Imagine a world without weekends. No days off at the end of a long string of days doing whatever it is you do to gather your thoughts and rest. (Or at least do a different kind of work. The lawn doesn't cut itself, after all.) The weekend is nice, because even if you aren't "doing anything," you still get a reprieve from your job or classes or whatever you do the rest of the week. But imagine that the weekend was taken away. You just became a professional baseball player.

Baseball is unique—and beautiful, at least from the perspective of the fan—for the fact that teams play almost every single day. I can imagine that players might have different adjectives for it. Players do get the occasional Monday or Thursday off, but it's often a travel day, and travel is hardly a relaxing activity. It's always game day. The term "everyday player" does not exist in basketball or football, because, well, football players have a game once a week. On top of that, the game of baseball itself is one that requires immense amounts of concentration. In the field, a player has to be ready on each pitch to spring into action, despite the fact that on most pitches, he won't be needed.

Baseball is also largely unique in being the one sport where individual players are given scheduled days off on days when the team is playing and they are healthy and would otherwise start. Bench players in baseball should more properly be called "occasional starters," because they get to fill in on the days when the regulars need a breather. Recently, I've been looking into the idea of roster construction and how teams might allocate their roster spaces (and payroll) in a more efficient manner, and this is an interesting area to look into. How important is it that a team have a good bench, not only for the inevitable injuries that will happen during the season, but for the purposes of giving its starters a rest every now and then?

First let's talk about why baseball players need days off. Baseball requires sustained attention in the absence of a stimulus. In other words, baseball players wait around a lot, but it's not like they can mentally check out of the game. (Oh, hi Manny.) Sustaining attention like that is the job of my favorite part of the brain, the pre-frontal cortex (PFC). Like anything else in the body, when the PFC has been overused (in this case, those neurons have been firing a few million too many times), it gets tired and needs some time for repair. We usually call that sleep, but sleep doesn't fix everything, especially when the PFC is going to get another workout in the next night's game. As I've mentioned in other articles that I've written, the PFC also controls things like pattern recognition, higher-order planning, and impulse control, things that after a long day at work, you lose the ability to do as well.

When the PFC is operating at a reduced capacity, the body does not shut down. In fact, the parts of the brain that are responsible for things like breathing, visual perception, and movement are located elsewhere. The PFC is the part of the brain that coordinates them in a logical way. A hitter will still see the ball and can still swing, but he may be working more from muscle memory than an ability to tend to what's going on and then execute a plan. That means that reaction times will be a little slower and decisions won't be as good. In a game that depends on quick reactions and good decisions, that can be a problem. Ever wonder why you're hearing more about the Adderall epidemic in baseball?

Let's look at what happens when players don't get the occasional day off.

Warning! Gory Mathematical Details Ahead!
I took a couple of different measures of how fatigued a player might be. I counted a player as having played in a game if he came to bat, which does exclude guys who played an inning or two as a defensive replacement. I counted the number of games that he played in the seven games prior to a given day’s game (and in the previous 14 and 21 days, although those both behaved in much the same way), the number of days since his last day off (even if a naturally occurring off-day), but capped it at 10 to make sure that it wasn't a DL stint (or a demotion to the minors). I also created a code for whether he had the day before off, whether through an act of the manager, an act of the schedule maker, or a snowstorm that happened to pop up. I also started all analyses in May to allow players to actually build up to being able to play three weeks in a row with little rest.

Using data from 2003-2012, I used the log odds method to control for the likelihood that a plate appearance would end in a specific event (strikeout, walk, HBP, single, extra base hit, HR, or out in play) given the pitcher-batter matchup, and whether the outcome was a general on-base event. I also controlled for the pitcher having a handedness advantage or not. I ran a bunch of binary logistic regressions and slipped the various "tiredness" indicators in to see what happened.

When running this many logit regressions, it's better to talk about variables that were consistently predictive, rather than individual results. For example, the number of days off before today was generally not a significant predictor, nor was having yesterday off. However, the number of games that a hitter had played in the previous seven days (or 14 or 21, it all came out significant) had an across-the-board negative effect on all sorts of hits (singles, doubles/triples, and home runs) and increased the number of outs in play that a hitter made. These variables actually had little effect on walks (there was some evidence that hitters walked a little more when tired) and surprisingly, strikeouts.

The difference between a hitter who had played five games in the last seven days (a day off, plus a Monday travel day?) and a hitter who had played in seven games in the last seven days was about three points in OBP. That's not a lot, but of course, over 600 plate appearances, it's about an extra two on-base events, and that's worth a run or so. Over multiple lineup spots, it can build into a lost win really quickly. A manager who does not rotate his players a bit may be bleeding away value and not even realizing it.

I also took a look at whether a player's age moderated any of these relationships. I ran a series of regressions in which I added in an interaction term between age (as of April 1st) and the tiredness factor under question. Maybe younger players are able to recover more quickly? The answer was no. While, for the most part, the "past seven days" factor continued to be predictive, the interaction term did not hold sway. Older players and younger players seem to experience this brain drain at equal levels.

What it Means
There are a couple of interesting takeaways from these analyses. One is the obvious implications for roster construction. A strong bench not only means that you have guys who can slot in and pick up most of the production that the regular would give you, but it also makes the regular better in the games that he does play.

But let's extend this a little more. Last week, I talked about players who are capable of playing more than one position and who hit well enough to start, and tried to measure the value their versatility can bring to a team. One way that a multi-positionalist can affect a team is to allow the better bench bats to get more playing time—the manager doesn't have to play the guy who hits barely over the Mendoza Line just because he's the only one who can handle shortstop. A multi-positionalist allows the manager to rotate guys in and out of the lineup with the knowledge that he doesn't have to give up as much today just to get his starter a rest. And the more he's able to give his starters a rest, the less value he bleeds away in tiredness. I also argued last week that multi-positional players make platoons easier to construct. While platoons have their own advantages, they also naturally rest the members of the platoon. Maybe we need to revise our valuation of multi-positional players upward.

Another interesting wrinkle (maybe for another episode) is the idea of AL teams forgoing a traditional designated hitter and using the position to give their regulars a rotating "day off," at least from field duties. It's not clear whether DHing actually has the same effect as a day off, but if it does, it's an interesting question as to whether a team would be better off paying for a regular DH (now the position with the highest salary in MLB) or for a really good 10th and 11th man to see semi-regular duty, whilst the team saves on the fatigue penalty paid by its regulars.

There are also the obvious lessons around managing players' workloads. Players need a day off every now and then. If not, they bleed away real value. But there were a few interesting properties of what happens when players don't get rest. One is that while playing more games in the space of seven or 14 or 21 days does sap a player of some of his mojo, there was little evidence that simply giving him the day off allowed him to press the reset button. You can't expect that the next day, he'll improve by leaps and bounds. Fatigue is a cumulative effect. A day off is more akin to recharging a battery than a reset button. The good news is that the extra battery life appears to be available for several days down the road.

Another interesting thing that came out of these analyses was the fact that the effect did not vary by age. Both older and younger players felt the effects of playing in a lot of games. While physical recovery times are generally shorter for younger men, the issue of pre-frontal cortex over-stimulation is one that would likely be felt equally across ages (at least within the 20-39 bracket). I can't state this definitively with the data set that I am working with now, but it leads me to wonder if this is a problem of the brain, rather than the brawn.

Finally, when the results were broken down by event type, strikeouts and walks, often the first outcomes that analysts look at, were unchanged from expectations by a heavy workload. In fact, players who have been playing a lot lately, if anything, tend to walk more. (The effect was not significant.) The effect came from hits that were turning into outs in play. Fly balls that used to go over the wall ending up in the right fielder's glove. Ground balls that used to be smacked hard enough to get past the shortstop now end up in a simple 6-3 putout. Baseball is a game where a small fraction of a second can make a huge difference on a ball. Miss on timing that swing by a little bit, and you'll roll the ball over to the second baseman.

Fatigue, whether physical or mental, is a real issue in the game of baseball. It's hard to do something every day that requires such intense concentration and not get a break from it, and it affects a player's performance. It's always been something that's been known in the game, and now we have some idea of how big the effect is.

Thank you for reading

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Love this article. As I read it, I was wondering if there was any position effect. Did not having a rest affect certain positions more (thinking up the middle more).

Also, it had me thinking about catchers who play more (Yes, Mr. Yost I'm looking at you), if the effect is greater for them. What will 140 games for Sal Perez mean compared to typical catchers who play a more reasonable 110-120.
brilliant analysis. Every time I get frustrated when I see Jim Leyland benching a couple key regulars for one of his AAAA bench role players in a game, I'm going to have to remember this article and cut him some slack.

The problem now is the current trends in specialized bullpens and starters going fewer innings has resulted in a need for larger pitching staffs. teams that got buy with 9 or 10 pitchers in now need 12 or 13. That's three more reserve not pitchers a team no longer has the luxury to carry. This reduces the ability to keep good reserves on hand which in turn reduces the ability to rest regulars which this article shows is a useful tool. For this reason, I believe the rosters should be expanded from 25 to 28 players.

Seems like managers have been practicing these kind of rest days for some time with Catchers.
Great stuff Russell! I have a few questions/comments:

You are comparing how they did in each game (one game only) compared to how they were expected to do given the pitchers they are facing that day (including platoon effects)?

To determine the expected outcome, what time frame did you use for the batter's expected stats - his season stats for that year?

The rest effect might be greater than observed in your regressions since I think we have to assume that players who get days off are sprinkled with players who got a day or two off because of some injury they are battling.

It would be nice to re-run your regressions and including only players who had a day or two off because his team did not play. Then do the same thing with players who had days off when their team DID play. Or code into your regression a dummy variable signifying whether a player had a day or two off by manager choice or because the team did not play.

Finally, whether a manager giving their players a day off or two is an advantage or not depends on how much they lose for that day in the substitution and for how long the rest has a positive effect.

Pizza, from your regressions are you able to estimate how long the effect lasts? You said that two days off in the last 7 was worth 3 points in OBP versus no days off in the last 7. But for how long is that advantage? You are referring only to the game immediately after the 7 day period, right? What about the the next week? Two weeks? Month? Will we see that 3 point advantage hold up? Do we see that advantage slowly bleed away? If so, what does the decay look like? How many days off per week or month of play would we need to sustain a 3 point OBP advantage compared to a player that never takes a day off?

Since all players get days off due to scheduling and rainouts, it is not clear to me that giving players extra days off is an advantage. On that day off, what if the sub is 30 or even 50 points in OBP worse than the regular? Now if you have to give the regular a day off every 2 weeks just to sustain that 3 point advantage, you are breaking even.

So, really, we need to know the gain in X number of games after a day off in order to see whether giving players days off is an advantage, and then, it depends on how much worse the substitute is. Just telling us that on one day there is a 3 point advantage is not very useful, unless I am misunderstanding what the regression says.

One reason why I don't particularly like these regressions, BTW, at least by themselves. You also need to show us a chart of some data that says, "Here is the OBP delta (actual minus expected) of all players who played the previous 7 days. Here is the OBP delta of players who had at least 2 days off in the previous 7."

You should also show us, "Here is the OBP delta for a week, and then 2 weeks, and then 3 weeks, etc., of all players who played 7 straight days prior to the first game in the week, 2 week, 3 week, etc., series, and compare that to the players who got the days off." Again, this enables us to see how long the rest effect is sustained without another day off.

Of course, even then, players who got a day off would likely tend to get more days off, so you would have to control for that as well.

This was a great notion and good research. At the same time, I think it requires a good deal more analysis before we start declaring that managers should or should not do this or that with regard to giving players days off...
I'm curious if any team has tried giving players a *real* day off. If they are supposed to report to the ballpark at 2pm for a 7pm game, let one position player a day report at 5pm. He can sleep in, relax, maybe spend time with the family. That would be the day he would not start, but obviously be in uniform for pinch hitting duties or extra innings. That might result in better recovery.
The difference is 3 points in OBP? No mention of a difference in BA or SLG.

Intuitively I expected a greater difference.

I suspect the difference between the starter and the bench player is more than 3 points in OBP.

True, but given the 60-80 PA that a bench player would need to adequately rest a starter, the two times on base per season lost translates to twenty or thirty points of OBP. In other words:

162 games of .327 OBP by a starting player
138 games of .330 OBP by a starting player +
24 games of .310 OBP by a backup player

What I'm insanely curious about is the degree to which we can quantify the effect Cal Ripken's streak had on his playing numbers. How pronounced is the effect when a player hasn't had a day off in the last month? Three months? Ten years?
I find it difficult to believe that 3 pts of OPB lost by playing more often is not more than lost by resting starters to play, what is often, players at or below replacement level. So maybe the point would be to rest/platoon/rotate the typical players and make damn sure your best players never ever rest. In my opinion, your study proves the opposite of your conclusion - if you are only losing 1 of a percent of your OBP, and starters a lot better than 1 percent of OBP better than reserves, don't EVER sit them. If Miggy loses 1% of his OBP, its 4 points, but he is probably 50-75 points better than his replacement. Perhaps if the data is handy you could look at slugging, which may exhibit more of the fatigue that most managers claim they are trying to avoid by sitting the regulars. But if its just 1% of OBP and everything else is the same, I'd wager you are losing games by replacing starters with replacement level players, whose effects will overwhelm a measily 3-4 points of OBP on average. The other thing you may want to consider is more play over longer periods of time than a week. I'd guess its when players are on the grind, not missing games for weeks at a time, is when you will really see a dip in performance. I'd wager this is particularly true in the dog days of July and August when temperature and humidity take its toll.
Sorry to Shkspr - didn't mean to reword your post - I didn't read down to the bottom before posting.
If a team didn't have a regular DH its top bench players would be the "9th and 10th men" not 10th and 11th
Ron Washington needed this article to be written last year. :-)
This study sounds complementary to the work done here:
I wasn't a big fan of the work done there.
Ah, thanks.