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

The results:

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 (). 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

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.

Like Hernandez, Zambrano has a very different batting line as a pinch hitter than as a pitcher. In 24 plate appearances as a pinch hitter, Big Z is hitting only .087 with a .087 OBP, compared to his .243/.249 line when hitting as a pitcher. Since we see the same effect for both of these pitchers, it seems like there’s some sort of difference in hitting as a pinch hitter that causes the pitchers to be less mentally prepared. Of course, these numbers come from a very small sample.

On the other hand, Micah Owings hits .307/.331 as a pitcher, and a quite similar .250/.298 as a pinch hitter. What’s the difference? Owings has almost double Zambrano’s plate appearances as a pinch hitter with 47. That seems to show that maybe Owings’ larger sample size is what causes the similarity. How can this be tested rigorously?

As we did with Kevin Youkilis and his title of Greek God of Take Your Base, we can use the binomial distribution to see if it’s reasonable for Owings, Hernandez and Zambrano to hit so differently as pinch hitters. To figure out whether it’s reasonable or not, let’s limit our inquiry to OBP just because it’s a more inclusive measure and then assume that the batting average as a pitcher (i.e. the one with a larger sample size) is the pitcher’s “true” batting average and use that to represent the probability of getting on base. Each plate appearance is a Bernoulli trial with a binary outcome – we’ll call it a success if the player gets on base and a failure otherwise.

Under the binomial distribution, the probability of a player with OBP p getting on base k times in n plate appearances is:

with

We’ll also need the margin of error for proportions. If p = OBP as pitcher, and we assume a t-distribution with over 100 plate appearances (i.e. degrees of freedom), then the margin of error is:

so that 95% of the time we’d expect the pinch hitting OBP to lie within

with

We’ll also need the margin of error for proportions. If p = OBP as pitcher, and we assume a t-distribution with over 100 plate appearances (i.e. degrees of freedom), then the margin of error is:

so that 95% of the time we’d expect the pinch hitting OBP to lie within

Let’s start with Owings. He has an OBP of .331 as a pitcher in 151 plate appearances, so the probability of having at most 14 times on base in 47 plate appearances is .3778. In other words, about 38% of the time, we’d expect a random string of 47 plate appearances to have 14 or fewer times on base. His 95% confidence interval is .254 to .408, so his .298 OBP as a pinch hitter is certainly statistically credible.

Owings is special, though. Hernandez, for example, has 994 plate appearances as a pitcher and a .237 OBP, with only one time on base in 15 plate appearances. It’s a very small sample, but the binomial distribution predicts he would have at most one time on base only about 9.8% of the time. His confidence interval is .210 to .264, which means that it’s very unlikely that he’d end up with an OBP of .077 unless there is some relevant difference between hitting as a pitcher and hitting as a pinch hitter.

Zambrano’s interval breaks down, too. He has 601 plate appearances as a pitcher with a .249 OBP, but an anemic .087 OBP (two hits) in 24 plate appearances as a pinch hitter. We’d expect 2 or fewer hits only 4% of the time, and 95% of the time we’d expect Big Z to hit between .214 and .284.

As a result, we can make two determinations.

1. Zambrano and Hernandez are hitting considerably below expectations as pinch hitters. It’s likely, though not proven, that this is a pattern among most pitchers.
2. Micah Owings is a statistical outlier from the pattern. It’s not clear why.

Obviously, comparing across these years involves making some assumptions, some of which are justified and some of which aren’t. They boil down to assuming that there isn’t any other factor that would explain Manny’s decline in performance. For example, I’m assuming that Manny didn’t play a tougher schedule in 2010 than in 2009, which may be suspect because the Dodgers were 20-7 in Manny’s games in 2009 but 17-10 in 2010. I’m also assuming that Manny’s position in the batting order didn’t change his numbers, which may or may not be a factor. Manny invariably hit third in 2009 and has more often been hitting 4th in 2010.

However, the line is very telling. In 2009, Manny hit 6 home runs in 120 plate appearances, or about .05 home runs per PA. This year, he hit 2 in 104 PA, or a little bit under .02 home runs per PA. This is a significant drop, and since he’s being intentionally walked significantly less (6.7% of the time in 2009 versus 1.9% of the time this year), it corroborates the idea that Manny is seen as less of a threat. This could also be a result of Manny being moved to the fourth slot instead of being protected by Andre Ethier, but Manny is often backed up by James Loney this year, so I’m not sure the protection issue is realistic.

Two of those home runs became sacrifice flies, and Manny is striking out slightly less this year, but overall his OBP and slugging average have fallen precipitously.

The margin of error for proportions is , so the margin for his OBP this year is , or about , which is about . That means that we can be 95% confident that Manny’s “true” OBP is somewhere within . This interval puts the bounds at on the low end and . In fact, the difference – – is about 1.14 standard  errors. That means we can only be about 75% certain that the difference is not due to chance.

Still, as the year plays out, I’ll be very interested in seeing whether this is a temporary trend or whether Manny’s numbers take a permanent dive.

Since he sat down 27 batters consecutively, the probability of that event happening is (.672)²⁷, or .0000218; equivalently, given his current stats, a bit over 2 in every 100,000 games that Braden pitches should be perfect games.

Over the same period (2007-2010), the American League OBP has hovered between .331 (this year) and .338 (2007). .336 was the mode (2008, 2009), so I’ll use it to estimate that the chance for a perfect game facing the league average team would be (.664)²⁷, or .0000157, or equivalently about 1.5 out of every 100,000 games should be a perfect game.As you can see, it’s more likely for Braden than the average pitcher, but not by much.

Nice job, Dallas!

As a side note, the Tampa Bay Rays were the victim of BOTH perfect games. Their team OBP was .343 in 2009, with a probability not to get on base of .657, meaning that the probability of getting 27 batters seated consecutively is about 1.2 in 100,000. Since many other teams have lower team OBPs, it’s very surprising that the Rays were the victims of both games.

I’m using a dataset I culled from baseball-reference.com’s Play Index to which I added Cy Young points for each year, as well as a number of binary variables for team division wins, team wildcard appearances, and so on. It includes every player who pitched from the 2005 through 2009 seasons, all told about 3000 observations. Using R, I tried a number of linear regression models to test their veracity.

First, I tried a variation of the James/Neyer formula, CYP = ((5*IP/9)-ER) + (SO/12) + (SV*2.5) + Shutouts + ((W*6)-(L*2)) + VB. I included IP, ER, SO, SV, SHO, W, L, and VB and got this result:

Call:
lm(formula = model <- cypoints ~ IP + ER + SO + SV + SHO + W +
L + VB)

Residuals:
Min       1Q   Median       3Q      Max
-31.2641  -1.4715   0.1084   0.9949 144.4079

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.1057887  0.2341857  -0.452    0.651
IP           0.0080245  0.0136774   0.587    0.557
ER          -0.0960892  0.0184517  -5.208 2.03e-07 ***
SO           0.0483835  0.0090107   5.370 8.45e-08 ***
SV           0.0001499  0.0218261   0.007    0.995
SHO          5.5749651  0.4340868  12.843  < 2e-16 ***
W            0.5653568  0.0899062   6.288 3.64e-10 ***
L           -0.3987691  0.0901410  -4.424 1.00e-05 ***
VB          -0.0191531  0.3781868  -0.051    0.960
—
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.977 on 3213 degrees of freedom
Multiple R-squared: 0.1952,     Adjusted R-squared: 0.1932
F-statistic: 97.43 on 8 and 3213 DF,  p-value: < 2.2e-16

This isn’t promising. Over the past five years, these factors aren’t very predictive at all – the model explains only about 19% of the variation in voting; innings pitched, saves, and the victory bonus aren’t statistically significant, and the victory bonus has a negative effect. The caveat, of course, is that James and Neyer aren’t predicting actual Cy Young voting points but rather a statistical construct that shows the relative likelihood that a given pitcher will receive the Cy. I’m predicting actual Cy Young points. Still, the effects should be similar.

In fact, the model grossly overestimates the proclivity of Cy Young voters for choosing relievers. A pitcher with Saves as his primary statistic hasn’t been given the Cy since Eric Gagne in 2003. This is a double-edged sword – on the one hand, saves have apparently been historically significant for the Cy, but on the other hand, the voting appears to be trending away from them. The five-year time set I used is a compromise to get enough data without compromising the trend.

After playing with R for a little while, I ended up creating a few extra measures that seem to capture the voting a little bit better (but not much). First, to approximate the relief effect, I created a “weighted saves” statistic that multiplies SV*GF and then takes the square root. To maximize the stat for a given number of games finished, all of those games would be saves. (Every save is a game finished, by definition.) Thus, it helps show that the pitcher was relied on as a clutch player. I did the same thing for Complete Games and Shutouts – weighted shutouts is the square root of CG*SHO. Again, to maximize this, every complete game should be a shutout. It ends up being far more predictive than CG or SHO alone. Finally, to capture the added value of each marginal win and marginal strikeout and the added penalty for each marginal home run and marginal walk, I included the squares of those terms. I also tried a dummy variable for previous year winner, since Lincecum’s so-so predicted points must have been bumped up by something.

After playing with the stats with parsimony in mind, I came up with a number of models, the best of which is:

Call:
lm(formula = model <- cypoints ~ W + Wsq + HR + HRsq + K + Ksq +
BB + BBsq + weightedsv + weightedsho)

Residuals:
Min       1Q   Median       3Q      Max
-40.7374  -1.0710  -0.1198   1.1044 122.7243

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  1.995e-03  2.795e-01   0.007   0.9943
W           -1.295e+00  1.315e-01  -9.844  < 2e-16 ***
Wsq          1.260e-01  7.371e-03  17.091  < 2e-16 ***
HR           1.807e-01  7.286e-02   2.480   0.0132 *
HRsq        -1.499e-02  2.143e-03  -6.996 3.19e-12 ***
K           -8.473e-02  1.642e-02  -5.161 2.61e-07 ***
Ksq          5.972e-04  6.734e-05   8.869  < 2e-16 ***
BB           2.292e-01  3.143e-02   7.292 3.82e-13 ***
BBsq        -2.826e-03  3.041e-04  -9.295  < 2e-16 ***
weightedsv   7.411e-02  1.652e-02   4.487 7.49e-06 ***
weightedsho  2.443e+00  3.252e-01   7.513 7.43e-14 ***
—
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.245 on 3211 degrees of freedom
Multiple R-squared: 0.3367,     Adjusted R-squared: 0.3346
F-statistic:   163 on 10 and 3211 DF,  p-value: < 2.2e-16

It’s not a great predictor, explaining only about 33% of the variation in points. However, all of the regressors are statistically significant at at leas the 99% level. Some of the other models I tried are here, so you can get an idea of how significant or insignificant other stats might have been at predicting the Cy Young winner.

The long and the short of it is, there appears to be very little predictive value for the Cy Young voting with respect to common statistical measures.

My third through nth thoughts involved my standard method of creating a needlessly complex spreadsheet using data culled from ESPN.com. The results were quite surprising.

I ran the usual 1.81 version of Wins Above Expectation numbers (Holy cow, Tampa and the Angels are way above!), and in addition I ran the Beane Count and checked how it correlated to the standings.

The Beane Count is the sum of ranks-in-league for home runs, walks, home runs allowed, and walks allowed (with lower obviously being better for the last two categories). The theory, according to Rob Neyer, is that teams that walk and hit home runs, and do not allow their opponents to walk or hit home runs, will do better on the whole than teams that fail to walk and hit homers, or allow their opponents to do the same.

According to the data (stored here in handy dandy PDF format), Beane Count actually correlates very strongly with league rank. In the American League, the correlation to actual wins is .658; in the NL, .736. The correlation to Pythagorean expected league rank was even stronger: .755 in the AL and a whopping .845 in the NL. I’m prepared to eat my words on this one: Beane Count is a solid stat, particularly when measured against highly predictive stats.

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.