Confidence

If one happens to be listening to a baseball game, one will often hear analysts talking about "confidence." For example, "He just doesn't have a lot of confidence in his fastball right now" or "He doesn't look very confident at the plate." Now, I normally try to take what broadcasters says with an epic grain of salt. However, in this case, their emphasis on confidence is well placed. Confidence is one of the most important aspects of analysis.

Unfortunately, we're talking about a different kind of confidence than they are.

When one deals in the realm of probability, which is to say, when one analyzes anything with unknowable information, one can never predict anything with 100% accuracy.

For example, let's say I have 1,000,000 consecutively numbered balls in a very large hopper. I draw a ball from the hopper and predict that the value on the ball I have drawn will not be exactly one. This is not a very dangerous prediction. In fact, I will be right 99.9999% of the time. However, I will also be wrong 0.0001% of the time. No matter how many balls I stick in the hopper, I will always have some chance of selecting that one ball that breaks my prediction.

The same is true in baseball. As the old adage goes, even a .300 hitter fails 70% of the time. Ignoring the banality of the adage, what it implies is important: we can never be 100% certain of a particular outcome in baseball.

Baseball involves two elements: skill and chance. Some prefer the term "luck" to "chance." I do not, because luck implies some moral element: "good" luck or "bad" luck. Rather, "chance" simply implies that there are things that are beyond a player's control.

Thus, whenever we are examining a player's performance record, we have to take care to account for the fact that some of that player's performance could be due to chance. So how much is due to chance and how much is due to skill? Again, we can never say for certain. However, we can leverage probability to tell us how much uncertainty there is in our estimate.

Let me illustrate what this looks like without any rigorous math, of which I assure you, Dear Reader, there is plenty. Suppose I have an apple sitting on my head and a bow and arrow with someone must attempt to shoot the apple. Having studied this subject repeatedly, I know that an average person can hit the apple without killing me (success!) only 25% of the time. I am in quite a fix.

However, the evil sadist that has put me in the gruesome predicament has given me a way out: I can pick who will be the shooter from a list of anonymous citizens, who have all provided me with their career attempts and successes in the common practice of shooting an apple off of one's head. How should I make my choice? Should I simply choose the person with the best ratio of success to failure?

Of course not! I'll end up choosing someone who shot one apple off of one head. This person will have taken one attempt in their entire life and just happened to succeed. Now, not knowing anything else about this person, should I conclude that they are a 100% apple-shooter? Only if I'm suicidal!

The odds are that this anonymous shooter is actually an average Joe: he probably hits the apple one out of every four tries and happened to get lucky this one time. To be sure, he could be the greatest apple shooter who ever lived. However, I cannot have any confidence in this conclusion based on only one trial.

Let's say I have another anonymous archer who has made 5,000 apple-shooting attempts and succeeded at 4,500 of them. The odds of this guy being a normal 25% shooter are close to zero. Amazingly, it is not guaranteed that this guy is not an average shooter. After all, in a universe of infinite possibilities, some average shooter will hit 4,500 of 5,000 shots. However, I can say with much, much more confidence that this man has a real skill at shooting apples of off heads.

Naturally, given a choice between the two, I'll pick the guy who's almost guaranteed to give me a 90% chance over the guy who has a ominously low chance of being 100% accurate.

So, how do we quantify this?

Well, that's where confidence comes into play. Confidence is the likelihood that our measured value is within a given range. For example, after that one trial, I can say with 99% confidence that our shooter falls withing the accuracy range of 100% ± 90%. Or I can say with 90% confidence that our shooter falls into the range of 100% ± 75%. Or I can say that 50% confidence that our shooter falls into the range of 100% ± 40%.

These numbers are made up, but they reflect the way confidence works: for any given sample we can increase our confidence in the range of values by increasing the size of the range. We can decrease the size of the range by decreasing our confidence level.

In the case of our apple shooters, because we only have one trial on which to base our conclusions about the man with 100% success, we will not be able to get a range of values useful enough to make a decision without destroying our confidence level. This ought to make intuitive sense to you. The more you increase the range, the more likely you are to be right, but the less useful the range is. The more you decrease the range, the more likely you are to be wrong, but the more useful the range is. With so little data, this guy is hardly more valuable than a shooter who's never taken a shot in his life. Without knowing anything else, it's very hard to conclude anything but that he's an average guy who got lucky once.

What of our man with 5,000 attempts? In his case, I might be able to say with 99% confidence that his skill is 90% ± 5%. Or I can say with 90% confidence that his skill falls into the range of 90% ± 2%. Or I can say that 50% confidence that his skill falls into the range of 90% ± 0.1%.

Why do we have more confidence with this guy? Because we have so many more attempts on which to base our conclusions. Again, it makes intuitive sense: the more data we have, the more confident we are.

Here is the important point: the only way to both increase confidence and decrease the range is to add more data points. Period.

Normally, when presenting statistics researchers will choose a confidence level and let the range fall where it is. A 50% confidence level just isn't that useful. Ninety-nine or ninety-five percent are probably the most common.

So why have I spent so many words belaboring this point? Because we're about to explore some bona fide objective analysis and confidence is a foundational concept. Confidence is why it means so little when a guy is 5 for 7 lifetime off of a given pitcher. Confidence is why analysts are so skeptical about a clutch hitting skill. Confidence is why it's so hard to judge how good a reliever is statistically.

So if you don't get confidence, you may want to reread the post (or find a better teacher), because if you don't, you may be the next guy to do this. Or at least, you won't get my upcoming posts.

Also

ESPN's player ratings currently rate Eric Byrnes ahead of Daisuke Matsuzaka.

You should not need anymore evidence than this to reject the player ratings wholesale. The idea that a left fielder who isn't slugging .500 or getting on base at a rate of 40% (or even close) is more valuable than a guy with 164 innings of 3.79 ERA in the American League East is just crazy.

Chapter Fourty-Four

In Which I Annoy My Paltry Readership By Returning To Highly Theoretical Baseball Musings And Pooh And Piglet Go Hunting And Nearly Catch A Woozle.

One of the things that always strikes me about statistical analysis is that it usually involves talking about baseball in the realm of the infinite. While this practice is theoretically correct, the thing that makes baseball interesting is that it is played in a finite series of events. A team does not have an infinite amount of time to allow its true talent to come through. It has only 162 games.

The result of this is that as soon as we start playing games, teams start accumulating "error" in terms of their actual performance versus their true talent or expected performance. In turn, the result of this is that it is actually quite easy for teams to vary wildly in their deviations from their expected performance.

(Keep in mind that when I talk about expected performance the assumption is that the expectation is accurate. I know Robert expects people who expect the Yankees to win 125 games a year, but that is not what we are talking about here.)

For example, let's say I have a coin that I will be flipping 10,000 times. We expect that this coin, if it is a fair coin, will come up heads 5000 times and tails 5000 times. Why? Because a fair coin is expected to come up heads half of the time and tails half of the time and I have 10,000 remaining flips.

So I make the first flip and it's heads. Two things happen immediately. First, and least consequential, is that I know have the world's smallest sliver of a fraction of a shadow of a glimmer of a doubt that the coin is biased towards heads. Naturally, this evidence would have a confidence level bordering on zero.

Secondly, and more importantly, my expectations have changed. Previously, I was expecting 5000 heads and 5000 tails. Now, I'm expecting 5000.5 heads and 4999.5 tails. Why? Because a fair coin is expected to come up heads half of the time and tails half of the time and I have 9,999 remaining flips.

Remember, the coin had to come up either tails or heads, and it is by no means surprising that it came up heads. Nevertheless, our expectation now must be different. Probability is only useful in discussing unknown outcomes. It has no role in discussing known outcomes. Information is highly influential, and therefore valuable, in probability.

This happens a lot in baseball too. The Yankees have grossly underperformed this year. We would expect them, base on runs scored and runs allowed, to be ahead of the Red Sox, not behind them. However, they don't get those "flips" back. That error from their true talent level has accumulated and now we have to adjust our prediction of their overall record downward, even though the team might not be at all worse than our original expectation. Crazy, huh? I think so.

Look at it in terms of a three game series against Baltimore. We would expect the Yankees to win about 1.8 of the games in that series (number derived from the patented "pulling a number out of my ass" technique). There's only one problem: you can't win 1.8 games. So after that series, the Yanks will be guaranteed to have overperformed or underperformed. If they underperformed, they can't expect to recoup that loss. If they have overperformed, that's money in the bank.

As with the coin flip, it's also true that the results of the team's games will influence our determination of their true talent. However, the sample needed before this starts having a noticeable effect is really, really large.

Anyways, the corollary to all of this is my (only?) favorite John Sterling cliché: you're never as good as you look when you're winning and you're never as bad as you look when you're losing.

So true. In fact, we've just proven it.

(The more theoretically minded will note that, other than perhaps having to adjust our "true talent," these results are still meaningless when we are making an infinite number of flips or playing an infinite number of games. No matter how many heads in a row I get, if the coin is known to be fair, the expected ratio of heads to tails after an infinite number of flips is always exactly 50/50.)

**EDIT** Fixed a really dumb typo.

Steroids Week, Part Two

I had originally intended to split steroids week into three parts, but then two things happened. First, I posted a lot of "bonus" content (if by "bonus," I mean, "reactionary"). Secondly, I remembered that I hate talking about steroids. So I've decided to roll my final two posts into this one, perhaps longer, post.

Essentially, I want to cover two topics. First, what can we say about the offensive era we live in and how much if it is driven by steroids? Second, what do I think about steroids and steroid users on a personal level.

As many of you are no doubt aware, from about 1995 through the current year, baseball has been on somewhat of an offensive kick. What you may not know is that the current offensive levels aren't entirely unprecedented. Baseball is always going through transitions between eras of more or less scoring. I expect this trend will lessen some as baseball tries to regulate its product more heavily. However, historically, the current level of offense is not highly unusual.

So what is the cause of this offensive boom? Good question. Let's examine the possible reasons and their likelihood of significance.

1. Steroids

You know all about this one, so let's get it out of the way. Essentially, the argument goes that athletes are bigger and stronger than they once were because of steroid use and therefore they are hitting better than ever.

I have no doubt that some baseball players have done steroids. I have no doubt that steroids make you stronger and that this will help you hit ball hard. I have great doubts that this effect is significant, relative to other factors. The truth about steroids, the one thing that must always be kept in mind, is that:

We just do not know what the effect of steroid use is on one's ability to hit a baseball.

Strength and baseball power are not perfectly related. The key with hitting for power is the efficient transfer of energy from your muscles to the ball via the bat. This requires the ability to make contact, proper mechanics, and, yes, some physical strength. Of these, physical strength is probably the least important when it comes to hitting a baseball hard.

2. Weight Lifting

While we're on the subject of strength, it's worth pointing out that not only is it hard to separate the effects of strength from other aspects of hitting, it's also almost impossible to separate the effects of weight-lifting without steroids from the effects of weight-lifting with steroids, with respect to baseball players.

Baseball players never used to lift weights. It was thought that weight lifting would screw up your swing and destroy your agility. Even as recently as the 1980's these things were viewed as more important than raw strength. It is not surprising that steroid use would become an issue at the same time as weight lifting; after all, the two are related. However, how do we know if Barry Bonds' strength gains are 10% due to weight lifting and 90% due to steroids or vice versa? We just don't know.

3. Smaller Ballparks

This has been rehashed elsewhere, but suffice to say that most of the modern ballparks that have been built are offensive havens. The stadiums built in the 1970's were often monstrosities that favored the pitcher. This trend has been reversing as teams have come to favor smaller, more intimate (read: pricey) ball parks.

4. Better Bats

The science of bats has come a long, long way. In the 1990's players switched to using a maple bat instead of traditional ash bat. These bats are made from better wood allowing them to be made much lighter. Lighter means faster. Faster means more bat speed. More bat speed means more power.

5. Juiced Balls

Yeah. When I first heard about this in 1998, I thought it was a conspiracy theory too. And yet, it may not have been wrong. There is good evidence that the balls in use today are much more lively than the balls used in previous generations. Some researchers claim that this may allow a ball to be hit as much as 30 feet farther. That's a lot. This also jives with the wonderment that many players and coaches often express when they see a little second baseman hit "what should have been an easy fly ball" for a home run. It's not just the big, burly men who are hitting more home runs.

6. Diluted Pitching

Expansion wouldn't cause league offense as a whole to go up for very long. For each crappy pitcher you add to the league, you also have to add a crappy hitter. However, it does mean that the good hitters that are already there get to face crappy pitchers in a higher percentage of their plate appearances. It works in reverse too: great pitchers get to face crappy hitters more often.

What's the result of this? The great ones play greater. You would see the frequency of both great individual hitting and great pitching performances go up. Anecdotally, this is the case. We've had more 20 strikeout games in the last 10 years than in all of MLB history. On a more quantitative note, almost all of the great single season strikeout performances (as a rate statistic) by pitchers are within the last 15 to 10 years. It should stand to reason that hitting would also see some bounce from this effect.

7. Different Offensive Philosophy

Joe Morgan may not like it, but guys do play for the three run homer. And they should, as it will lead to more wins. The natural strategic evolution of baseball is seeing the death of widespread use of inefficient tactics like the sacrifice bunt. It's a slow process, but as people become more educated as to the value of outs, they will be less likely to play for one run and more likely to play for three. And that's a good thing.

8. Modern Medicine

You know how many players in the 1960's had successful reconstructive elbow surgery? Zero. Nutrition and medicine are leaps and bounds better now than when even just 20 years ago. Why should it surprise is that this helps players perform better?


9. Fewer Pitchers Pitching Inside

This allegation is, of course, purely speculative, but many believe that the presence of protective gear has allowed hitters to crowd the plate more than ever, taking away half of the plate and forcing pitchers to work predominantly outside. Naturally, when you only have to worry about half as many pitch locations, it's easier to hit.

Now the point of listing all these factors isn't to convince you that steroids are meaningless. In fact, it's not even to convince you that each of the listed factors are meaningful.

No, the point is that we just do not know how much these individual factors have contributed to the offensive boom. The media has turned this into a black and white issue: steroids, used by cheaters, have caused the offensive explosion. The truth isn't nearly that simple. There are a plethora of factors that may or may not have contributed to the offensive explosion in baseball. Trying to separate one from the other is likely a fool's errand.

(Interestingly, in the case of Barry Lamar Bonds, you can also add the existence of a marvelous hitting aid as a potential factor in his power surge. I'm not sure how much I take this claim seriously, but it is fascinating to think about.)

So. What does all this mean? What do I think about steroids in baseball and Barry Bonds in particular.

First, because it's so hard to separate which factors were instrumental in the increased offense in baseball, we shouldn't be giving players asterisks or discounting their records in the record book. We should be letting history write its own narrative about the players involved. Each era in baseball has to be properly adjusted for before you can properly appreciate the achievements that took place during that era. Hitters in 1968 struggle to hit even .300. A bunch of guys will clear that number this year. Context, as always, is everything, and it is this context, completely neutral and completely scientific, that will cause us to look back on this era and properly discount the value of the offensive numbers that were put up.

It's ironic then that the media hates the very people that advocate these adjustments. We're the geeks, those guys with a slide rule. We're ruining baseball with our numbers. You'd think that the politics of steroids would have made for the strange bedfellows of geeks and writers. What does that say about how much the media hates geeks relative to steroids?

Yes, I do think steroids are cheating. I do think that they violate the spirit of the game. I do not want them to have anything to do with baseball. We now have a system in place for this. Guys who are caught cheating should serve their suspensions and then be allowed to return to baseball. They should not be banned from the Hall of Fame and they should not have their records invalidated.

That being said, the fact that they cheated should weigh into the Hall of Fame process. It speaks to their lack of character and does call their performance into question to some degree. However, this criteria should only be used in borderline cases. I don't think steroids can make a guy who belongs outside the Hall into a no doubt Hall of Famer.

As for Barry Bonds, I think that there's an excellent chance that the man used steroids. Unfortunately, he did so at a time when there were no consequences and now that there are consequences, he has not failed a test. There's absolutely no way you can invalidate what he's done officially without some kind of failed test.

Is Barry the home run king? Well, yeah, if you mean that he's hit more home runs than anyone in Major League Baseball history. Of course, that would be true even if he had failed a test. The games still count. If you mean that Barry is the greatest slugger of all time, you'd be wrong and will continue to be wrong unless Barry can belt another 150 home runs. That's about what he'd need to surpass the Babe after appropriate context adjustments.

Barry Bonds is a first ballot, no doubt, surefire, inner-circle Hall of Famer. He should get 100% of the Hall of Fame vote. He won't because some will let the steroid issue convince them that somehow he isn't worthy, despite his pre-2001 numbers. It won't matter, because no one gets 100% of the vote anyway, not even Cal Ripken.

Most importantly, I think that steroid use in baseball is not rampant. I think that most guys do not use steroids. I think that the steroid issue is a media driven frenzy designed to give themselves something about which to moralize without fear of reproach. I just want the issue to die, to go away and never return. I want baseball to be about baseball again. I want to be able to listen to a baseball game on TV without the announcers bringing up steroids.

That will be a good day.

Steroids Week: Bonus Coverage 5

Chipper Jones answered a question about A-Rod and steroid use and the answer has been blown way out of proportion. Peter Abraham, at his excellent blog, addresses just how out of proportion the reaction is, and in doing so, says this:

I cannot tell you categorically that Rodriguez has never used PEDs. But I can tell you this: I’ve never witnessed a baseball player who works harder at being good. I used to get to Legends Field around 7:45 a.m. most mornings in spring training and A-Rod was always on a back field taking grounders. Not sometimes. Always.

During the season, we’re allowed in the clubhouse 3.5 hours before the game and he is always there, either having finished working out or on his way. He also lifts weights after games. The Yankee coaches marvel at this work ethic and hold him up as an example to their younger players.

Alex was heavier last season, especially in his upper body, and it cost him on the field. He has dropped at least 15 pounds this season and has more range and quickness at third base and much quicker bat speed.

Along with the working out, he radically changed his diet. It’s to the point where he brings his own food to the stadium and calls ahead to restaurants to find out what’s on their menus.

There’s no reason to feel badly for a guy with that kind of talent and a contract that will change the lives of his great, great grandkids. But in this case, Rodriguez is being treated unfairly.

Do you see what Mr. Abraham has done? A-Rod works hard, therefore he is a good guy and less likely to use steroids. A-Rod's suspicion level is mitigated by Peter Abraham's perception of how hard A-Rod works at his craft.

Now, if I had to venture a guess, I'd wager that Barry Bonds works harder than one can possibly imagine to be a great hitter. In fact, as I hope to touch on at a later date, if Bonds is using steroids, he may be working harder than he normally would because steroids enable him to push his body harder when training.

PED use and hard work are not mutually exclusive. PED use is that strange kind of cheating where you actually have to work harder when cheating than you do when not cheating. It can be driven by fear of failure or by desire to succeed, but either way you have to be driven to care enough to use PEDs.

Yet, we often throw up guys as examples of non-users just because they are hard working, affable guys. Because we perceive these guys playing the right way and being gritty and hustling and working hard, we elevate them beyond reproach. Bonds is a surly son of a bitch, so he doesn't get the benefit of the doubt.

But Pete Rose? He was a baseball hero, until it turned out he actually wasn't.

Steroids Week: Bonus Coverage 4

This article contains all you really need to know about 756, including what Hank Aaron thinks and what Willie Mays thinks.

I really don't have anything more to say.

Steroids Week: Bonus Coverage 3

Just to be fair to Mr. Kurkjian, this is a much more reasonable attitude (from the same article as our friend Pedro Gomez):

I will acknowledge Barry Bonds for what he has done: hit more home runs than anyone in history. It is a fascinating accomplishment, one that's worthy, on some level, of celebration. We have never taken records away in baseball history, and we should not take this one away unless we're prepared to take away a whole bunch of records and achievements during this era. We shouldn't put an asterisk next to it, either. There already is -- and always will be -- an imaginary asterisk next to this era. We should do what baseball has always done with its records and controversies: attach a story to them, and then let our best baseball fans -- they believe something fishy went on here -- decide how to recognize this achievement. As for Hank Aaron, he no longer will have the most home runs of anyone in history, but his legacy will not be lessened. Bonds' chase has given us another chance to celebrate the greatness of Aaron's career, and the strength of his purpose. His legacy might even be strengthened because, as far as we know, he hit 755 home runs naturally, legally and honestly.

I can respect this opinion. People don't think less of Ruth for what he did. People haven't forgotten Roger Maris. And we're all learning how to put the context of the players' accomplishments in their proper context. We don't need witch hunt to extricate players' records and we don't need an asterisk.