Runs are a function of total bases, certainly, but total bases are functions of things like hits, home runs, and walks. Basically, runs are a function of getting on base and of advancing people who are already on base. Obviously, the best measure of getting on base is On-Base Percentage, and Slugging Average (expected number of bases per at-bat) is a good measure of advancement.

OBP wraps up a lot of things – walks, hits, and hit-by-pitch appearances – and SLG corrects for the greater effects of doubles, triples, and home runs. That doesn’t account for a few other things, though, like stolen bases, sacrifice flies, and sacrifice hits. It also doesn’t reflect batter ability directly, but that’s okay – the stats we have should represent batter ability since the defensive side is trying to prevent run production. The model might look something like this, then:

This is the simplest model we can start with – each factor contributes a discrete number of runs. If we need to (and we probably will), we can add terms to capture concavity of the marginal effect of different stats, or (more likely) an interaction term for SLG and, say, SB, so that a stolen base is worth more on a team where you’re more likely to be brought home by a batter because he’s more likely to give you extra bases. As it is, however, we can test this model with linear regression. The details of it are behind the cut.

I’m using a dataset (available on request) of American League data pulled from Baseball-Reference.com’s Leagues page.  I’m using the AL only because I don’t want to correct for the designated hitter’s differential runs.

The first thing I need to do is decide whether to add a trend correction.

I don’t have to account for a time trend, so I’m just going to use the team-level data. Using linear regression, I fitted the model above and got the following output:

Multiple R-squared: 0.9164,     Adjusted R-squared: 0.9132

It looks like OBP and SLG are in fact highly significant, with each sac fly corresponding to about two-thirds of a run scored, a sac bunt corresponding to about .28 runs scored, and a stolen base actually having a negative effect (but it’s only significant at about the 23% level, so we can’t be sure it’s actually different from zero). This model explains about 91% of the variation in run scoring, which is reasonable since it ignores pitching and defense entirely.

This could be tightened up a bit, but as it stands it gives us a reasonable idea of how runs are produced.

That pinged my “Stuff Keith Hernandez says” meter, and so I decided to crunch some numbers and see if that’s true. I interpreted Keith’s statement as implying that the number of designated hitters should be increasing, since managers are moving away from an everyday DH and toward spreading the DH assignments around a bit more. The crunching also needs to account for interleague play, which should obviously increase the number of DHes. So, after controlling for interleague play, does DH use show an increasing trend with time?

To set up the regression, I modified an existing data set I had to include a variable for the number of people with at least one at-bat as a designated hitter (culled from baseball-reference.com/play-index). B-R.com didn’t have a listing for 1973, so I noted that 1974 had 106 DHs and 1975 had 107 and made an educated guess (that would be consistent with Keith’s statement) that 1973 had 105. Then, I added a binary variable Inter which took value 1 if there was interleague play that year and value 0 otherwise. Finally, I created time variables DHt (starts at 1 in 1973 and increases with each year), Intert (starts at 1 in 1997 and increases with each year), and squares of both of the time variables. My dependent variable is the number of players with at least one at-bat as a designated hitter (DHes) divided by the number of teams playing with the DH rule (DHTms). Finally, armed with this dataset, I pushed the numbers through R and came out with this result:

Some caveats are in order. First of all, according to a Breusch-Pagan test, the error terms are absolutely heteroskedastic (that is, they’re correlated to something that I haven’t accounted for in my data). Second, I have an R[sup]2[/sup] of .9884, meaning that this data explains almost 99% of the variance in the number of designated hitters used. That’s a lot of explanatory value, and usually means you’re doing a regression that looks like “Right shoes = B0 + B1 Price + B2 Left shoes + error term” – that is, one where you’re missing some obvious highly correlated term. I’m not sure what that term might be, though. Also, there isn’t really enough data from interleague play to run robust time series analysis on it.

However, we can make some statements. First of all, interleague play adds about 43 designated hitters, or about 2.68 per National League team although that probably varies by the number of series played. Second, DHes per team decreased until they hit a minimum in 1989 and then began increasing again in terms of time series. What do you know? Keith might have been right after all.

Is hatred-based investing rational? Making the usual assumptions (people are rational utility maximizers, etc), and assuming that you gain some utility from seeing a company you hate losing money, and that you lose a commensurate amount of utility from seeing that company make money, then it’s absolutely rational under certain circumstances. Mainly, it would serve as a hedge strategy against emotional distress. In Adams’ example, he’s talking about BP and their recent oil spill. Owning BP provides a hedge against the disutility of watching BP potentially recover and begin to profit again – you get paid an amount that should offset some of your lost utility. Conversely, if you lose money, at least your money loss is offset by a gain in utility.

Obviously, it’s not something to do with all of your money. The optimal hedge ratio will also vary consumer-by-consumer.

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.

The Conventional Wisdom™ is that the DH should increase hit batsman. The argument is that pitchers don’t bear the costs of hitting a batter with a pitch because they don’t bat, so they’ll be less careful to avoid hitting a batter or more likely to plunk a batter out of malice. Do the numbers bear that out?

To attack this question, I’m using the same dataset I used in the earlier post – the per-game average data for each league since 1954, with an added dummy variable for whether the DH rule was in effect that year, and with time normalized to begin with 1955 and an added quadratic term. (I pulled it from Baseball-Reference.com.) I started using the same variables as the previous post:

That is, check for a trend and then after controlling for that check to see if there is a significant effect based on the DH rule. However, it occurred to me that there might be an experience effect – if more players are showing up in the league, you might get matching effects for pitchers with no control hitting batters and for batter with no experience crowding the plate because they haven’t been trained not to. I added a term for the number of batters in the league to control for that:

The regression output was:

Residual standard error: 0.03256 on 107 degrees of freedom
Multiple R-squared: 0.8038, Adjusted R-squared: 0.7965
F-statistic: 109.6 on 4 and 107 DF, p-value: < 2.2e-16

The Batters term (and the other three terms) are all statistically significant at the 99% level. These variables explain around 80% of the variation in HBP per game, based on the R-squared statistic. The Breusch-Pagan test, with a null hypothesis of no heteroskedasticity, has a p-value of .2 – not enough to reject that null hypothesis, so ordinary least squares are appropriate here.

After controlling for time and the effect of talent pool dilution, the designated hitter rule represents about .08 hit batsmen per game, or roughly one hit batsman every 12.5 games, which translates to about 13 additional hit batsmen over the course of a team’s season. (Of course, that effect could be almost entirely explained by Kevin Youkilis stubbornly refusing to back off home plate.)

Making the standard statistical assumptions, the margin of error using proportions is . Three of the proportions were .46, meaning that the margin of error would be which simplifies to . Using 12 degrees of freedom, a t-table shows that the critical value for 95% confidence  is 2.18. Thus, the binomial confidence interval method, tells us we can be 95% sure that the true value of the proportion lies within the range .46 ± 2.18*.1382 = .46 ± .30 = .16 … .76. Clearly, this range is far too large to reject the conclusion that the proportion is significantly different from .5.

For the simple measure of more runs, the proportion was .31, meaning that the margin of error is or . The 95% confidence interval around .31 is .31 ± 2.18*.1283 = .31 ± .2797 = .03 … .59. Again, .5 is included in this range.

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.

I haven’t had time to run the numbers, but I’m a bit dissatisfied with the Misery Index in this case. The most obvious issue is that while inflation is a bad thing, so is deflation; however, under the Index, a high deflation rate is seen to mitigate high unemployment. The second is that steady, targeted inflation is a sign that the economy is growing smoothly and under control.

Again, without crunching the numbers, I can’t say anything specific, but it seems to me that a formula with nicer properties might measure either the absolute rate of change from one period to the next (capturing volatility fairly cleanly) or, for the less mathematically inclined, the absolute value of the change. The problem of measuring a rate of change is that you’d need to correct for unemployment as well; measuring rates of change also leaves you sensitive to different lengths of time being measured, whereas the misery index as it stands can be seen as a snapshot.

So, a compromise: set a benchmark – perhaps 3% for inflation and 5% for unemployment, since those are numbers that are bandied about as “targets.” Snapshot the measure by measuring the absolute value of the rate minus the benchmark figure.

Misery Index
Accurate, but hamfisted
Plausible? Who knows?

Barry Bonds is, without a doubt, one of the most controversial figures in baseball. He’s currently trying, again, what he tried last year – shopping himself around for the league’s minimum salary. (Thanks to the Sports Law Blog for the link.) Inside, I’d like to briefly discuss collusion and look at the incentives involved with this situation.

Collusion in the economic sense involves an agreement to act inefficiently. To give an oversimplified stock example, suppose a competitive market for chili peppers. That is, there are many people growing chili peppers and many people who want to buy chili peppers. Each bushel of peppers costs about $1 to produce and process from seed to sale. The market is big enough that no one producer or consumer can affect the overall price of the peppers. If one person raises his prices, the others can easily undercut him, and if he wants to sell his peppers he’ll have to meet them. As a result, the sale price approaches the (marginal) cost of production ($1).

Now suppose that only one person grows chilies. They still cost $1 to produce, but the chili monopolist can maximize his profits however he chooses. (The standard assumption is that he’ll set the price equal to marginal revenue, not marginal cost.) If I had made up a demand function and were in the mood to do calculus, I could figure out exactly how many bushels he would produce, what their price would be, and how many fewer bushels that would be than the perfectly competitive market would yield, but the important thing is that monopolies derive more profit than competitive firms, at a cost to the consumer. (Let’s leave aside the idea of a natural monopoly, where high production costs make it more efficient for only one firm to produce.)

So, what’s collusion? Simply, it’s an agreement by two or more firms to act like a monopoly and split the much higher monopoly profits. This is good for the firms and (generally) bad for the consumers because there’s a loss of welfare – some people want the product but at the higher monopoly price they can’t afford it and the monopoly produces fewer goods than the competitive market.

And what does this have to do with Bonds? Consider that Bonds is a producer of runs, and production can be roughly measured by the OPS stat. (There are other methods, but I’ll follow SI’s lead.) Assume that there’s a positive relation between previous OPS and salary (that is, that when negotiating contracts you can put OPS into a formula and that formula will spit out a salary, and that most salaries more or less line up with it), and that Bonds would play either right field or DH, so defense wouldn’t impose an additional cost. A rational team, when offered Bonds’ OPS number for a significantly smaller salary than it would otherwise have to shell out for the same number, should sign Bonds for $400,000. It would be fairly easy to make the leap from this statement of rational behavior to declaring that the teams are behaving irrationally and therefore must be colluding.

SI make a good counterpoint – in economic terms, Bonds’ clubhouse demeanor and notoriety would cause high amounts of negative utility in terms of unhappy and distracted teammates in the first case and customers unwilling to buy tickets to see Bonds in the second. Both of these can affect the profit of the firms and would have to be accounted for in any analysis of the value the team can expect to derive from Bonds. Suppose the most extreme case – that all fans are so disgusted by Bonds that they refuse to attend games he plays in, and that his teammates are so distracted by his attitude that they produce zero runs. In that case Bonds will still be a high producer, but he would still have a decidedly negative effect on profit and on his team’s record.

SI also makes two suggestions I don’t agree with – that Bonds’ felony charges might provide a disincentive in that the team has a high probability of losing him, and that Bonds’ age makes him unattractive. The first assertion ignores the fact that Bonds will still probably produce more than $400,000 worth of runs in a partial season, so in a strict runs-to-salary relation he still represents a net profit. The second doesn’t take into account that at age 42 in 2007 Bonds was worth $15.5 million dollars and that he would have to have declined at an almost impossibly high rate to fail to produce $400,000 worth of runs. Bonds will be out of shape and fragile, but the comparison need not be Bonds now to Bonds in 2007. I’d much rather have an out-of-shape and fragile Barry Bonds as my designated hitter than a $400,000 wet-behind-the-ears rookie. Hell, considering how fragile David Ortiz is, I’d rather take Bonds at $400,000 than Ortiz at his market salary. (The Red Sox could spend the difference on a new fifth starter now that Curt Schilling retired.)

Collusion thus doesn’t make much sense to me in this case, since the incentive to break collusion would be so strong. Stable collusion requires some method for the colluders to punish a fellow colluder who cheats. In a market this would be by simply returning to competitive prices, depriving the undercutter of his excess profits. Here, unless Selig is directly involved (unlikely), there’s no way for the teams to punish anyone for signing Bonds. (Yes, I suppose they could start plunking the batters every time, but that’s not sustainable.) The alternative explanation that makes the most sense to me is the disutility argument – that production and ticket sales will suffer for any team that employs Bonds. If, however, the SI article is correct in its idle suggestion that Bud Selig has ordered teams not to sign Bonds, Selig should be ashamed of himself.

Inexpensive runs
will not placate angry fans.
“Cheap offense” indeed.

What I take from Sauer’s post is that plus-minus is a surrogate variable for ability to be a team player. That opens the broader question of what can be measured and whether nonmeasurable statistics are ever useful in building a team.

If it’s not measurable, does it exist? Many people believe so. Think, for example, of clutch hitting, the vaunted (alleged) ability of certain players, such as Derek Jeter and David Ortiz, to hit more reliably in situations where the team’s expectation of winning or losing is weighted more heavily. The usual clutch situation is described as “close and late.” Clutch hitting is widely regarded by statistical analysts to be a myth, largely an artifact of small sample size. (In some cases, such as Jeter’s as identified in Wikipedia,the reputation for clutch hitting isn’t supported by any stats at all, but just a long memory for isolated incidences of close-and-late production.)

Jeter, predictably, thinks people who find statistical evidence that he doesn’t live up to his reputation should be defenestrated.

My opinion is that measurable and objective statistics are essential for valuing players (and coaching staff). The article on Battier is a red herring – Battier has what is presumably an anomalous plus-minus, particularly when accounting for distortions based on the quality of his teammates. Battier does, in fact, “have stats.” “Team player ability” isn’t a soft, undefined concept, but rather the ability to act as a multiplier for other players’ ability, and it’s measurable in final production numbers (probably as a factor of the player’s specific marginal product of labor). Battier’s plus-minus stat is evidence of an increased marginal product of labor.

Jeter, on the other hand, is a defensive liability who happens to be productive offensively. He’s a perfectly good hitter, but his numbers should be allowed to stand on their own rather than being propped up by some ephemeral idea that he occasionally produces at full ability and runs the rest of the season at some lower value of his optimal productivity.

High plus-minus is
evidence of production;
clutch hitting is not.