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More on Home Runs Per Game July 9, 2010

Posted by tomflesher in Baseball, Economics.
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In the previous post, I looked at the trend in home runs per game in the Major Leagues and suggested that the recent deviation from the increasing trend might have been due to the development of strong farm systems like the Tampa Bay Rays’. That means that if the same data analysis process is used on data in an otherwise identical league, we should see similar trends but no dropoff around 1995. As usual, for replication purposes I’m going to use Japan’s Pro Baseball leagues, the Pacific and Central Leagues. They’re ideal because, just like the American Major Leagues, one league uses the designated hitter and one does not. There are some differences – the talent pool is a bit smaller because of the lower population base that the leagues draw from, and there are only 6 teams in each league as opposed to MLB’s 14 and 16.

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

\hat{HR} = .957 - .0188 \times t + .0004 \times t^2 + .0911 \times DH

where \hat{HR} is the predicted number of home runs per game, t is a time variable starting at t=1 in 1954, 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:

Estimate Std. Error t-value p-value Signif
B0 0.2462 0.0992 2.481 0.0148 0.9852
t 0.0478 0.0062 7.64 1.63E-11 1
tsq -0.0006 0.00009 -7.463 3.82E-11 1
DH 0.0052 0.0359 0.144 0.8855 0.1145

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 R^2 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 (Residual = \hat{HR} - HR. 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

\frac{(S_C -(S_1+S_2))/(k)}{(S_1+S_2)/(N_1+N_2-2k)} \sim\ F_{k,N_1+N_2-2k}

where S_C = 6.3666 is the combined sum of squared residuals, S_1 = 1.2074 and S_2 = 2.2983 are the individual (i.e. MLB and Japan) sum of squared residuals, k=4 is the number of parameters, and N_1 = 100 and N_2 = 100 are the number of observations in each group.

\frac{(6.3666 -(1.2074 + 2.2983))/(4)}{(100+100)/(100+100-2\times 4)} \sim\ F_{4,100+100-2 \times 4}

\frac{(6.3666 -(3.5057))/(4)}{(200)/(192)} \sim\ F_{4,192}

\frac{2.8609/4}{1.0417)} \sim\ F_{4,192}

\frac{.7152}{1.0417)} \sim\ F_{4,192}

.6866 \sim\ F_{4,192}

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.

In the previous post, I looked at the trend in home runs per game in the Major Leagues and suggested that the recent deviation from the increasing trend might have been due to the development of strong farm systems like the Tampa Bay Rays’. That means that if the same data analysis process is used on data in an otherwise identical league, we should see similar trends but no dropoff around 1995. As usual, for replication purposes I’m going to use Japan’s Pro Baseball leagues, the Pacific and Central Leagues. They’re ideal because, just like the American Major Leagues, one league uses the designated hitter and one does not. There are some differences – the talent pool is a bit smaller because of the lower population base that the leagues draw from, and there are only 6 teams in each league as opposed to MLB’s 14 and 16.

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

\hat{HR} = .957 - .0188 \times t + .0004 \times t^2 + .0911 \times DH

where \hat{HR} is the predicted number of home runs per game, t is a time variable starting at t=1 in 1954, 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:

Estimate Std. Error t-value p-value Signif
B0 0.2462 0.0992 2.481 0.0148 0.9852
t 0.0478 0.0062 7.64 1.63E-11 1
tsq -0.0006 0.00009 -7.463 3.82E-11 1
DH 0.0052 0.0359 0.144 0.8855 0.1145

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 R^2 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 (Residual = \hat{HR} - HR. 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

\frac{(S_C -(S_1+S_2))/(k)}{(S_1+S_2)/(N_1+N_2-2k)} ~ F

Back when it was hard to hit 55… July 8, 2010

Posted by tomflesher in Baseball, Economics.
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Last night was one of those classic Keith Hernandez moments where he started talking and then stopped abruptly, which I always like to assume is because the guys in the truck are telling him to shut the hell up. He was talking about Willie Mays for some reason, and said that Mays hit 55 home runs “back when it was hard to hit 55.” Keith coyly said that, while it was easy for a while, it was “getting hard again,” at which point he abruptly stopped talking.

Keith’s unusual candor about drug use and Mays’ career best of 52 home runs aside, this pinged my “Stuff Keith Hernandez Says” meter. After accounting for any time trend and other factors that might explain home run hitting, is there an upward trend? If so, is there a pattern to the remaining home runs?

The first step is to examine the data to see if there appears to be any trend. Just looking at it, there appears to be a messy U shape with a minimum around t=20, which indicates a quadratic trend. That means I want to include a term for time and a term for time squared.

Using the per-game averages for home runs from 1955 to 2009, I detrended the data using t=1 in 1955. I also had to correct for the effect of the designated hitter. That gives us an equation of the form

\hat{HR} = \hat{\beta_{0}} + \hat{\beta_{1}}t + \hat{\beta_{2}} t^{2} + \hat{\beta_{3}} DH

The results:

Estimate Std. Error t-value p-value Signif
B0 0.957 0.0328 29.189 0.0001 0.9999
t -0.0188 0.0028 -6.738 0.0001 0.9999
tsq 0.0004 0.00005 8.599 0.0001 0.9999
DH 0.0911 0.0246 3.706 0.0003 0.9997

We can see that there’s an upward quadratic trend in predicted home runs that together with the DH rule account for about 56% of the variation in the number of home runs per game in a season (R^2 = .5618). The Breusch-Pagan test has a p-value of .1610, indicating a possibility of mild homoskedasticity but nothing we should get concerned about.

Then, I needed to look at the difference between the predicted number of home runs per game and the actual number of home runs per game, which is accessible by subtracting

Residual = HR - \hat{HR}

This represents the “abnormal” number of home runs per year. The question then becomes, “Is there a pattern to the number of abnormal home runs?”  There are two ways to answer this. The first way is to look at the abnormal home runs. Up until about t=40 (the mid-1990s), the abnormal home runs are pretty much scattershot above and below 0. However, at t=40, the residual jumps up for both leagues and then begins a downward trend. It’s not clear what the cause of this is, but the knee-jerk reaction is that there might be a drug use effect. On the other hand, there are a couple of other explanations.

The most obvious is a boring old expansion effect. In 1993, the National League added two teams (the Marlins and the Rockies), and in 1998 each league added a team (the AL’s Rays and the NL’s Diamondbacks). Talent pool dilution has shown up in our discussion of hit batsmen, and I believe that it can be a real effect. It would be mitigated over time, however, by the establishment and development of farm systems, in particular strong systems like the one that’s producing good, cheap talent for the Rays.

How often should Youk take his base? June 30, 2010

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Kevin Youkilis is sometimes called “The Greek God of Walks.” I prefer to think of him as “The Greek God of Take Your Base,” since he seems to get hit by pitches at an alarming rate. In fact, this year, he’s been hit 7 times in 313 plate appearances. (Rickie Weeks, however, is leading the pack with 13 in 362 plate appearances. We’ll look at him, too.) There are three explanations for this:

  1. There’s something about Youk’s batting or his hitting stance that causes him to be hit. This is my preferred explanation. Youkilis has an unusual batting grip that thrusts his lead elbow over the plate, and as he swings, he lunges forward, which exposes him to being plunked more often.
  2. Youkilis is such a hitting machine that the gets hit often in order to keep him from swinging for the fences. This doesn’t hold water, to me. A pitcher could just as easily put him on base safely with an intentional walk, so unless there’s some other incentive to hit him, there’s no reason to risk ejection by throwing at Youkilis. This leads directly to…
  3. Youk is a jerk. This is pretty self-explanatory, and is probably a factor.

First of all, we need to figure out whether it’s likely that Kevin is being hit by chance. To figure that out, we need to make some assumptions about hit batsmen and evaluate them using the binomial distribution. I’m also excited to point out that Youk has been overtaken as the Greek God of Take Your Base by someone new: Brett Carroll. (more…)

What is the effect of the Designated Hitter? May 30, 2010

Posted by tomflesher in Baseball.
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Intuitively, the designated hitter rule seems like it should increase scoring. By getting on base more often than the pitcher would have, the designated hitter helps produce runs by hitting, by being on base so that other players can drive him in, and by not accumulating outs by bunting or striking out as often as the pitcher does. However, there should be a corresponding effect from having pitchers left in the game longer: a better pitcher who remains in the game might get more outs than a reliever who came in simply because the manager pinch-hit for the starting pitcher because he needed offense.

Behind the cut, I’ll explain the testing I did to determine whether the effect of a DH is positive (hint: it is) and look at how big an effect is actually there.

(more…)

Cy Young gives me a headache. January 15, 2010

Posted by tomflesher in Baseball, Economics.
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As usual, I’ve started my yearly struggle against a Cy Young predictor. Bill James and Rob Neyer’s predictor (which I’ve preserved for posterity here) did a pretty poor job this year, having predicted the wrong winner in both leagues and even getting the order very wrong compared to the actual results. Inside, I’d like to share some of my pain, since I can’t seem to do much better.

(more…)