Brandon McCarthy thinks that Brandon Inge is worth 10 wins or so to a team behind closed doors. Jonny Gomes, too. Participating in a player panel at the SABR Analytics Conference earlier this month, McCarthy posited that if Inge and Gomes had been removed from the 2012 Oakland A's, they might have fallen from a 94-win team to a 70-win team, purely by virtue of being deprived of the effect the two players had in the clubhouse. According to WARP, Gomes was worth 2.2 wins last year, while Inge was worth 0.6. So, assuming that if neither had been on the team, they would have been replaced by… well, replacement level players, that means that Inge and Gomes somehow combined for 21.2 wins just by being good guys in the clubhouse.

Okay, so maybe McCarthy was exaggerating. Maybe the point that he wanted to make was that Inge and Gomes were fun to be around in the clubhouse and that that helped him and other players out quite a bit. Maybe he wasn't trying to be accurate to the third decimal place—or even the tens place. He just wanted to say that he believes that these sorts of things can make a difference on the field. But it does raise a question that I seem to be visiting a lot lately. What measurable difference can a player make behind the scenes?

McCarthy, now with the Arizona Diamondbacks, spoke to AZ Central's Nick Peicoro afterward and explained the effect like this:

It sound[sic] stupid, but if you have a rookie that comes up and rookies are filled with self-doubt, filled with worry, and now you’re in the big leagues and you come to a team where nobody makes you feel welcome. So now you’re already nervous, you’re kind of worried about your lot, and then the guys around you, you’re not comfortable and you don’t feel like you’re one of them. You don’t feel kind of free and like you can do what you do. But if you have a guy like Jonny Gomes or Brandon Inge or someone who just comes up and is just kind of (BS-ing) with you and it just sort of loosens you up and then everyone else can kind of get in the mix… That loosens you up, which in turn the person you interactive[sic] with — there’s a whole trickle down effect to it that’s impossible to quantify but it does exist in there.” (emphasis is mine).

Was that a challenge?

Warning! Gory Mathematical Details Ahead!
This is going to be tricky (and very gory). We don't know what Inge or Gomes (or anyone else) did behind the scenes, except in the most general terms. Whom did they help? On what day? (MLBAM folks, we really need BFFf/x up and running soon.) We do, however, know what clubhouses they've been in and who else was in there. And we know, in general, how those guys did from year to year. It's a very rough-hewn method, and we'll talk about the limitations in a bit, but we're not totally in the dark when it comes to measuring the effect of a single player on his teammates over time.

To do this, I used a mixed-linear model approach. For all player seasons (batters) from 2003-2012, I coded for whether the player had Inge as a teammate (so basically, everyone who played on the Tigers for most of the last decade) in that season. I did the same for Gomes.

Now, to guard against the fact that I don't want to look at raw stats and mistake the fact that Inge or Gomes just happened to play with talented players (or really bad ones), I used an AR(1) covariance matrix. The idea here is that since I'm taking repeated measures for each player, the model will adjust for the fact that if a player hit a lot of home runs last year, we will expect him to do so again next year. For the initiated, I pegged the covariance matrix to the player's age on April 1st of the year in question.

In addition, Inge spent most of that time in Detroit with Comerica Park as his home base. To that end, I entered the player's team (as a proxy for home park effects) as a fixed factor in the model. Where a player played for more than one team, I entered the team for whom he had the greatest number of PA. I also entered the calendar year as a fixed factor, since offense has been slowly declining from10 years ago until now. This will correct (some) for the declining offensive environment over time.

Age also went into the model as a fixed (categorical) factor, to make sure that the effects weren't just due to the aging curve. I restricted the sample to players who were in their age 23 to age 35 seasons, and also to hitters who had more than 250 PA in the season in question. I should say that these aren't perfect adjustments, but they will do the job for the moment. I should also point out that I eliminated any actual stats belonging to either Brandon Inge or Jonny Gomes.

Finally, I entered whether the player had Brandon Inge as a teammate during that season. I modeled each player's strikeout rate (per PA) for the season, and looked to see, once the rest of these things had been controlled for, whether the Inge effect was significant or not. I then did the same for walk rate, HBP rate, singles rate, double/triple rate, HR rate, and outs on balls in play rate.

Theoretically, this model should give us the answer to the question, "If you took an average hitter from this population that you've selected, adjusted for park, year, and age, what is the extra effect of having Brandon Inge as a teammate?"

The answer for Inge was surprising. Players who played with Inge had strikeout rates about 2.3 percent higher than what might be expected and about 2.2 percent fewer singles. Uh oh. But not all hope is lost. There were some marginally significant effects for home run rate, which increased by about 0.7 percent (p = .09), walk rate, which increased by about 1.1 percent (p = .14), and outs in play, which decreased by 1.6 percent (p = .27). (Note: yes, I'm fully aware that those numbers don't fully reconcile.) If anything, the players around Brandon Inge leaned more to a three-true-outcome philosophy of hitting than their previous (and subsequent) performance would have predicted for them, adjusted roughly for park, year, and age.

For Gomes, none of the effects came out significant. It doesn't look like Jonny Gomes's teammates showed systematic improvement from his mere presence.

Was it Inge?
Before we go further, let's acknowledge something. "I was teammates with Brandon Inge" is a pretty good proxy for "I was a member of the Detroit Tigers." While the model attempts to correct for home park (and by some manner of indirect association, team philosophy), it's important to point out that the Tigers have had a lot of well-known TTO guys who have come through Comerica Park over the past few years (Dmitri Young, Curtis Granderson, Austin Jackson, Miguel Cabrera, and um… Brandon Inge.) Because Inge and "Detroit Tiger" are nearly synonymous, the model may simply be splitting the blame for the fact that all of these TTO guys were around between Inge and the Tigers. Maybe this was a case where the Tigers brass liked to acquire guys who were already in the TTO mold and then encouraged them to become more so, either directly or indirectly.

Did Inge cause them to become TTO hitters? Probably not. One statistic that the AR(1) covariance matrix gives us is the AR(1) rho, which is kind of like a multiple year-to-year correlation. TTO outcomes are among the most stable of batting statistics, and in this sample, all were above .70. When I switched the "Inge factor" over to a random effect to look at the variance composition stats, it barely registered as a driver of the variance.

In defense of the thought that Inge might have had some effect, the reason that many of the effects observed were marginally significant was not due to a small observed effect, but because of a large error bar. Suppose that Brandon Inge really did have an effect on some of the players with whom he played, but not all of them. Instead of having some random noise distributed around a mean change of zero, it would be that random noise stretched out by the true effects for some of the players in the Inge sample. Measures of variance, such as standard error, would increase. Maybe Inge isn't helping everyone on the team, but maybe he's not completely inert. It's possible (although this is hardly proof) that the fact that we don't know what Inge said to whom and when makes our measure too rough to pick up the signal that's lurking underneath.

It's really hard to tell exactly what's going on in this model. In a perfect world, Inge would have moved around to a few teams (much in the way Gomes has), and we might have seen the effects at each of his stops. However, we don't have that luxury here. Major League Baseball continues to ignore my requests to randomly shift players from team to team and to assign playing time to its players randomly. It would make sabermetrics a lot easier.

Maybe Brandon McCarthy is right. Given what data we have, we can't really quantify the effect that Brandon Inge had on the 2012 A's. There's too much methodological noise, and I don't know that there's a way around this one.

Value Above Cardboard Cutout
Before we even entertain the idea of Brandon Inge being a 10-win personality, let's for a moment do a quick thought experiment. Suppose that, as Brandon McCarthy suggested, Brandon Inge really had been removed from the 2012 Oakland A's. Not completely. Let's pretend that nothing changed about the way that he played in the field, but when he crossed into the dugout and clubhouse, he turned into a cardboard cutout. He didn't talk to anyone, and did nothing, good or bad, to affect the team's morale or chemistry.

From the sound of it, McCarthy said that Inge's value was in his ability to engage younger players, and to make them feel more relaxed. There's no reason not to believe McCarthy, so I'll accept that this really was the case. Now, with Inge rendered a cardboard cutout, what would happen? In the same way that we have replacement level on the field, we should look at things similarly off the field.

Let's assume that there is some skill for helping other people to better acclimate to a new situation, or inspiring people, or keeping them loose. Let's assume that some players are good at it and others not. Most are somewhere in the middle. A baseball team is a small society and, like any society, it must have roles that people fill to accomplish the tasks it needs to survive. Inge filled the role of "guy who kept people loose and feeling happy." If he were replaced with a cardboard cutout, would no one have stepped into that role? Someone probably would have, even if they weren't as skilled, in the same way that someone would have stepped into the role of right fielder had Josh Reddick gotten hurt. Maybe the new clubhouse clown wouldn't have gotten quite the effect that Brandon McCarthy alleges that Brandon Inge and Jonny Gomes got, but it wouldn't be zero.

And that's going to be important in both quantifying on-the-field value and the value of a clubhouse presence. We need a reasonable baseline. Assuming for a moment that McCarthy's pronouncement that Inge produced oodles of collateral on-the-field value with his people skills is completely true, let's not credit him with all of that value. If the next most-qualified guy would have done 70 percent as good a job, then Inge is "only" worth three wins. It's similar to the way that replacement level taught us that we shouldn't be massively impressed by 15 HR from a first baseman when we'd estimate that some scrub with the same amount of playing time would have hit 12. To be fair, we know what replacement level looks like on the field, but we have no idea what it is off the field. As this search for the effect of the clubhouse factor continues, we do need to keep the idea of a proper baseline in mind.

Why the Search Should Continue
I don't buy the idea that Brandon Inge is a 10-win player behind the scenes. But I do think that Brandon McCarthy is onto something here. There might indeed be a very rich vein of hidden value that can be mined, and it can be found when McCarthy says, "there’s a whole trickle-down effect…" Consider that to improve his own on-the-field value, a player has perhaps 500 or 600 PA to make a difference. To change the outcome of one plate appearance, his outcome rates must change by around 0.2 percent, and even going from the worst possible outcome (a strikeout) to the best (a home run) is worth roughly 1.5 runs. To produce one win, he'll need change his own individual outcome rates on both stats by more than a percentage point. That's a tall order. Not impossible, but tall.

The reason that a better understanding of how team chemistry works holds great promise is the issue of scale. A player can change only himself on the field, but the trickle-down effect in the clubhouse might touch everyone on the team. Suppose that Inge was able to help several guys on the A's, and that they accounted for only half the plate appearances that the team had during the season. Since the average team sends about 6,000 hitters to the plate in a season, Inge suddenly has his finger on the scale for 3,000 PA, rather than 500. If he can move the needle on the combined strikeout rates and home run rates for the players over whom he holds sway by 0.2 percent on the whole (one PA in 500), he produces most of a win.

The effects of chemistry might be small, but they can be spread over a wider swath of players. And if they are, then suddenly there's a multiplier effect, because a nice personality can affect more than just one person. Now, is it reasonable to believe that a player might be able to, once or twice in a season, pick up a friend on the team who is feeling down, and through just being a nice guy, give him a little extra boost that turns what would have been an out into a hit? If that seems reasonable, then suddenly, the value of those few extra hits starts to add up.

Maybe we don't know how to measure chemistry or clubhouse behavior yet, and maybe we'll never really have the kind of data that we need to do it. But if we take a small leap of faith and trust the people who have been in clubhouses, we might then have the final piece to make a rational, statistically/numerically based argument for team chemistry being not only something that's real, but something that can be really powerful.

I should properly acknowledge the inspiration of Jay Jaffe, whose piece in Sports Illustrated inspired this investigation.