As a reminder, the MLB regression gave us a regression equation of

where is the predicted number of home runs per game, t is a time variable starting at t=1 in 1955, and DH is a binary variable that takes value 1 if the league uses the designated hitter in the season in question.

Just examining the data on home runs per game from the Japanese leagues, the trend looks significantly different.  Instead of the rough U-shape that the MLB data showed, the Japanese data looks almost M-shaped with a maximum around 1984. (Why, I’m not sure – I’m not knowledgeable enough about Japanese baseball to know what might have caused that spike.) It reaches a minimum again and then keeps rising.

After running the same regression with t=1 in 1950, I got these results:

This equation shows two things, one that surprises me and one that doesn’t. The unsurprising factor is the switching of signs for the t variables – we expected that based on the shape of the data. The surprising factor is that the designated hitter rule is insignificant. We can only be about 11% sure it’s significant. In addition, this model explains less of the variation than the MLB version – while that explained about 56% of the variation, the Japanese model has an value of .4045, meaning it explains about 40% of the variation in home runs per game.

There’s a slightly interesting pattern to the residual home runs per game (. Although it isn’t as pronounced, this data also shows a spike – but the spike is at t=55, so instead of showing up in 1995, the Japan leagues spiked around the early 2000s. Clearly the same effect is not in play, but why might the Japanese leagues see the same effect later than the MLB teams? It can’t be an expansion effect, since the Japanese leagues have stayed constant at 6 teams since their inception.

Incidentally, the Japanese league data is heteroskedastic (Breusch-Pagan test p-value .0796), so it might be better modeled using a generalized least squares formula, but doing so would have skewed the results of the replication.

In order to show that the parameters really are different, the appropriate test is Chow’s test for structural change. To clean it up, I’m using only the data from 1960 on. (It’s quick and dirty, but it’ll do the job.) Chow’s test takes

where is the combined sum of squared residuals, and are the individual (i.e. MLB and Japan) sum of squared residuals, is the number of parameters, and and are the number of observations in each group.

The critical value for 90% significance at 4 and 192 degrees of freedom would be 1.974 according to Texas A&M’s F calculator. That means we don’t have enough evidence that the parameters are different to treat them differently. This is probably an artifact of the small amount of data we have.

Two questions come to mind immediately:

1. How likely is a season with 4 no-hitters?
2. Does this mean we’re on pace for a lot more?

The second question is pretty easy to dispense with. Taking a look at the list of all no-hitters (which interestingly enough includes several losses), it’s hard to predict a pattern. No-hitters aren’t uniformly distributed over time, so saying that we’ve had 4 no-hitters in x games doesn’t tell us anything meaningful about a pace.

The first is a bit more interesting. I’m interested in the frequency of no-hitters, so I’m going to take a look at the list of frequencies here and take a page from Martin over at BayesBall in using the Poisson distribution to figure out whether this is something we can expect.

The Poisson distribution takes the form

where is the expected number of occurrences and we want to know how likely it would be to have occurrences based on that.

Using Martin’s numbers – 201506 opportunities for no-hitters and an average of 4112 games per season from 1961 to 2009 – I looked at the number of no-hitters since 1961 (120) and determined that an average season should return about 2.44876 no-hitters. That means

and

Above is the distribution. p is the probability of exactly n no-hitters being thrown in a single season of 4112 games; cdf is the cumulative probability, or the probability of n or fewer no-hitters; p49 is the predicted number of seasons out of 49 (1961-2009) that we would expect to have n no-hitters; obs is the observed number of seasons with n no-hitters; cp49 is the predicted number of seasons with n or fewer no-hitters; and cobs is the observed number of seasons with n or fewer no-hitters.

It’s clear that 4 or even 5 no-hitters is a perfectly reasonable number to expect.

Wins Above Expectation (WAE) is then the difference between Wins and Expected Wins, which are simply the Pythagorean Expectation multiplied by Games played. It’s a useful measure because it can be interpreted as wins that are due to efficiency (in economic terms) or, more simply, play that’s some combination of smart, clutch, and non-wasteful. It rewards winning close games and penalizes teams that win lots of laughers but lose close games, since the big wins predict more games will be won when all those runs are spent winning only one game.

Using Baseball-Reference.com, I crunched the numbers for AL teams up to June 15. As usual, the Los Angeles Angels of Anaheim lead the league in WAE with 3.68, with Detroit’s 2.39 a close second,  but the Tampa Bay Rays are a surprising last with -1.96 WAE. Obviously, this early in the season it’s too soon to conclude anything based on this, but the complete data is behind the cut.

No, of course not. Find out why behind the cut.

It’s a tempting story – that a secret, untapped pitching ability lurks inside players known more for their bats, and the idea that someone playing in the outfield could be the world’s greatest reliever if only they’d give him the chance. Scott Spiezio pitched once for the St. Louis Cardinals and has a facebook group dedicated to his pitching prowess.

The problem is that a position player pitching has two advantages, one much stronger than the other. The weak advantage is that there’s no chance to scout a position player before he pitches, with the possible exception of a known pitching threat like Wade Boggs. Even then, it’s difficult to know what the player has been holding back. The strong advantage is that, well, position players aren’t very good pitchers.

How does that work? Intuitively, a major-league batter is used to a pitcher performing at a high level. Once he’s warmed up, he has a set of skills maximized for hitting a 90-plus-mile-per-hour ball thrown at him. Timing has become second nature. This is why changeups are so effective – a player isn’t expecting a ball being hurled slowly at him, and so he swings as if a fastball were coming. Being thrown nonstop changeups (which is effectively what a position player will do, given that he doesn’t regularly practice pitching) is jarring and will throw off the batter’s concentration. To a lesser extent, this is seen when a left-handed pitcher relieves a right-handed pitcher.

Does that make sense? Let’s make the assumption that a player at the major league level will be used where his manager assumes he will make the strongest contribution to the team, as constrained by the rest of the talent available. Thus, while Swish would make a perfectly cromulent designated hitter on some teams, and plays enough first base to be a starter for some clubs, his best fit for the Yankees is playing the corners in the outfield. It would be economically inefficient and thus irrational for Joe Girardi to start him at, say, shortstop, because he has a better shortstop (Cody Ransom).

So, almost entirely because Swisher is an outfielder, we can assume that he cannot pitch at the major league level. Unpacking this, he lacks some quality – consistency, endurance, speed, control, something like that – and therefore cannot be a consistently good pitcher. However, the payoff of using a player who can’t pitch consistently shrinks in emergency relief situations, since the cost of exhausting a real reliever outweighs the expected cost of using a non-pitcher to pitch (in most cases, giving up a few runs). However, as an outfielder, we know he has the arm strength to throw the ball. (This also explains why catchers, who have to have strong throwing arms and throwing reflexes, are often used as emergency relievers.) So, given that it makes economic sense to use Swisher instead of using, say, Mariano Rivera simply to fulfill the idea that only a relief pitcher should be used as a relief pitcher, it also makes sense that Swisher will perform somewhat well. He lacks only some of the qualities of a good pitcher, not all of them. Once you factor in the lack of preparation that the Rays had to face a jarring series of changeups, and the difficulty of making that mental adjustment, it is perfectly sensible to expect Swisher to have a good outing.

So why doesn’t Swish pitch every night? For the simple reason that if players expect to face a slow-hurling outfielder every night, there would be practice time dedicated to hitting 75-mile-per-hour fastballs. It would then become inefficient to use Swisher, when a harder-throwing real reliever could get outs with greater predictability.

Sorry, Swish. Great outing, but we won’t be using you again for a while.

Nick Swisher can pitch
Struck out Kapler with a change
Now stay in the field.

According to the MLB Players Association FAQ, about 12% of filed arbitration requests actually went to arbitration since 1990. From 1974 to 2006 (inclusive), the overall record was 269 arbitration decisions in favor of the club to 199 in favor of the player. Whether that represents better statistical methods by the clubs, players overvaluing their contributions systemically, an inefficiency on the part of the arbitrators, or simply more effective rent-seeking on the part of the clubs is difficult to determine.

The 88% of cases that were filed but not arbitrated since 1990 were settled – that is, either the player and the team came to an agreement or the player was otherwise disposed of (say, in a trade).

A very simple model of negotiation involving arbitration might work like this:

1. Player and agent prepare request.
2. Team prepares first offer.
3. Agent and team negotiate.
4. If the numbers are close, a compromise can be reached through negotiation with a given amount of transaction costs.
5. If not, further negotiation, with additional costs, can reach a compromise.
6. If no compromise can be reached, arbitration is filed with team offer T and player offer P; negotiations continue, with marginal transaction costs accruing. T is considered the minimum, and P represents a marginal increase. The expected salary value is T + [(P-T)/2].
7. Each side argues at arbitration, and a solution is reached which favors one side completely over the other. Thus, (P-T) is allocated entirely either to the player in salary or to the team in surplus.

Under File-and-Go arbitration, step 6 is truncated; there is no marginal cost of additional negotiation because negotiation is stopped. Otherwise, the marginal costs are similar to the standard arbitration strategy.

Edmonds points out that the two players who are in arbitration with Tampa Bay this year, Willy Aybar and Dioner Navarro, made offers that were very close to the team’s offer. My conjecture is that they understand that the team’s probability of winning arbitration is greater than the player’s, all else being equal, and have made an effort to come close to the team’s offer. Tampa Bay, meanwhile, does not seem to be extracting rent in the form of very low offers; thinking at the margin, one of the most efficient teams in the league will likely save $550,000 by making reasonable but low offers.

The arbitrators are also likely to rely on traditional statistics, rather than more recently developed sabermetric statistics. It would be interesting to see what kind of record the Red Sox and the As have in arbitration.

Navarro and Aybar are making a departure from the strategy of my second-favorite player, inept catcher Josh Paul, who employed the tactic known to the Greeks as “asking for way too much given his history.” Paul was offered salaries near the league minimum after being best known for failing to tag AJ Pierzynski in the 2005 ALCS. The play led to Chicago defeating Paul’s Angels; Chicago then went on to win the World Series. He asked, in both cases, for roughly double the league minimum and was denied both times.

Good-faith team offers
Rewarded by arbiters
Josh Paul is inept

Q.E.D.

Has Scotty suffered the greatest postseason indignity ever? Nope. Not even close. That honor belongs to Mike Mussina of the 1997 Orioles.

Using Baseball Reference’s Play Index, I took a look at pitchers with 6.0 innings or more and 0 earned runs, but whose team lost the game, and sorted by game score. I was surprised how many there were – 38 since 1903, when Cy Young pitched a seven-inning, one-run, 0-earned-run performance. They didn’t track game score at the time, but if my cocktail napkin is correct, he would have come in at 71 (owing mostly to the lack of strikeouts). In 15 of those, the starting pitcher lose the game when his defense gave up an unearned run on his watch. I’ll disqualify them, because I think the indignity is greater when you leave eligible for the win. (The highest game score in that case was John Smoltz, who score 79 points in the 1996 World Series for the Braves. He gave up 4 hits and one unearned run, striking out 10, in an 8-inning performance. Criminally unsupported, he lost the game 1-0.)

In 1997′s ALCS game 6, Mike Mussina pitched 8 scoreless innings, giving up 1 hit and 2 walks, striking out 10. In 8 innings, he managed to hold a lineup including Manny Ramirez, Jim Thome and David Justice scoreless. The game went to the 11th inning, at which point Todd Fernandez hit a two-out solo home run. Jose Mesa saved the game, 1-0. Mussina’s game score was 88, his highest postseason game score ever (although he had set the mark at 80 just 4 days prior, also in a 1-run loss). In the regular season, Moose has gone to 88 or better 13 times, with two one-hit shutouts topping the list at 98 each.

Kazmir, meanwhile, has made three postseason appearances, at game scores of 30, 46, and 72. His top regular season game score is 91, which is also the only time he’s beaten Moose’s 88. (His second-best game score is 84.) He’s still young. He’s got a lot of time to be as criminally unsupported as Mussina.

Today’s project: find Pythagorean expectations for each team, then find the difference between the actual and expected win percentages (“pythagorean difference”). Apply the pythagorean difference to the total number of games played to determine a team’s Wins Above Expectation by multiplying the total number of games by the pythagorean difference.

Practical application: none.

Discussion and numbers behind the cut.

I’m using OpenOffice.org Calc and numbers that I copypasted from ESPN.com, cited above. The formatting is a little wonky because this was just a play session for me, but here are my data in handy dandy PDF format (original spreadsheet format available upon request).

Wins Above Expectation is a measure of the economy of a particular baseball team – the ability to win more games than the number of runs scored would predict. I would therefore hypothesize it to be useful as a measure of defensive ability, though of limited use from a predictive standpoint because anything it would predict could be more elegantly but less surely predicted using other measures.

Speaking of elegance, I can’t help but notice that the pythagorean expected rank and the rank by run differential are the same in all cases. The elegance of run differential as a method of ranking teams’ performance hadn’t occurred to me before, but I’ll probably continue to use it. The use of pythagorean expectation is to predict a number of wins given a particular number of games. (The correlation between expected rank and rank by run differential is obvious after thinking it through, since pythagorean expectation is basically just a special case of differential.)

The Rays’ exceedingly economical performance isn’t surprising, since they’ve been squeaking out wins regularly all season. As a team, they rank second in MLB in the teams’ number of saves (a stand-in variable for close wins), behind only the LA Angels (who beat them in terms of WAE as well). There’s also no surprise that one of this season’s biggest punching bags, Toronto, is last in the league in negative WAE (that is, wins below expectation).  This is what happens when you lose tight games and get your wins by enormous margins. Lesson to be learned: consistent performance creates measuring statistics that closely match performance.

Edit to Add: I added an additional page of stats using the 1.81 exponent cited in the Wikipedia article. The numbers changed; the ranks didn’t.