# How to make your team better by firing your third base coach

April 3, 2008 17 Comments

If there is one position on the baseball diamond that is overlooked and under-appreciated by the fans, and under-exploited by Sabrmetricians, it’s… the third base coach? Wait a minute, all he really does is wave his arm at the runner. (And he relays the signs about the bunt that we all know is coming.) Well, I suppose the position is a little harder than it looks. After all, on a hit to the outfield, that’s a big decision to make. Should the coach send the runner who is now approaching third or hold him? Runs are at stake, and if he waves his arm, there’s likely to be a play at the plate. That’s always high drama. If the runner is out, the third base coach is criticized as a crazy man with a wildly waving arm. (Wendell Kim, are you out there?) If the runner is safe, the batter gets an RBI, the runner gets a run scored, and the third base coach gets…. nothing. Not even an atta-boy. Where’s the justice in the world?

How can we tell whether a third base coach is doing his job well? When making the decision of whether to send the runner on a ball in play (first to home on a double?), he must take into account a few factors.

- Where is the runner currently located and how much further would he have to go to make it home. 60 feet vs. 120 feet could make a great deal of difference.
- Where is the ball currently located. Is it still in the outfield? Is it already to the cutoff man?
- How fast is this runner?
- How strong is the outfielder’s arm (and to some extent, the cutoff man’s)?
- And I suppose “who’s on deck?” (although I’ll be ignoring it. Sacrificing precision for direction.)

The situation that best controls these factors is the would-be sacrifice fly. With a runner on third and less than two out, a fly ball is hit to the outfield. When the runner starts running, we know he’s got exactly 90 feet to cover. And we know the ball is in the outfielder’s glove. We have speed metrics for runners (and I’ve looked at them before, more below), and finally some good metrics for arms (although I found out about them too late to fully incorporate them).

So, should this particular runner on third try to tag up on this particular fly ball to this particular outfielder in this particular situation? Here’s how we can start to build to the answer. First off, I suppose that given how far away the ball is when it’s caught, how fast the runner is, and how good an outfielder’s arm is, there is a certain probability that the runner will be safe if he attempts to run. I’ve shown previously that the chances for success or failure are *much* more influenced by the distance the ball was hit than by the speed of the runner. The question is “deep enough”, not “fast enough.” The third base coach is surely mentally sizing up what he thinks this probability is before he decides whether or not to wave the runner home. What *should* be his threshhold of probability for whether not he should send the runner? The answer is always the threshhold should be placed where the benefits (in terms of runs scored) outweigh the risks.

Let’s do a little math. The run expectancy matrix will be a big help here. I used the run expectancy matrix from 1993 (more on why I used ’93 in a minute) to calculate what the break-even point would be for sending the runner vs. keeping him at third. A sacrifice fly can only happen when there are zero or one outs on the board, and there’s a runner on third. (Yeah, I suppose a sac fly can happen from second, technically… shut up.) At the moment that the ball is caught, the third base coach can pick between being assured of whatever run expectancy is present (with the runner still on third) or can take a chance on what might happen if he sends the runner. He might be safe or out, but those two outcome have very different run expectancies. To make this a little more concrete, let’s say that we start off with a runner on third (no one else on base) and no outs. The batter lofts a flyball to left field and it’s caught. There is now one out and a runner on third, which (in 1993) had a run expectancy of .989 runs. The runner could tag and try for home at which point he could either be safe (now one out, no runners, 1 run in, so a run expectancy of 1.275) or out (now two out, no runners, no runs in, so a run expectancy of .102). We can figure out what the break-even point is on this wager, which is “given a million trials, what percentage of the time would the runner have to be safe in order that we would neither gain nor lose runs?”

The formula is: current RE = (probability of being safe) (RE if safe) + (probability of being out) (RE if out). Since the probability of being out is just (1 – probability of being safe), we can substitute that into the equation and solve for the probability of being safe which is the breakeven point. Turns out that it’s 75.6% in that particular situation that I mentioned. So, if the third base coach thinks that the runner has about a three in four chance (or better) of scoring on the play, he should send the runner. If he consistently does this, his team will benefit in the long-term by scoring more runs.

But how to tell what the actual chances of success are. For that, we go to the data and use some statistical maneuvering. From 1993-1998, Retrosheet PBP files actually have pretty good data on hit location. It’s not complete, and I suppose that it’s not exactly 100% perfect, but it’s a good place to start. Plus, it’s free. Why is it important to have hit location data? It’s easier to score on a deep fly ball than one caught right behind second base. Well, if I know where the fielder is when he catches the ball and how far away that is from home plate (something that can be done with a few trigonometry-based tricks), then I know how far his throw must travel, if the outfielder even bothers to make a throw.

I took all fly balls caught by an outfielder from 1993-1998 where there was a runner standing on third base with less than two outs when the ball was hit. I calculated the distance the ball was hit (or a rough estimate thereof) away from home plate, and checked to see whether the runner on third in question made an attempt for home, and if so, whether he was successful. (I only used events in which I had a hit location to reference distance from.) Since success is most determined by distance, I ran a binary logit regression looking at success or failure in all the instances in which a runner made an attempt at scoring using distance the ball was hit as a covariate. This generates an equation predicting the probability of his success based on the distance the ball was hit from home plate. I used this equation to generate the probabilities for being safe for all of these flyballs, both those in which the runner made an attempt or and those in which he was held. Then, I compared the model’s probability estimate of whether the runner would have made it or not to the break-even probability estimate.

It is actually very rare, in this model, for it to make sense to hold the runner at third, assuming that the ball is caught by an outfielder. In 1993, there were 22 fly balls in which the model’s probability estimate was below the break-even point, this out of 1322 fly balls for which we had appropriate distance data. (Not all flyballs had hit location data… I politely excused them from the data set for a moment.) There were 232 cases (out of 1300 exactly) in which the model said to send the runner, but the runner held. Third base coaches are erring on the side of caution and it’s costing their team. I’ve shown before that 97% of the time a runner who is sent from third on a sac fly makes it home safely. It seems that third base coaches are only going for the sure things. They should be a little more daring.

I ran a small experiment. I took the 251 cases in the data set in which the runner stayed (The 19 extras were “proper” times, within the model, for the runner to have stayed at third) and calculated what would happen if I replaced the third base coach with a sign on a stick saying “If the gentleman currently holding the ball is an outfielder, please run an additional 90 feet at this time.” What would be the gain in run expectancy if I sent every runner, even when it was a stupid idea to do so? Over those 251 cases where I had distance numbers to use, the league would have gained 91.70 runs, meaning that holding the runner at third on a would-be sac fly costs a team an average of 0.365 runs each time it happens.

In 1993, there were 329 times when a runner was held at third on a fly ball. (I only had hit location data on 251 of them). So league-wide, third base coaches cost their teams about 120 runs on would-have-been sac flies by being too conservative. Over 28 teams (in 1993), that’s a little more than 4 runs per season. And that’s just *one of the calls that a third base coach has to make*.

I’ve shown elsewhere that third base coaches are far too conservative in other decision-making situations of this type (first to third on a single? first to home on a double? second to home on a single?). If they’re leaving four runs per season on the table just with sac flies, I wonder what the cumulative effect is when you consider everything that the third base coach does… It’s harder to get a read on those situations statistically, because you can’t really tell from Retrosheet data where/when the ball is picked up on a double and where the runner was on the basepaths when it finally was picked up, but it stands to reason that there’s some inefficiency on the part of the third base coaches of America (and Toronto) and that it may very well be around the same magnitude.

My solution: fire the third base coach. Don’t have one. Why pay someone good money when he’s actually making your team worse? The reason that the third base coach is so conservative goes back to the point I made at the beginning of the article. If the runner gets gunned down, the third base coach gets criticized. If the runner is safe, he rarely gets praised. He has every reason in his own personal cost/benefit analysis to err on the side of holding the runners and no reason to send the runners unless he’s sure of their making it, even though it’s costing the team in the long run. Third base coaches are human and surely, they feel the sting of the criticism that follows from a runner being gunned down in a key situation. But, that means that your favorite team is being robbed of runs by a man’s own psychological need for approval. A sign on a stick doesn’t care if it gets criticized. Instead of a third base coach, just tell the players to run like crazy, playground style. It sounds a little weird, but I think I’ve got some decent evidence here that it might actually improve scoring.

I get the feeling that this type of work wouldn’t go over well with traditionalists (aka anti-moneyballers) who feel the need to have an third base coach. But I certainly feel smarter after reading this, it’s something I hadn’t thought about.

I’ve actually felt this way ever since Steve Smith cost the Phillies some runs last year. In the beginning of the year he sent guys and some were thrown out and, in the attempts to evade criticism then became ultra-conservative. They approached 900 runs and might have gotten to 950 or 960 without him.

I’ve also wondered about the need for a hitting coach. Lloyd McClendon of the Tigers this year, for instance. An announcer on opening day made a similar point of yours in that Lloyd will not be heralded if the team hits well because, on paper, they should score 121,291,192 runs this year. However, if they do not hit it is suddenly his fault?

Is it really worth it for a team to pay someone money for this? With all of the advanced video scouting reports and technology today do we need to have hitting coaches?

Awesome, Cutter.

Why did you use 1993?

Great article, Pizza!

This one of those articles that makes you wonder why it hasn’t been written already. I like the reference to running around the bases playground style. My favorite part of softball games was always putting pressure on the defense by taking an extra base on plays in the field. One obviously can’t do it to quite the same extent in the major leagues, but it’s interesting to see how far coaches have gone to the opposite extreme.

I wonder to what extent the outfield defense would be able to compensate if runners were more aggressive. Right now there’s not much incentive to improve the outfield defense if most of the time they are not being tested.

Retrosheet only has hit location data for 1993-1998, so I picked 1993 because it was first.

I could imagine that the response would be to prioritize outfield arm strength, although at what cost, I don’t know. There would be no incentive to re-position the outfielders, because the outfielder’s first duty is to catch the ball, and has to position himself optimally for that.

Perhaps the response would be simply “the best defense is a good offense” strategy and tell your own 3B coach to start waving his arm. Like Brazil’s national soccer team, the motto would be “You score one, we’ll score three.”

Great study. Can you tell in the cases when a runner should not have been sent and wasn’t, how did the team do in the inning? Did they score more or fewer runs than expected? It looks like you had variables in there for the arm of the fielder and the speed of the runner. But maybe the next hitter coming up was good or the pitcher was lousy or it was in the early innings. Just trying to give the human and his judgement the benefit of the doubt (but my guess is that they won’t be helped if you could get those other variables in).

Cyril, I didn’t have speed in there (although I’ve previously shown it doesn’t matter much). I also didn’t have arm variables in there because they’re harder to come by. THT just put together a fantastic system (I believe John Walsh’s work), but his work doesn’t go back into the years that I needed. I might try to duplicate his work on my files.

Hi Pizza, I was linked over here from Neyer’s blog and I’ll post basically the same thing I did there. You have a big problem with your method. You’re calculating the probability that a runner scores based on the distance the ball was hit using the success rate of players that attempted to score. But only people the 3rd base coach decided had a good chance to score attempted it. Thus, it is extremely hard, actually impossible, to untie the general chance of the average runner scoring on the average fielder given the distance the ball must travel. Ie. 3rd base coaches are probably holding slow runners on good throwers when it is a shallow fly, but they are more likely sending fast runners on poor throwers on shallow flies. So, the success rate is artificially high because of the the third base coach. And thus, your data set is completely thrown off for what you want to test.

Notice the only way to actually get a true answer is to put a sign that says: “If the gentleman currently holding the ball is an outfielder, please run an additional 90 feet at this time,” for a few months or a season, or however long it takes to build a good sample, then do this study again. And I’m guessing the basic result will remain, send the damned run almost always.

Anyway, I strongly believe questions like these need answers and questions like this don’t get answered unless someone starts trying. So keep it up.

Neyer linked me? Oooooh. I’m tingling.

I’ve looked at the impact of speed in this situation before. It is actually rather minimal, as is the speed-distance interaction term. The one thing that I’m not able to control for is outfielder arm, which certainly could be in play here. However, I also know that distance from the plate explained some crazy amount of variance (around half, I believe, in my previous study) in the decision of whether or not the 3B coach sent the runner. It looks like 3B coaches are simply looking at how deep it is. A few methodological tweaks would give me a slightly more clear answer, but we also know that success rates for all extra-base advances are well above the breakeven points. I’m guessing it owes to the conservative nature of the average 3B coach.

This would be a really interesting study to conduct by combining intuitive scouting with the actual numerical results. Combining the two would seem to solve Wally’s problems. I know for a fact that Ryan Howard was held at third today on a double by Pat Burrell when I strongly felt Howard would have had a good chance of scoring; looking at it from a removed standpoint, it would seem like a good move to hold him since he’s slow and fat and the fielder had at worst an average arm. Despite this, it would have been close at the plate and due to him eventually being stranded and hindsight being 20/20 it would have been to take the risk.

We should get a group together and study 3B coaches for a bit, combining our intuitive scouting as big fans and numerical skills.

A few minor corrections on some assumptions. I have tested speed in the past and found it rather inconsequential. I haven’t been able to test arm strength as a possibility, and I can’t say yea or nay to whether it has an effect. I agree that’s the weakest link in my argument.

Concerning the speed issue, at one point, I calculated that the world’s fastest sprinter could cover 90 feet within about 3 seconds, and that I could probably do it in 4.5-5. Figure that MLB players are slower than a sprinter but faster than me. So, we’re looking at a window of 3.5-4.5 seconds. Reaction times probably cancel out, as the runner has to react to the thought that the fielder caught the ball and the fielder has to react to the fact that it’s now time to throw toward home plate (or the cutoff man).

I’m no expert on whether your assumptions about the throwing speed of an outfielder (and the resulting calculations that go with it) are correct. I would caution you that there’s one variable that isn’t being accounted for. Aiming. Outfield arms are one part raw strength and one part aiming in the right direction. The problem with long throws is that over a short distance, a miscue of three degrees in proper angle (or perhaps a few gusts of wind/air resistance/ball spin) wouldn’t have as big deal in the final destination of the ball, but over a 250-300 foot distance, it’s going to be an issue.

Consider how hard it is for even a well-trained sniper with a rifle to hit a target consistently at 250-300 feet when he has time to think, a projectile that is much more aerodynamic and travels at faster speeds, and for what it’s worth, several chances to get it right.

On the other issue that you bring up concerning the marginal cases, binary logit is specifically set up to handle S-shaped functions like that.

My basic point is that third base coaches are entirely too conservative. If you don’t believe these analyses, then perhaps you’ll consider this: the success rates for extra-base advancement is consistently above the break-even point for such things. In this article, my goal was to see exactly how much of a difference that made.

This is an interesting idea, but the assumptions and conclusions drawn from them are in my opinion unwarranted.

Your stats models say that runner speed and OF arm aren’t very important in determining success rate? How about we sanity test that by using a physical model? The difference in home-to-first times between fast runners and slow runners is at least .6 seconds — home to first spreads being a very close proxy for third-to-home spreads. Outfielders routinely throw at 90 mph, or 132 feet per second. Of course, the ball slows in flight, so let’s assume an average velocity of 75 mph or 110 feet per second (keep in mind that the shorter the throw, the less it will slow).

Therefore, the difference between a fast runner and a slow one is roughly equal to .6*110 = 66 feet of throwing distance. Throwing distance is not the same thing as distance from plate since the ball is thrown with arc, but even at an angle of 30 degrees , 66 feet of throwing distance is equivalent to about 55 feet of distance from plate. Put simply, a slow runner can require the ball to be hit up to 50 feet further to score than a fast runner.

What about OF arm? Well, we can get a decent proxy on that from catchers’ pop time. MLB catchers have pop times that vary by up to .2 seconds. But that’s only on a throw of 132 feet. The variance on a longer throw would be more than that. Also, there should be a larger spread between OFs pop time because pop time is much less important skill for an OF than a catchers. So it’s not unreasonable to assume that an OFs pop time can vary by up to .5 seconds. Again, using the calculations above, that could equate to another 50 feet or so of distance to plate.

Judging by this, a fast runner running on a poor arm should be able to score on a ball hit about 100 feet shallower than a slow runner on a good arm. Let’s say I’ve overstated things by a factor of 2 — that still gives a 50 foot spread. That’s very significant, given that the fly balls in question probably vary at most by 150 feet or so (i.e. everybody scores on a ball hit 320 feet; nobody scores on a ball hit 170 feet).

I don’t what significance these rough calculations have on the ultimate issue here. They do suggest that the model presented by Pizza Cutter is flawed, since he ignores speed and arm strength.

My uninformed speculation as to the source of the flaw is that he regresses over the full range of fly balls rather than the marginal range where speed and arm can be expected to make a difference. It would be like using this model to evaluate whether park effects make a difference on home run frequency. Since the vast majority of balls hit aren’t HRs in any park, running a regression on all balls in play would show that the speed and angle of the ball of the bat explains nearly all the variance in whether the ball is a HR or not. That’s true. But it doesn’t mean that park effects aren’t real or significant. They are very real within the subset of balls that MIGHT be HRs.

It’s not that ard to find evidence concerning third-to-home time and OF arm strength. These things are measured by scouts. We don’t need to use speculation about what a world-class sprinter can do versus what you can do, etc.

I don’t take issue with the idea that third-base coaches are too conservative. I have not seen direct evidence of that, but I think that coaches generally are too conservative and in any event I have no reason to doubt your conclusion. In my opinion, though, you have exaggerated your case, making third base coaches appear to be far more conservative than perhaps they are.

I’ve seen too many runners nailed at home in crucial situations to believe that tagging should be nearly automatic. Did you see the decisive play in the deciding game of the NL West last season between Rockies and Padres? If Holliday was one iota slower (say, if he had been Todd Helton), he would have been out for sure.

So count me as one of the outs-are-precious crowd. I don’t like to see outs thrown away base-running. For instance, your analysis doesn’t take into account pitch counts. Every out you lose at the plate represents on average 7-9 pitches that you don’t make the opponent throw, and can represent the difference between a tough inning and a grueling one. Or the difference between the starter going 7 versus getting into middle relief.

However, I would add to your point in the following way: you haven’t taken into account the possibility of throwing error. Sometimes the runner is safe because the ball gets past the catcher, and if there is another runner on base he can advance. This would counsel in favor of tagging even more when there is another runner on base.

As for throwing accuracy, why use analogies to rifle fire? Third basemen and catchers routinely make long throws, usually with good or excellent accuracy. A 132 foot throw isn’t so much shorter than a 250 foot throw that it can’t be used as a rough proxy (i.e. you’d expect the 250 foot throw to be maybe half as accurate). True, catchers have the advantage of making the exact same throw every time, but greater accuracy is required.

In any event, I don’t think throwing accuracy makes much difference to my point. It doesn’t influence the differential between throws made by good arms and/or against slow runners versus those made by bad arms and/or against fast runners. It’s that differential where I lose you. I think your argument proves too much.

P.S. No desire to get into a math discussion, but here’s what I wonder about your regression. Leave S-curves aside for the moment. Your goodness-of-fit tests are going to be affected by whether you include a lot of obvious cases or not (i.e. tag for 400 foot fly, don’t tag for 100 foot fly). You said that runner speed is significant but explains little of the variance. Isn’t that because you’ve designed the study so that little of the variance is really at question? Yes, over all fly balls ever hit in a game, runner speed doesn’t make much difference. But over the class of cases we’re concerned with, i.e. short-medium fly-balls, maybe the runner speed does make a lot of difference?

Outs are precious, but there are some situations that are a good gamble. If we played a game where you paid a dollar to call a coin flip. If you won $2 for getting it right (and nothing if you got it wrong), that’s a fair bet. If I raised the reward to $3, then suddenly, you’d be a fool not to take that bet. You have a 50/50 shot of losing, but the possible rewards outweigh the risks.

PC, I think what BW is saying is that you haven’t fully accounted for the negatives of being thrown it. It’s more than just run expectancy.

I think his estimate of 7-9 pitches is off (that would mean an average game has 189-243 pitches – I haven’t looked in awhile, but I’m pretty sure it isn’t that much – but it is significant – not to mention losing a plate appearance, which keeps the lineup from cycling further.

I agree that runners should be sent more often, but that your study most likely overstates the case.

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