlet syndrome - pitcher John Hudgins (2005).

The definition of an epidemic, according to Wikipedia, is "when new cases of a certain disease, in a given human population, and during a given period, substantially exceed what is 'expected,' based on recent experience."

Recent experience tells us that roughly 10 players have been diagnosed with TOS in the past 9 years. More than half of those players belong(ed) to a very specific population: the Texas Rangers.

Thoracic outlet syndrome (TOS) is fairly common in overhead athletes like swimmers and baseball players. The overhead movement of the arm changes the orientation of the clavicle (collar bone) in such a manner that it may compress the brachial plexus (the nerve bundle the leads into the arm from the neck) and/or the subclavian artery and vein against the first rib.

The compression usually leads to numbness or pain in the affected arm, but it can also lead to blood clots like it did with Aaron Cook in 2004.

Undiagnosed TOS can have very serious health implications. In Cook's case, a clot broke away from the compression site in his shoulder and traveled to his lungs resulting in a pulmonary embolism.

Diagnosis is clearly very important when it comes to TOS. The Texas Rangers, however, have experienced quite a large number of TOS cases in recent years. Here's a brief look at a few reasons why this may be the case.

Access to expert opinion

Dr. Gregory Pearl, of Texas Vascular Associates, is a well-respected vascular surgeon who happens to live in the Metroplex. Dr. Pearl was involved with the TOS cases for Rogers, Blalock, Harrison, and Saltalamacchia - and likely Hudgins and Rheinecker as well. This relationship history and his proximity to the ballclub makes it far easier for Texas Rangers to be diagnosed with TOS.

Kenny Rogers provided the club with a first-hand example of what TOS can do to a pitcher's performance. When Rogers returned with an extra 4-5 mph on his fastball, Dr. Pearl was probably put on speed dial.

Throwing mechanics

Pitchers are a high risk group for TOS compared to position players because of the quantity and intensity of their throws but also because of the way they turn their heads toward the plate. With the previous image in mind, take a look at Matt Harrison.

When the head and neck turn away from the compression site, the brachial plexus and subclavian blood vessels are pulled into the narrowing gap between the rib and clavicle.

For low intensity throws where the head doesn't turn, TOS is less of a concern.

Training methods

Of particular note are position players Hank Blalock and Jarrod Saltamacchia, each of whom had TOS in his throwing shoulder. To discount their mechanics entirely would be foolish, but I found no reports of TOS diagnosis in any other position player. This suggests, perhaps incorrectly, that something behind the scenes has made a significant contribution.

Weight lifting can produce stress far in excess of what an intense throw can produce, but it's practically impossible to properly perform any exercise and cause thoracic outlet compression at the same time. When bad form enters the equation, though, all bets are off.

Dynamic exercises may contribute an intertial element in a manner similar to that of throwing a baseball. Even these, when performed properly, aren't likely to be significant contributors. As with weight lifting, if they are not performed correctly, they become a risk for TOS and a number of other potential problems.

If training is to blame, it's likely a series of exercises rather than a single one that results in thoracic outlet compression.

Blind, stinking luck

Not to be overlooked is random chance. It is entirely possible that the Texas Rangers have simply been unlucky. It is possible that each affected player was genetically at risk for TOS and would have been diagnosed no matter what team he was playing for. It may be nothing more than luck that has brought these players to Arlington.

So which is it?

In truth, it's most likely a combination of these factors. Given the current state of exercise science, training methods are probably the least likely to blame.

Throwing mechanics and luck combined with having a "resident" expert have likely been equally responsible for the Rangers' having lapped the rest of Major Leage baseball in thoracic outlet syndrome diagnoses.

[Historical TOS Note: David Cone and J.R. Richard, both pitchers, were also known/beleived to have suffered from thoracic outlet syndrome, but both diagnoses were well before the "Dr. Pearl era."]