Home field advantage is naturally higher in a hitter's park

The Rockies have always had a huge home-field advantage (HFA) at Coors. From 1993 to 2001, Colorado has played .545 at home, but only .395 on the road. That's the equivalent of the difference between going 89-73 and 64-98. 

Why such a big difference? I have some ideas I'm working on, but the most obvious one -- although it's not that big, as we will see -- is that higher scoring naturally, mathematically, leads to a bigger HFA.

When teams play better at home than on the road -- for whatever reason --the manifestation of "better" is in physical performance, not winning percentage as such. The translation from performance to winning percentage depends on the characteristics of the game. 

In MLB, historically, the home team plays around .540. But if the commissioner decreed that now games were going to be 36 innings long instead of 9, the home advantage would roughly double, with the home team now winning at a .580 pace.

(Why? With the game four times as long, the SD of the score difference by luck would double. But the home team's run advantage would quadruple. So the run differential by talent would double compared to luck. Since the normal distribution is almost linear at such small differences (roughly, from 0.1 SD to 0.2 SD), HFA would approximately double.)

But it's not *always* that a higher score number increases HFA. If it was decided that all runs now count as 2 points, like in basketball, scoring would double, but, obviously, HFA would stay the same. 

Roughly speaking, increased scoring increases the home advantage only if it also increases the "signal to noise ratio" of performance to luck. Increasing the length of the game does that; doubling all the scores does not.

In 2000, Coors Field increased scoring by about 40%. If that forty percent was obtained by increasing games from 9 innings to 13 innings, HFA would be around 20% higher. If the forty percent was obtained by making every run count as 1.4 runs, HFA would be 0% higher. In reality, the increase could be anywhere  between 0% and 20%, or beyond.

We probably have the tools available to get a pretty good estimate of the true increase.

------

Let's start with the overall average HFA. My subscription to Baseball Reference allowed me to obtain home and road batting records, all teams combined, for the 1980-2022 seasons:

         AB        H     2B    3B    HR     BB     SO
------------------------------------------------------
home   3209469 846723 161290 19928 95790 321178 612545
road   3363640 859813 163954 17203 96043 308047 668363

What's the run differential between those two batting lines? We can look at actual runs, or even the difference in run statistics like Runs Created or Extrapolated Runs. But, for better accuracy, I used Tom Tango's on-line Markov Calculator (the version modified by Bill Skelton, found here). It turns out the home batting line leads to 4.79 runs per nine innings, and the road batting line works out to 4.36 R/9.

         AB        H     2B    3B    HR     BB     SO    R/9
-------------------------------------------------------------
home   3209469 846723 161290 19928 95790 321178 612545  4.79
road   3363640 859813 163954 17203 96043 308047 668363  4.36
-------------------------------------------------------------
difference                                              0.43

That's a difference of 0.43 runs per game. Using the rule of thumb that 10 runs equals one win, a rough estimate is that the home team should have a win advantage of 0.043 wins per game, for a winning percentage of .543. 

That's a pretty good estimate -- home teams actually went .539 in that span (51832-44409). But, we'll actually need to be more accurate than that, because the "10 runs per win" figure will change significantly for higher-scoring environments such as Coors. 

So let's calculate an estimate of the actual runs per win for this scoring environment.

The Tango/Skelton Markov calculator includes a feature where, given the batting line, it will show the probability of a team scoring any particular number of runs in a nine-inning game. Here's part of that output:

          home   road
----------------------
2 runs:  .1201  .1342
3 runs:  .1315  .1404
4 runs:  .1282  .1309

From this table, which actually extends from 0 to 30+ runs, we can calculate how many runs it would take for the road team to turn a loss into a win.

Case 1:  If the road team is tied after 9 innings, it has about a 50% chance of winning. With one additional run, it turns that into 100%. So an additional run in a tie game is worth half a win.

How often is the game tied? Well, the chance of a 2-2 tie is .1202*.1342, or about 1.6%. The chance of a 3-3 tie is .1315*.1404, or 1.8%. Adding up the 2-2 and the 3-3 and the 0-0 and the 1-1 and the 4-4 and the 5-5, and so on all the way down the line, the overall chance is 9.7%.
 
Case 2:  If the road team is down a run after 9 innings, it loses, which is a 0% chance of winning. With one additional run, it's tied, and turns that into a 50% chance. So, an additional run there is also worth half a win.

How often is the road team down a run? Well, the chance of a 3-2 result is .1315*.1342, or about 1.8%. The chance of 4-3 is .1282*.1404, another 1.8%. And so on.

The total: a 9.54% chance the road team winds up losing by one run.

What's the chance that the additional run will give the *home* team the extra half win? We can repeat the calculation, but instead of 3-2, we'll calculate 2-3. Instead of 4-3, we'll calculate 3-4. And so on.

The total: only 8.54%. It makes sense that it's smaller, because the better team is less likely to be behind by a run than ahead by a run.

We'll average the home and road numbers to get 9.04%. 

So, we have:

9.7% chance of a tie
9.0% chance of behind one run
----------------------------------------------
18.7% chance that a run will create half a win

Converting that 18.7% chance to R/W:

    0.187 half-wins per run  
=   5.35 runs per half-win 
=   10.7 runs per win

So, we'll use 10.7 runs per win for our calculation.

(Why, by the way, do we get 10.7 runs per win instead of the rule of thumb that it should be 10.0 flat? I think it's becuase the Markov simulation always plays the bottom of the ninth, even when the home team is already up. It therefore includes a bunch of meaningless runs that don't occur in reality. When some of the run currency is randomly useless, it pushes the price of a win higher.

We'd expect that roughly 1/18 of all runs scored are in the bottom of the ninth with the home team having already won. If we discount those by multiplying 10.7 by 17/18, we get ... 10.1 runs per win. Bingo.)

We saw earlier that the home team had an advantage of 0.43 runs per game. Dividing that by 10.3 runs per win, gives us

Predicted: HFA of .42 wins per game (.542)
Actual:    HFA of .39 wins per game (.539)

We're off a bit. The difference is about 2 SD. My guess is that the Markov calculation, which is necessarily simplified, is very slightly off, and we only notice because of the huge sample size of almost 100,000 actual games. 

-------

OK, now let's do the same thing, but this time for Coors Field only.

I could do the same thing I did for MLB as a whole: split the combined Coors batting line into home and road, and calculate those individually. The problem with that is ... well, if I do that, I'll be getting the Rockies' actual HFA at Coors, which is huge, because it includes all kinds of factors that we're not concerned with, like altitude acclimatization, tailoring of personnel to field, etc.

So, I'm going to try to convert the Coors line into an approximation of what the split would look like if it were similar to MLB as a whole.

Here's that 1980-2022 MLB split from above, except I've added the percentage difference between home and road (on a per-AB basis) below:

         AB        H     2B      3B     HR     BB     SO
---------------------------------------------------------
home   3209469 846723 161290   19928  95790 321178 612545
road   3363640 859813 163954   17203  96043 308047 668363
---------------------------------------------------------
diff            +3.2%  +3.5%  +21.4%  +4.5%  +9.3%  -3.9%

I'll try to create something similar for 2000 Coors.  The overall batting line, for both teams, looked like this:

         AB    H   2B 3B  HR  BB  SO     R/9   
---------------------------------------------
Coors  5843  1860 359 56 245 633 933    7.43

Here's my arbitrary split, into Rockies vs. road team, in such a way to keep roughly the same percentage differences as in MLB overall, while also keeping the R/9 roughly 7.43. Here's what I came up with:
     

          AB      H      2B     3B      HR     BB     SO  
--------------------------------------------------------
  home   5843   1884    362     66     249    672    936
  road   5843   1826    350     54     238    615    974
--------------------------------------------------------
  diff         +3.2%  +3.4%  +22.2%  +4.6%  +9.3%  -3.9%

I ran those through Tango's calculator to get runs per 9 innings:

          AB     H    2B     3B   HR    BB    SO     R/9
---------------------------------------------------------
  home   5843  1884  362     66  249   672   936    7.783
  road   5843  1826  350     54  238   615   974    7.071
---------------------------------------------------------
  avg                                               7.427
---------------------------------------------------------
  diff                                              +.712

Next, I ran the runs-per-game distribution calculation to get a runs-per-win estimate. (I won't go through the details here, but it's the same thing as before: calculate the probability of a tie, then a one-run home win, then a one-run road win, etc.)

The result: 14.37 runs per win. 

As expected, that's significantly higher than the 10.7 we calculated for MLB overall. (Adjusting 14.37 for the superfluous bottom-of-the-ninth gives about 13.6, so, if you prefer, you can compare 13.6 Coors to 10.1 overall.)

The difference of .712 runs per game, divided by 14.43 runs per win, gives an HFA of 

0.0495 wins per game

Which translates to a home winning percentage of .5495. 

Comparing the two results:

.542 home field winning percentage normal
.549 home field winning percentage Coors
-----------------------------------------
.007 difference

The difference of .007 is worth only about half a win per home season. Sure, half a win is half a win, but I'm a little disappointed that's all we wind up with after all this work. 

It's certainly not as much of an effect as I thought there would be before I started. Even if you deducted this inherent .007, it would barely make a dent in the Rockies' 150 percentage point difference between Coors and road. The Rockies would still be in first place on the FanGraphs chart by a sizeable margin -- 42 points instead of 49.

Looked at another way, an additional .007 would move an average team from the middle of the 29-year standings, to about halfway to the top. So maybe it's not that small after all.

Still, our conclusion has to be that the Rockies' huge HFA over the years is maybe 10 percent a mathematical inevitability of all those extra runs, and 90 percent other causes.

A study on NBA home court advantage

Economist Tyler Cowen often links to NBA studies in his "Marginal Revolution" blog ... here's a recent one, from an August post. (Follow his link to download the study ... you can also find a press release by Googling the title.)

The study used a neural network to try to figure out what factors are most important for home (court) advantage (which I'll call "HCA"). The best fit model used twelve variables: two-point shots made, three-point shots made, and free throws made -- repeated for team at home, opposition on road, team on road, and opposition at home.

The authors write, 

"Networks that include shot attempts, shooting percentage, total points scored, field goals, attendance statistics, elevation and market size as predictors added no improvement in performance. ...

"Contrary to previous work, attendance, elevation and market size were not relevant to understanding home advantage, nor were shot attempts, shooting percentage, overall W-L%, and total points scored."

On reflection, it's not surprising that those other variables don't add anything ... the ones they used, shots made, are enough to actually compute points scored and allowed. Once you have that, what does it matter what the attendance was? If attendance matters at all, it would affect wins through points scored and allowed, not something independent of scoring. And "total points scored" weren't "relevant" because they were redundant, given shots made.

------

The study then proceeds to a "sensitivity analysis," where they increase the various factors, separately, to see what happens to HCA. It turns out that when you increase two-point shots made by 10 percent, you get three to four times the impact on HCA compared to when you increase three-point shots made by the same 10 percent.

The authors write,

"[This] suggests teams can maximize their advantage -- and hence their odds of winning -- by employing different shot selection strategies when home versus away. When playing at home, teams can maximize their advantage by shooting more 2P and forcing opponents to take more 2P shots. When playing away, teams can minimize an opponent's home advantage by shooting more 3P and forcing opponents to take more 3P shots."

Well, yes, but, at the same time, no. 

The reason increasing 2P by 10 percent leads to a bigger effect than increasing 3P by 10 percent is ... that 10 percent of 2P is a lot more points! Eyeball the graph of "late era" seasons the authors used (I assume it's the sixteen seasons ending with 2015-16). Per team-season, it looks like the average is maybe 2500 two-point shots made, but only 500 three-point shots.

Adding 10 percent more 2P is 250 shots for 500 points. Adding 10 percent more 3P is 50 shots for 150 points. 500 divided by 150 gives a factor of three-and-a-third -- almost exactly what the paper shows!

I'd argue that what the study discovered is that points seem to affect HCA and winning percentage equally, regardless of how they are scored. 

------

Even so, the argument in the paper doesn't work. By the authors' own choice of variables, HCA is increased by *making* 2P shots, not my *taking* 2P shots. Rephrasing the above quote, what the study really shows is,

"When playing at home, teams can maximize their advantage by concentrating on *making* more 2P and on forcing opponents to *miss* more 2P. That's assuming that it's just as easy to impact 2P percentages by 10 percent than to impact 3P percentages by 10 percent."

But we could have figured that out easily, just by noticing that 10 percent of 2P is more points than 10 percent of 3P.

------

The authors found that you increase your HCA more with a 10 percent increase in road three-pointers than by a 10 percent increase in road two-pointers. 

Sure. But that's because, with the 3P, you actually wind up scoring fewer road points. Which means you win fewer road games. Which makes your HCA larger, since winning fewer road games increases the difference between home and road. 

It's because the worse you do on the road, the bigger your home court advantage!

Needless to say, you don't really want to increase your HCA by tanking road games. The authors didn't notice that's what they were suggesting.

I think the issue is that the paper assumes that increasing your HCA is always a good thing. It's not. It's actually neutral. The object isn't to increase or decrease your HCA. It's  to *win more games*. You can do that by winning more games at home, increasing your home court advantage, or by winning more games on the road, decreasing your home court advantage.

It's one of those word biases we all have if we don't think too hard. "Increasing your advantage" sounds like something we should strive for. The problem is, in this context, the home "advantage" is relative to *your own performance* on the road. So it really isn't an "advantage," in the sense of something that makes you more likely to beat the other team. 

In fact, if you rotate "Home Court Advantage" 360 degrees and call it "Road Court Disadvantage," now it feels like you want to *decrease* it -- even though it's exactly the same number!

But HCA isn't something you should want to increase or decrease for its own sake. It's just a description of how your wins are distributed.

A model for explaining home field advantage between sports

It occurred to me that it might be possible to predict, or explain, the difference in home field advantage between different sports, based on their rules and outcomes.  In this post, I'll just talk about the "absolute" home field advantage, in terms of goals or points.  (The translation to winning percentage is easy after that, but I'll save that for a future post.)

Take a look, and let me know what you think.

----

Here's the home field advantage (HFA) for three different sports, in terms of goals or points.

0.453 - Premier League Soccer (2010-11)
4.000 - NBA (estimate)
0.783 - NHL (1980-81 to 1984-85)

As expected, the HFA is highest for basketball, where the most points are scored, and lowest for soccer, where the fewest "points" are scored.  Still, they're very different in terms of rates.  Here are the percentages by which the home team outscores the visiting team:

38% - Premier League (+0.453 home goals per 1.1737 road goals)
4% -- NBA (roughly, +4 per 100)
27% - NHL (+.783 per 2.937)

These HFAs are all over the place.  In basketball, the home team only outscores the visiting team by 4 percent.  But, in the NHL, it jumps to 27%, and in soccer, it's almost 40%!

Why the big differences? 

It has nothing to do with the length of the game.  If the home team scores 4% more points over 48 minutes, you can also expect it to score 4% more points over a minute, a quarter, or a season.  (It's the percentage of how many more *wins* the home team gets that depends on game length, but, again, that's not what we're discussing in this post.)

So, what is it then?  There are probably many contributing factors, but I think the biggest has to do with the structure of the individual games.  That's because it's easy to get a higher or lower HFA just by changing the rules.

------

Start by looking at the NBA, where the HFA is about 4 points a game.  Let's change the way basketball works, to move that difference away from 4 points.

In fact, let's do that while keeping many aspects of the game constant.  We'll stay with a game where each team gets 100 possessions, has to throw a ball through a hoop on a basketball court, and has an average score of 100.  We'll just change the "internal" rules.

------

Suppose we change the game to consist of only foul shooting.  Each possession, the team gets two foul shots.  If it sinks them both, it gets two points.  Otherwise, it gets zero.  We can assume the average player shoots 71%, so that the probability of two straight makes is almost exactly 50%.  (If 71% seems a bit low, just imagine that we make the hoop a bit smaller at the same time we change the rules.)

We have empirical data that lets us figure out what HFA would be, thanks to King Yao, who compiled home and road free-throw percentages for a few recent NBA seasons.  The numbers were:

75.95% home team
75.72% visiting team

For the chance of making two straight shots, then, we can just square those numbers:

57.68% home team
57.34% visiting team

The difference is 0.34%.  Over 100 possessions, that's .34 extra scores, or around 0.7 extra points. That's much smaller than the 4 point HFA in "real" basketball.  We've reduced HFA by 80 percent just by changing the rules!

------

How can we construct a game where HFA is higher?  Again, that's easy -- try "double or nothing" basketball. 

In that game, when you score a field goal, you don't get the two points yet.  Instead, you immediately get a second possession, and you have to score on that one too.  If you do get two in a row, you get 4 points.  If you don't get the second one too, you get zero.

A game consists of 100 possessions for each team (so each team will get somewhere between 100 and 200 attempts to make a field goal).

In this game, the HFA will be roughly double.  How do we know?  Well, the real life HFA is 4 points, and each team gets roughly 100 attempts.  So, we can guess that, in normal basketball, the home team might score on around 52% of attempts, while the visiting team will score only on 50%. 

But, now, each team has to make two in a row.  The home team will do that around 27% of the time (52% squared), while the visiting team will be at 25% (50% squared).  That's still two extra scores per game, but now each score is worth four points.  So, instead of winning 104-100, the home team will win 108-100.

The change in the rules has increased the HFA from 4 points to 8 points -- from 4% to 8%.

------

So: three different hoops games, three different HFAs.  That shows that you have to examine the rules in order to understand where HFA is coming from.  It can't be just crowd influence, or referee bias, or familiarity with the home court, or anything like that.  Those are things that could *cause home field advantage to exist*.  But they aren't things that could, on their own, cause the level of home field advantage to *vary between sports*.  For that, you need to examine the rules.

------

So, is there a factor that explains how the HFAs change for the three games? 

It seems to me that the answer is: the level of "compounding" of events, the number and difficulty of the things that all have to go right for you to score.

For the "two in a row" game, you need to score twice in a row, not just once.  If you have a 4% advantage on each one, you'll have an 8% advantage on two of them compounded.  (1.04 squared is about 1.08.)

For the foul shooting game: In "real" basketball, there's more than just shooting.  To score a field goal, you have to do a whole bunch of things right.  For instance, you have to (a) pass the ball around accurately; (b) deke out a defender enough to get a good shot; (c) have the other members of your team distract the other defenders so they can't block; and then (d), take an accurate shot.

That's four things that might all have to go right.  From the discussion above, we know the HFA for two consecutive foul shots is 0.34%.  Suppose each of those four of those things, from (a) to (d), have that same 0.34% advantage.  Then the home team gets an advantage of roughly four times 0.34%, or roughly 1.4 percent.

That hypothetical only gets us to 1.4 percent, not to 4 percent.  That suggests that more than four compoundings are necessary -- maybe as many as 10.  That's certainly reasonable.  Foul shooting seems to be something that's simpler than most other basketball skills. We assumed, for convenience, that "taking an accurate shot" was exactly as complex as foul shooting ... but it might be twice as complex.  You have to shoot accurately, but first you also have to judge the shot.  If the other three things are also twice as complex, which doesn't seem implausible, then we have eight compoundings. 

Anyway, the point is not to get this particular example to work out perfectly, but to show that it's at least a decent approximation.

-------

The "compounding" explanation also seems to work if you compare hockey to basketball.

In basketball, the net is unguarded.  In hockey, there's a goalie trying to stop the puck.  The goalie has his own HFA, while the hoop presumably does not.  So, that's at least one extra compounding in favor of hockey.

In basketball, it's difficult to force a turnover; it's a fairly rare event.  In hockey, it's easy, since physical play is allowed -- you just run into the puck carrier, if you can, and dislodge him.

So, in hockey, part of the goal scoring process is avoiding checks from the opposition.  If four players touch the puck before a shot, and each one has the same chance of losing the puck as the entire team has in a basketball possession ... then you have three extra compoundings, since there's an additional HFA for each of the four players.

In soccer, it's even more extreme: it isn't unusual to take 10 or 15 passes before you get a decent chance at a shot.  So, ten things have to go right.  If each pass in soccer has the same chance of being intercepted as a pass in basketball, but soccer requires three times as many passes before a goal ... well, now you have three more compoundings.

This is all theoretical, of course, but we can check the numbers to see if they're reasonable.

In soccer, the absolute HFA was 38 percent more goals.  In basketball, it was only 4 percent.  To get from 4 percent to 38 percent, you need about eight times as many compoundings (since 1.04 to the eighth power equals approximately 1.38).

Is eight compoundings reasonable?  I think it is, because we can get a similar result another way.

In the NBA, about 50 percent of possessions result in a score.  In soccer, it's probably, what, around 2 percent?

If soccer had two compoundings to every one basketball compounding, the scoring rate would still be 25 percent (you'd have to do something with a 50 percent success rate, twice).  If it had three compoundings, it would be 12.5 percent.  Four compoundings, about 6 percent.  Five compoundings, 3 percent.  Six compoundings, 1.6 percent, and we're there.

So, a naive estimate is that it to score a goal in soccer, you have to be skilled enough to do what it takes to score a goal in basketball, six consecutive times.

We were expecting 8 compoundings from the "compare HFA" argument.  The "probability of scoring" method suggests 6 compoundings. 

Not bad!  Those two estimates, 8 and 6, are pretty close.  Why aren't they closer? 

Well, it could be that some of my estimates were off, like the one where I guessed that 2 percent of soccer possessions score.

Or, it could be that there's a large difference in competitive balance between the two leagues.  The more lopsided the talent, the lower the HFA (when the better team is so good that it always wins, the HFA is obviously zero). 

But, most importantly, it could be that there are factors outside of "compoundings".  For instance, referee bias, which need not be anywhere near the same order of magnitude as player HFA.  Actually, that could very well be it: in soccer, the referee can have a very large impact on the game.  In the NBA, a blown call is worth a couple of points out of 100.  But, in soccer, a blown penalty call could be one goal out of two.

------

Anyway, if you buy the idea that these two estimates of compounding should be the same, that suggests a way to get a rough estimate of what HFA should be in other sports, at least for sports that are similar to basketball/hockey/soccer, in the respects we used in our arguments.  Specifically:

-- you can divide the game into possessions

-- each team gets roughly an equal number of possessions

-- you can only score once per possession

-- you can estimate a probability of an average team scoring on each possession

-- what keeps you from scoring at will is a defense that's similar to defenses in basketball/hockey/soccer and also subject to HFA (which excludes, say, foul shooting or skills competitions)

-- referee bias is roughly the same order of magnitude as for basketball/hockey/soccer.

To get an estimate, you start with a known sport and a known HFA, and you adjust it by the differences.  Let's use the NBA as our reference point.  It has a 50% percent chance of scoring on each possession (0.5), and an absolute HFA of 4% (1.04). 

We now adjust for the number of compoundings based on the difference in probability of scoring on a possession.  That leads to this formula:

Let p be the probability of scoring on a single possession in your particular sport.  Then:

HFA = 1.04 ^ [log(p)/log(0.5)]

(Checking that it works for the "double or nothing" basketball variation: p=0.25 gives HFA=1.08, which is 8% more points, which is correct.)

That's your rough estimate of HFA.  I emphasize it's *rough*.  You'd still need to adjust it (slightly) up if your league has more competitive balance than the NBA, or down if it has less.  And you'd have to adjust it up (perhaps substantially) if you think the effect of referee bias on HFA is higher, or down if you think referee bias is lower.

And, of course, there might be other factors I haven't thought of.


-------

Does someone want to try this for other leagues or other sports, and see how close it comes?  I'd be curious to see the NLL, which has, maybe, 12 goals per team per game.  The problem is estimating possessions, which is hard for lacrosse but easier for, say, the WNBA.

Factors influencing home field advantage

The more games in a season, the more likely the best teams will rise to the top, and the worse teams will fall to the bottom.  That's just common sense, and the law of large numbers.

Similarly, the more innings in a game, the more likely the best team will win.  If you put the whole season into a single 1,458-inning game, there's no doubt that (for instance) the Yankees would beat the Twins.

Home field advantage (HFA) is one of those things that makes teams better.  And so, the longer the game, the more likely the home team's advantage will show up in the results.  HFA for a 3-inning game would be smaller than HFA for a 9-inning game.

So, it's easy to compare HFA within one given sport.  But how do you compare two?  According to "Scorecasting," from 1989 to 1999, in the NBA, home teams went .605.  In the NHL, they went .557.  Why the difference?

In the past, I've used an argument that I now somewhat regret.  It went something like this: "We know that a longer game means a higher HFA.  Therefore, basketball must be "longer" than hockey in some sense.  Perhaps there are more confrontations between players, or something, which allows the HFA to expose itself more easily."

But now, I think, that line of thinking is too vague.  It's almost a circular argument.  "Why is the NBA higher?"  "Because the game is longer."  "What do you mean by longer?"  "I don't know exactly, but it's the attribute of NBA games that makes home field advantage bigger."

It's like, suppose we don't know what causes lung cancer, except smoking.  And then we find a country that has a high rate of lung cancer, even though they don't smoke much.  Do we say, "that country must be somehow 'cigarettier'?"  That would be silly.

And, in any case, we can do better.  There are identifiable reasons why the NBA home record is higher than the NHL home record.  They don't solve the problem entirely, but at least they're concrete factors.

-------

I'm going to start by calculating the theoretical HFA for the National Hockey League, step by step.

From 1980-81 to 1984-85, the home team outscored the visiting team by .70619 goals per game. 

The home team scored an average 4.264 goals per game.  Since it's typically assumed that goals have a Poisson distribution, the SD of goals per game is the square root of that, or 2.065.  (It's a property of the Poisson distribution that the SD is the square root of the mean.)

The visiting team scored an average of 3.557 goals, for an SD of 1.886.

So, the SD of (home team - visiting team) is 2.797.

Therefore, if the two teams were exactly equal, the goal differential would be a normal curve with mean 0, and SD 2.797.

But the HFA makes those two teams unequal, by .76019 goals.  Divide that by 2.797 and we see that they're unequal by 0.271 of an SD.  Therefore, the home team wins if the random outcome is greater than -0.271 SDs.

Going to a normal distribution table, that probability is 0.607.

In those actual NHL games, the home team actually had a winning percentage of .592.  Not bad!

Why is our theoretical estimate too high?  Well, one reason is that our calculation assumed two equally talented teams.  But in real life, there are always differences in talent, sometimes large ones..  And HFA decreases as the talent gets more uneven.  (If an .000 team plays a 1.000 team, the HFA is obviously zero.)

So, that's one reason our estimate is too high.  It's probably not all of it.

-------

Now, let's go back to game length.  We all agree that if we increased the length of an NHL game, say from 60 minutes to 120, the HFA would increase. 

Suppose the league does that.  But, at the same time, it decides to also reduce the number of goals scored.  Now, every time a goal is scored in the six-period game, the referee flips a coin.  If it's heads, the goal stands.  If it's tails, the goal doesn't count.

That means the average game score is the same.  The distribution of goal differential is the same.  The only thing that's different is the length of the game -- the number of confrontations between players, and the length of time one team has to show it's superior to the other team.

So, we should expect the HFA to go up, right?

It doesn't.  It stays the same.  (actually, it goes down a bit, but never mind.)

In the old NHL, home goals had a Poisson distribution with mean 3.557.  And in the new NHL, home goals *also* has a Poisson distribution with mean 3.557.  The distributions are identical, because Poisson applies (as an approximation) to any rare events.  Whether it's over 60 minutes or 120 minutes, 3.557 goals qualifies as rare.

So if we repeat the calculation for HFA, every step is exactly the same as before!  And so we get the same answer.

-------

So if "length of the game," in terms of confrontations or action, doesn't matter, what *does* matter?

Goals.  The more goals scored, the higher the Poisson mean, and so the higher the SD of game results.  That means more randomness, a wider spread.  If there's a wider spread, that means the HFA of 0.706 goals is smaller relative to luck.  And so, it has less opportunity to express itself, and we get a lower HFA.

Just to give you an example: suppose the average goals per team increases to 6.  That means the SD of a game difference is 3.46.  The difference of 0.706 goals is now only .204 of an SD, which gives you 58.1 percent of a normal curve.  So the HFA drops to .581.

Goal difference is part of the reason that I got a theoretical HFA of .592, but "Scorecasting" showed an actual HFA of only .557.  The Scorecasting study used the ten seasons ending 2009, when scoring was historically low.  I used 1980 to 1984, when scoring was historically high.

-------

So, have we found an answer?  Can we say that one reason why basketball HFA is higher than hockey HFA, is that basketball has so much more scoring?  Well, yes and no.  Yes, scoring is part of it, but, no, we can't use this particular argument, because basketball is not Poisson.

Indeed, non-Poisson-ness is one of the factors boosting the NBA home field advantage.  As it turns out, the farther the distribution gets from ideal Poisson, the lower the random variance.  And lower randomness boosts HFA, by providing less noise to drown out the HFA's signal.

If the NBA switched to Poisson, by making the game 20 times longer, and making baskets 20 times harder to achieve, HFA would go down, even though scoring would stay the same. 

Well, not necessarily.  It depends whether teams change the way they play under the new system.  The home advantage in the NBA is, what, 3 or 4 points a game?  If the hoop became 20 times harder to hit, that 3 or 4 points might change to something else entirely, and we'd have to recalculate.

--------

So we have two factors affecting HFA so far, all else being equal:

1.  Non-Poisson-ness increases HFA.
2.  More scoring decreases HFA.

--------

You can probably think of more factors.  I've got a couple I'll save for a future post.

Presentation on home-field advantage

I've posted the slides for my SABR presentation on home-field advantage (.ppt).

Nothing new here ... everything in the slides I've posted about previously.

Home field advantage on pitch calls, by count

Did a bit of last minute research before putting together my presentation on home field advantage (HFA) for the SABR convention.

In "Scorecasting," Toby Moskowitz and Jon Wertheim wrote that HFA on ball-strike calls varies with the importance of the situation. They said that in clutch plate appearances, HFA is very high -- but, when it doesn't matter much, HFA actually goes the *other* way, and visiting pitchers actually get the benefit of more called strikes than home batters. They concluded that biased umpires are favoring the home team, but trying to compensate the visiting team by calling more strikes for them when it's not as important.

A few months ago, MGL did a study, and found some confirmation for Scorecasting's results. He did find that HFA went up with leverage. However, he didn't find any situations in which the home team actually had an advantage -- just situations in which they had less of an advantage than usual.

So I tried the same thing today (but not as rigorously). I used 2000-2009 Retrosheet data, and I got similar results.

Overall, not looking at leverage yet, the home pitchers got 0.6 percentage points more strikes than the visiting pitchers. (Specifically, the visiting team had 31.2% of their called pitches ruled strikes, but the home team had 31.8% of theirs ruled strikes.)

In certain higher-leverage situations (for which I used 8th inning or later, score tied), the difference was higher -- 1.2 percentage points. In another higher-leverage situation (9th inning or later, tying run at the plate), the difference was also higher -- 1.0 percentage points.

But in lower-leverage situations (one team leads by 5 runs or more), the difference was only 0.3 points.

Summary:

0.3 -- one team leading by 5+ runs
0.6 -- all situations
1.0 -- ninth inning+, tying run at bat
1.2 -- eighth inning+, score tied

Another thing I did is, for all these situations, I computed the HFA in terms of the outcomes of the plate appearances. Here are the home team advantages by wOBA points:

.0010 -- one team leading by 5+ runs
.0013 -- all situations
.0018 -- ninth inning+, tying run at bat
.0025 -- eighth inning+, score tied

As expected, an excellent correlation between HFA on ball/strike, and HFA on eventual outcome.

But here's something interesting: the home/road difference on what percentage of pitches were swung at (including foul balls and balls in play):

0.48 -- one team leading by 5+ runs
0.54 -- all situations
0.88 -- ninth inning+, tying run at bat
0.59 -- eighth inning+, score tied

For instance, overall, home teams swung at 44.9 percent of pitches, but road teams swung at 45.4 percent of pitches.

So, not only did home teams have fewer strikes *called* against them (first table), but they also had fewer *swings* (third table). That suggests that visiting teams actually throw fewer strikes than home teams, since this results holds even on pitches where the umpire has no say.

But, you could argue otherwise. It's possible that the swing difference is because the home batters know they're going to get more marginal calls in their favor, so they don't swing on iffy pitches in order to work a walk. It's also possible that batters are worse on the road, and they can't tell a good pitch from a bad pitch quite as well as they can at home.

So I'm not sure if we can draw any conclusions from this, but I thought it was worth mentioning.

------

Anyway, that does confirm the "Scorecasting" basic findings. But it occurs to me that what might be causing this is just the different ball/strike counts.

As I said above, when an umpire called a home pitch, it was a strike 31.8 percent of the time. But, I checked, and when an umpire called a home pitch *on an 0-0 count*, it was a strike 43.0 percent of the time. That's a big difference. Maybe it extends to home/road differences too?

It does. The HFA was also much bigger on 0-0: instead of 0.6 percentage points, it was 0.9 percentage points.

So maybe HFA is lower in low-leverage situations just because when a game is a blowout, teams pitch differently and you get a different frequency of the different counts. So, what I did was break down all pitches by count, and by leverage group. The leverage groups were:

High ..... 8th inning or later, 0-1 run difference
Low ...... One team leading by 6+ runs
Average .. All other plate appearances.

Here are the results, in percentage points of HFA (called strikes as a percentage of all called pitches). Standard errors are in parentheses.

-------------------------------- Leverage -----------------
----------------- Average -------- High ----------- Low ---
-----------------------------------------------------------
0-0 count ..... 1.03 (0.09) ... 1.02 (0.20) ... 0.66 (0.28)
0-1 count ..... 0.63 (0.14) ... 0.77 (0.31) ... 0.79 (0.43)
0-2 count ..... 0.51 (0.20) ... 0.62 (0.45) .. -0.13 (0.64)
1-0 count ..... 0.92 (0.14) ... 1.54 (0.31) ... 0.38 (0.44)
1-1 count ..... 0.65 (0.15) ... 0.35 (0.34) .. -0.22 (0.48)
1-2 count ..... 0.83 (0.17) ... 0.72 (0.37) ... 0.56 (0.53)
2-0 count ..... 0.98 (0.23) ... 1.11 (0.52) ... 1.18 (0.75)
2-1 count ..... 0.80 (0.20) ... 0.75 (0.46) ... 0.22 (0.65)
2-2 count ..... 0.56 (0.18) ... 0.44 (0.40) ... 1.05 (0.58)
3-0 count ..... 1.07 (0.37) ... 2.35 (0.86) ... 1.15 (1.20)
3-1 count ..... 0.90 (0.30) ... 1.31 (0.69) ... 0.64 (0.97)
3-2 count ..... 0.56 (0.22) ... 1.25 (0.50) ... 1.77 (0.72)

Is there evidence here that HFA depends on leverage? If you compare the average leverage to the high leverage, you get that the high-leverage situations have a higher HFA in 7 out of the 12 cases -- not much more than average. Comparing average to low, you get more HFA for the average situations again 7 out of 12 times. And, comparing high to low, the "high" only win 8 out of 12.

Doesn't seem like much. But I think I've just diced up the data so finely that you can't see the real pattern any more. It looks like all three of the three lowest differences do appear in the low-leverage column (although that could be partly because the SDs are high there, so you expect extreme more values than in the other columns).

Here's the equally weighted average of all three columns, each column weighted by the smaller of the frequencies (home, road) in column 1:

0.82 (~ 0.04 SD) overall
0.92 (~ 0.10 SD) high leverage
0.58 (~ 0.14 SD) low leverage

So there is something there, although smaller than it looked before adjusting for count. But the differences are not statistically significant, although the low-leverage one is close.

Conclusion: from 2000 to 2009, home teams were somewhat more likely to get a strike call in higher-leverage situations than in lower-leverage situations. This is significant only at approximately p=0.1.

How can we separate referee bias from other sources of home-field advantage?

Are umpires and referees biased in favor of home teams?

The evidence seems to say they are. In "Scorecasting," Tobias Moskowitz and Jon Wertheim listed many bits of evidence that suggest such bias. While I don't agree with them that refereeing is the *only* thing that causes home-field advantage, they make a pretty good case that it at least causes *some* of it.

Next month, I'll be giving a presentation on this topic at the SABR convention in Long Beach. My tentative plan is, first, to show evidence (mostly baseball) showing umpire bias. For that, I'll use examples from "Scorecasting." In addition, there's a great study from John Walsh in the 2011 Hardball Times book, which uses Pitch f/x data to show that umpires miscall the strike zone in favor of the home team by about 0.8 pitches a game (which is worth .015 in winning percentage, or about one-third of observed HFA). And, there was some work done my Mitchel Lichtman at "The Book" blog.

Second, I plan to show evidence that some of the performance differential is very unlikely to be caused by umpires or referees. For instance, suppose I find out that when putting the ball in play with nobody on and nobody out on a 2-1 count, home teams have significantly better outcomes than road teams. If that's the case, it can't have much to do with umpires, right? Because if the ball is put in play, the umpire doesn't have to make a call, except, perhaps, on a safe/out play, and we know those are rarely missed.

That sounds reasonable, but it's not necessarily right. Perhaps the reason road teams have worse ball-in-play outcomes (if in fact, they do) is that, knowing that umpires are biased against them, they have to swing at worse pitches to avoid taking an unwarranted strike. For that reason, it could be that the effect of umpires is much more than the 0.8 pitches a game. Indeed, it's logically possible that the *entire* home field advantage is caused by umpires. For instance, if the batter knows that he'll *never* get a called ball on a certain outside pitch, he'll swing at all of them. The bias will then show up only in the outcome of balls in play, and in extra (swinging) strikes, but not in extra called strikes.

So, you can't really be sure that the 0.8 figure isn't a biased, lowball estimate of the umpire's bias.

Still, you can probably argue that a large extra effect of this sort is implausible. If two-thirds of umpire bias was hidden by batters compensating, it would leave some evidence. You'd see a lot more swinging strikes for the visiting batters. You'd see a lot more balls in play for the visiting batters. You'd see lower pitch counts for the home pitcher, because the visiting hitters would be making contact more often.

Or would you? Maybe even those effects are too small to be seen with the naked eye, as it were. If it took many years of Pitch f/x data, and many years of waiting for John Walsh to notice an umpire bias of .015, who's to say that an additional umpire bias of .030 can immediately be seen in other stats?

Does anyone see a way out of this dilemma, or have any suggestions on what kind of evidence would help resolve the issue? The only one I can think of is NBA foul shooting, where there's absolutely no referee influence, but still a significant HFA. For baseball, though ... there's nothing like that that I can think of.

I might just have to bite the bullet and go with the argument I'm making here: that a large hidden umpire bias effect is implausible, but not impossible.

But if you have any ideas or comments, let me know.

Are soccer referees biased for the home team?

A little while ago, one of the economics blogs I read (I forget which one) posted a link to a recent (2007) home field advantage paper. The paper is "Referee bias contributes to home advantage in English premiership football," by Ryan H. Boyko, Adam R. Boyko, and Mark G. Boyko. It's a free download.

It's actually pretty clever what they did. What they figured is this: if home field advantage (HFA) is caused by referee bias, it stands to reason that different referees would have different levels of such bias. So they checked the individual HFAs of different referees, to see if the distribution matched what would be expected if they were the same, and any differences were random error.

At least I *think* that's how they did it. They used an "ordinal multinomial regression model," and, unfortunately, I can't explain that because I don't know how it works. The results look pretty much like a normal regression. They tried to predict goal differential for every game. To do that, they used crowd size and percentage of available seats filled. They had dummy variables for year. Most importantly, they also included expected goals for and against for the home and visiting teams, where "expected" means average for the games of that season, not including that game (but not adjusting for the fact that both averages will be one game biased for home/road). And, of course, they used the identity of the referee for the game.

From all that, they got that referees were collectively statistically significant, at p=0.0263. But that was the result of a Chi-squared test on the entire group of 50 referees, so there's no coefficient we can look at. So, we know the referees have statistically significant differences in HFA, but we can't figure out *by how much*.

It turns out, however, that the significance goes away if you omit one outlier referee from the study. That referee's HFA is a huge 1.2 goals. The mean was 0.412, and no other referee was higher than 0.7. When the authors exclude that one referee from the study, the statistical significance jumps to p=.18.

The authors provide a chart (Figure 1) with the HFAs of all fifty referees. From that chart, you can't really tell if the referees are all the same or not -- you need the significance test. To the naked eye, the differences look pretty consistent with what you'd expect if the differences are just random (except, of course, for the one outlier).

Only two out of fifty referees have negative HFAs (that is, they refereed games where the visiting team outscored the home team, on average). However, it does appear that the referees with lower HFAs are a little farther from the average han the referees with higher HFAs, for what that's worth.

So the question remains: *how much* is the difference in referees, as compared to HFA? We don't know. It would have been nice if the authors had given us some variances: how much would the variance be if there were no bias? Then we could subtract the theoretical from the observed, and conclude something like, "the variance of HFA bias amongst referees is X".

But, as I said before, the authors were so concerned about showing there IS bias that they didn't calculate HOW MUCH bias they actually observed.

-------

One thing to keep in mind is that while it's possible to estimate the *differences* among referees, there's no way to know the actual level of bias. It could be that all referees are biased, but a bunch just happen to be a little less biased. Or it could be that almost all referees are unbiased, but a select few were able to overcome that, and it's those unbiased referees that are causing the statistical significance.

It's like, suppose one interviewer wants to higher all three of the black candidates interviewed for a position, and another wants to hire none of them. You can tell one or both of them is biased. But is it that one interviewer doesn't like blacks? Is it that the other interviewer is practicing affirmative action? Or is it a combination of both? You can't tell unless you know enough about the candidates to figure out which of them "should" have been hired by an unbiased interviewer.

Same thing here. We need to know what the HFA "really" would be if all referees were unbiased. But that, we don't know, and there's no real way to know from this study.

The authors of the paper acknowledge that, but nonetheless argue for the position that HFA is all refereeing, and that most referees are biased:

"Certainly, the [many referees biased option] seems more reasonable, especially given the floor near gD = 0 (no home advantage)."

I don't really understand that, and I don't really agree with it ... I think the most plausible assumption is to assume bias among the fewest number of referees that seems reasonable. Actually, I think the most plausible assumption is to note the huge outlier, and the lack of significance of the distribution of the others, and reach the tentative conclusion that (a) there's no evidence of bias in general, but (b) we should really look closely at the outlier referee, to see if we can figure out what's going on there.

-------

Finally, even if there IS a difference in referees, it might not be bias in favor of the home team. It could just be style of refereeing. According to Table I of the paper, home teams score a lot more goals on penalty kicks than visiting teams do. Overall, the difference was 0.044 goals per game.

Suppose that certain referees are just less likely to call penalties -- say, 1/3 less likely. That would reduce the HFA on penalties from 0.044 goals, down to 0.03 goals -- a difference of 0.015 goals.

It's not much, but add in differences in yellow cards, red cards, free kicks, and so on, and see what you get. It could turn out that a significant part of variability in referee HFA could be referee characteristics that have nothing to do with the home team at all.

-------

Hang on -- maybe we *can* get at least an upper limit for how much HFA the referees could cause. In Table 1 of the paper, the authors give home and road stats for yellow cards, red cards, and penalty kick goals.

-- Yellow cards: road teams get 0.45 more per game than home teams.

-- Red cards: road teams get 0.038 more per game than home teams.

-- Penalty goals: home teams get 0.044 more per game than road teams.

A red card sends the player off for the rest of the game (and the team plays a man short). I remember reading somewhere what that's worth, but I don't remember where I saw it. Let's say it's an entire goal.

A yellow card is a warning. It doesn't cost the team anything (other than a free kick), but, since a second yellow card leads to a red card, the player affected might play with a bit more caution. It looks like there are about 20 yellow cards for one red card. So, let's suppose the player with the yellow card would get the red card one time in 10 if he didn't adjust his play. That means the first yellow card gives him a 10% chance of costing his team a goal. If he "spends" the entire 0.1 goals on more cautious play, we could say a yellow card is worth 0.1 goals.

A penalty goal, obviously, is a goal. I think I read somewhere that there's very little HFA on penalty kicks, so we can assume that the difference is the number of penalty kicks awarded.

So, let's add this up:

0.45 yellow cards times 0.1 goals equals 0.045 goals;
0.038 red cards times 1 goal equals 0.038 goals;
0.044 penalty successes times 1 goal equals 0.044 goals.

The total: 0.127 goals.

What else could the referees be doing to influence the outcome? Well, there's free kicks. And there's extra time -- some studies have suggested that the refs allow more injury time when the home team needs it. But those seem like they'd be much smaller factors than the ones above. Let's bump up the total from 0.127 to 0.15.

Also, it could be that visiting teams have to play more cautiously because of referee bias, and those numbers are artificially low because they don't include the effects of that. We included the effects of caution in the yellow card calculation, but not in the others. I don't know how to estimate that. It could be anything, really, from 0 percent to 1000%. However, if it were seriously high, someone would have noticed how teams play so much more aggressively at home than on the road. Certainly it would be mostly a conscious decision, so players would talk about it all the time, how they have to play so much more timidly to avoid provoking the referee.

Since that doesn't happen much (or does it?), it seems reasonable to assume that there's not much of that going on. Still, I'm going to set it high, and assume the effect of unpenalized cautious play is 50 percent of the total. That unrealistic assumption brings us up to about 0.22 goals.

We're still at just a little over half of observed HFA.

And, to get to half, we had to make some seriously unrealistic assumptions -- that ALL of the difference in yellow cards, red cards, and penalty kicks was due to referee bias against the visiting team, and that players are compensating with another 50 percent on top of that.

So, Table 1 of the paper is the strongest evidence I've seen that referees can't be causing much of HFA. And no regression is required -- it's just simple arithmetic!

Why is there no home-court advantage in foul shooting?

There's a home-site advantage in every sport.

Why is that? Nobody knows. One hypothesis is that it's officials favoring the home team. One piece of data that appears to support that hypothesis is that when you look at situations that don't involve referee decisions, the home field advantage (HFA) tends to disappear. In "Scorecasting," for instance, the authors report that, in the NBA, the overall home and road free-throw percentages are an identical .759. Also, in the NHL, shootout results seem to be the same for home and road teams, and likewise for penalty kick results in soccer.

However, there's a good reason for the results to look close to identical even if HFA is caused by something completely unrelated to refereeing.

The reason is that free-throw shooting involves only one player. At the simplest level, you could argue that foul shooting is offense. All other points scored in basketball are a combination of offense and defense. Not only is the offense playing at home, but the defense is playing on the road, which, in a sense, "doubles" the advantage. Therefore, if the home free-throw shooting advantage is X, the home field-goal shooting advantage should be at least 2X.

That's an oversimplification. A better way to think about it is that a foul shot attempt is the work of one player. A field goal attempt, on the other hand, is the end result of the efforts of *ten* players. Not every player is directly involved in the eventual shot attempt, but every player has the potential to be. A missed defensive assignment could lead to an easy two points, and the offense will take advantage regardless of which of the five defensive players is at fault. The same for offense: if a player beats his man and gets open, he's much more likely to be the one who gets the shot away. The weakest or strongest link could be any one of the ten players on the court.

So it might be better to guess that the HFA for a possession is 10X, rather than just X. We can't say that for sure -- it could be that the things a player has to do on a normal possession are so much more complex than a free throw, that the correct number is 20X. Or it could be that a normal possession is less complex than a free throw, so perhaps 5X is better. I don't know the answer to this, but 10X seems like a reasonable first approximation.

------

What would the actual numbers look like?

The home court advantage in basketball is about three points. That means that instead of (say) 100-100, the average game winds up 101.5 to 98.5.

Three points per game, divided by 10 players, is 0.3 points per game per player. Over (say) 200 possessions, that's 0.0015 points per possession per player.

If home-court advantage were made up only of serious mistakes, mistakes that turn a normal 50 percent possession into a 100 percent or zero percent possession, then that works out to exactly one point per mistake. In that case, the average player would make one such extra mistake every 667 possessions. That's a little less than one every three games. If you assume that a mistake is worth only half a point, then it's one mistake per player for every 333 team possessions.

In reality, of course, it's probably not nearly as granular as "mistakes" or "good plays". It's probably something like this: a player plays his role with an overall average of 50 effectiveness units, random between possessions, plus or minus some variation. But that's an average of home, where he plays with an average of 51 effectiveness units, and road, with an average 49 effectiveness units.

Still, that doesn't matter to the argument: the important thing is HFA is one point per player for every 667 total team possessions, regardless of how it manifests itself.

------

Now, let's go back to free throws. I'm going to assume that a player's HFA on a single possession should be about the same as a player's HFA on a single free throw. Is that OK? It's a big assumption. I don't have any formal justification for it, but it doesn't seem unreasonable. I'd have to admit, though, that there are a lot of alternative assumptions that also wouldn't seem unreasonable.

But the point of this post is that it is NOT reasonable to assume that a player's HFA on a free throw should be the same as the overall HFA for an entire game. That wouldn't make any sense at all. That would be like seeing that the average team wins 50 percent of games, and therefore expecting that the average team should win 50 percent of championships. It would be like seeing that the Cavaliers are winning 17 percent of their games, and expecting that they score 17 percent of the total points.

In any case, the overall argument stays the same even if you argue that the HFA on a single possession should be twice that of a single free throw, or half, or three times. But I'll proceed anyway with the assumption that it's one time.

If the HFA on a free-throw is the same 0.0015 points per player as on a possession, then you'd expect the difference between home and road free throw percentages to be 0.15%. Instead of the observed .759 home and road, it should be something like .75975 home, and .75825 road.

Why don't we see this? Well, here's one possible explanation. Visiting teams are behind more often, so will commit more deliberate fouls late in the game. They will try to foul home players who are worse foul shooters. Therefore, the pool of home foul shooters is worse than the pool of road foul shooters, which is why it looks like there's no home field advantage in foul shooting.

Since we're talking about such a very small HFA in the first place, this doesn't seem like an unreasonable explanation. It would be interesting to run the numbers, but controlling for who the shooter is. I suspect if you have enough data, you'd spot a very small home-court advantage in foul shooting.

"Scorecasting:" is home field advantage just biased officiating?

There's a new book that's about to come out, that you might have heard about. It's called "Scorecasting," and the publisher was kind enough to send me a review copy. It's basically a Freakonomics for sports, in intent, in tone, in writing style, and even down to its similar authorship -- one academic economist (Tobias Moskowitz, a finance professor) and one journalist (Sports Illustrated's L. Jon Wertheim).

The book's website is here; it has an excerpt from one of the chapters, and you'll find other excerpts online if you search the authors' names.

The topics will be very familiar to sabermetricians and regular readers of this blog. There are chapters on the Hot Hand, on competitive balance, on NBA refereeing, on steroids, and so on. There isn't a huge amount of breakthrough stuff there, although there are certainly a few new insights. Mostly, the authors summarize what they've learned from academic articles on sports, and they add the results of a few little studies they did themselves.

Alas, by concentrating on the journals, they've missed much of the scholarship of us amateurs. For instance, in the chapter of competitive balance, they argue that baseball is less balanced than the football because MLB teams play 162 games, while NFL teams only play 16. That, of course, is only a small part of the story. There are other parts, such as the distribution of talent in the league, and the internal details of the game itself. Tom Tango has effectively solved the problem of comparing different sports (here's just one of his many posts), but the authors seem unaware of that (although they do mention Tango in the book once, in a mention of leverage, referring to him as a "stats whiz.")

And they're occasionally completely off, as when they say the sample size of the MLB playoffs is enough that the best team ought to win the series.

Still, a lot of the material is solid; the authors are at their strongest when they're reviewing one of the more famous and established studies, like the Romer "fourth down" paper, and the Massey/Thaler NFL draft study. (I'll probably do a full review of the book later, but, for now, just picture a sports Freakonomics that's not as rigorous as most of the websites, but does mention a few things that you didn't know before.)

Anyway, I'm going through the book, and suddenly I see that the authors claim to have solved the problem of what causes home field advantage (HFA). That was sort of shocking. My personal subjective view is that HFA is the biggest unsolved problem in sabermetrics, and very little progress has been made. There's so little progress, in fact, that I've started to take seriously a hypothesis that seems way off the wall -- the theory that humans have built in evolutionary programming that makes them more physically and mentally effective when defending their own turf. (I'm not saying that it's necessarily true, just that I have a bizarre attraction to it.) In that light, finding these HFA claims was a bit like picking up a newspaper article on math, and finding that the reporter has proved Fermat's Last Theorem.

So what's the authors' solution to the long-standing HFA conundrum? Refereeing. After dismissing most of the usual suspects (fan enthusiasm, travel, tailoring the team to the park), Moskowitz and Wertheim believe that most, or all, of HFA can be explained by biased officiating.

They list a bunch of supporting evidence, which I'll summarize here. If you want to follow along, some of this stuff is also in a long excerpt from the book that appeared a couple of weeks ago in the Jan. 17 issue of Sports Illustrated (the article, unfortunately, does not appear to be online).

------

Soccer

1. In soccer, the referee controls how much extra "injury time" is added to the end of a match. It turns out that injury time is longer when the home team would benefit. When the home side was ahead by a goal, there were two minutes of injury time, on average, in a sample of Spanish league games. But when they were *behind* by a goal, it was four minutes.

2. In 1998, the point structure changed to give the winning team three standings points instead of two. Immediately, the above injury time bias increased.

3. The same bias exists in England, Italy, Germany, Scotland, and the US.

4. "... home teams receive many fewer red and yellow cards even after controlling for the number of penalties and fouls on both teams."


Baseball

5. In baseball, the authors looked at the percentage of called pitches that are strikes. In crucial situations (high leverage), home teams got a lot more favorable calls. But in low-leverage situations, it was *road* teams that got more favorable calls. "This makes sense," the authors write. "If the umpire is going to show favoritism to the home team, he or she will do it when it is most valuable -- when the outcome of the game is affected the most. You might even contend that it noncrucial situations the umpire might be biased against the home team to maintain an overall appearance of fairness."

6. " ... the success rates of home teams in scoring from second base on a single or scoring from third base on an out -- typically close plays at the plate -- are much higher than they are for their visitors in high-leverage/crucial situations. yet they are no different or even slightly less successful in noncrucial situations."

7. Over a large sample of 5.5 million pitches, "called strikes and balls went the home team's way, *but only* in stadiums without QuesTec ... Not only did umpires not favor the home team when QuesTec was watching them, they actually gave *more* strikes and *fewer* balls to the home team. In short, when umpires knew they were being monitored, home field advantage on balls and strikes didn't simply vanish; the advantage swung all the way to the visiting team."

8. In low-leverage situations, even in non-QuesTec parks, there was no bias at all.

9. The authors then analyzed pitches using Pitch f/x data, to see how many pitches were miscalled based on the recorded location. For pitches on the corner of the strike zone, there were more miscalls in the home team's favor than in the visiting team's favor. The home advantage was largest on full-count pitches, followed by other three-ball counts, other two-strike counts, and, lastly, all other counts. So, the more crucial the pitch, the greater the HFA.

10. "Over the course of the season, all of this adds up to 516 more strikeouts called on away teams, and 195 more walks awarded to home teams than there otherwise should be, thanks to the home plate umpire's bias. And that includes only terminal pitches -- where the next called pitch will result in either a strikeout or a walk. Errant calls given earlier in the pitch count could confer an even greater advantage on the home team."

11. "This adds up to an extra 7.3 runs per season given to each home team by the plate umpire alone. That might not sound significant, but cumulatively, home teams outscore their visitors by only 10.5 runs in a season." [That latter number isn't correct ... in 2010, it was 23.5 runs. 23.5 runs equals 2.35 wins out of 81, which is a .530 winning percentage. (UPDATE: Oops! I forgot to adjust for the home team not batting in the bottom of the ninth when leading. If you adjust for that, the home advantage is a lot bigger than 10.5 or 23.5 runs.)]


Football

12. In the NFL, "Home teams receive fewer penalties than away teams -- about half a penalty less per game -- and are charged with fewer yards per penalty. Of course, this does not necessarily mean officials are biased. But when we looked at more crucial situations in the NFL ... we found that the penalty bias [increases]."

13. When instant replay came to the NFL, the home winning percentage declined from 58.5 percent (1985-98) to 56 percent (1999-2008). "Before instant replay, home teams enjoyed more than an 8 percent edge in turnovers ... When instant replay came along ... the turnover advantage was cut in half." Also, "the home team does not actually fumble or drop the ball less often than the away team ... they simply lose fewer fumbles than away teams. After instant replay was installed, however, the home team advantage of *losing* fewer fumbles miraculously disappeared, whereas the frequency of fumbles remained the same. ... In close games, where referees' decisions may *really* matter ... home teams enjoyed a healthy 12 percent advantage in recovering fumbles. After instant replay was installed, that advantage simply vanished."

14. After instant replay, there was no change in the relative frequency of home and away penalties. That might be because penalties can't be challenged.

15. Away teams have their challenges upheld 37 percent of the time, versus 35 percent for home teams. But when the home team is losing, the visiting team wins 40 percent, versus only 28 percent for the home team. So it looks like the referees favor the home team more when they need it more.


Basketball

16. In the NBA, fouls and turnovers that are not subjective referee calls (like shot clock violations) are equal for home and road teams. But for subjective calls, away teams get between 1 and 1.5 more of those per game. Visiting players are 15 percent more likely to be called for traveling than home players.

17. "How much of the [HFA] in the NBA is due to referee bias? If we attribute the differences in free throw attempts to referee bias, this would account for 0.8 points per game. If we gave credit to the referees for the more ambiguous turnover differences ... this would also capture another quarter of the home team's advantage. Attributing some of the other foul differences to the referees and adding the effects of those fouls (other than free throws) ... brings the total to about three-quarters of the home team's advantage. And, remember, scheduling in the NBA [visiting teams play more back-to-back games than home teams] explained about 21 percent of [HFA]. This adds up to nearly all of the NBA home court advantage."


Hockey

18. In the NHL, home teams get 20 percent fewer penalties and receive fewer minutes per penalty. "On average, home teams get two and a half more minutes of power play opportunities ... than away teams. That is a *huge* advantage." If you multiply that by a 20 percent success rate, you get an extra 0.25 goals per game for the home team. Since the average overall differential is only 0.3 goals for the home team, "this alone accounts for more than 80 percent of the home ice advantage in hockey."

19. There is no apparent HFA in shootouts, where refereeing makes no difference. Also, in NBA foul shooting. And, even in Pitch f/x data. Visiting pitchers throw no worse, according to Pitch f/x, than home teams do. It's only the umpires' calls that are different.

-----

It's an impressive array of evidence and argument. But, at least some of it doesn't hold up.

Look at number 5: in baseball, in low leverage situations (I believe this means the bottom 50%), the authors say that umpires favor the visiting team. That would mean that, in less critical situations, we should find a "visiting field advantage." But home teams outscore visiting teams even in medium-leverage situations. For instance, here's the breakdown of home and road runs scored by inning (1954 to 2007). The last column is the percentage by which the home team outscored the visiting team:

1 61872-52071 +18%
2 46823-42539 +10%
3 53590-48188 +11%
4 53357-49593 +8%
5 53203-48448 +10%
6 54401-50603 +8%
7 52231-48641 +7%
8 50451-47781 +6%

You would think that you'd have more high-leverage events in the later innings -- but the HFA goes *down* in the last few innings, not up.

But I might be wrong about that, maybe the eighth inning has no more high-leverage situations than the first inning (after all, there are more 8-1 games in the eighth than in the first). So, let's look at innings where, at the start, one team was at least four runs ahead of the other. Those should all be low leverage, for the most part, and should show the visiting team having the advantage.

Nope:

2 2543-2139 +19%
3 4583-4176 +10%
4 8817-7801 +13%
5 10940-10057 +9%
6 14371-13279 +8%
7 15698-14583 +8%
8 16935-16180 +5%

Now, this could be just because, in a four-run game, the home teams are a lot better than the visiting teams. What if we look at situations when the *visiting* team is ahead by at least four runs? Then, we should see a huge effect in favor of the visiting team: first, they're probably a much better team, and, second, the low leverage means the umpire should still be favoring them.

But, no. Even in those situations, the home team still performs a little better, on average, having the advantage in five of the seven cases:

2 957-1022 -6%
3 1974-1799 +10%
4 3609-3355 +8%
5 4435-4645 -5%
6 6269-5705 +10%
7 6627-6562 +1%
8 7309-7179 +2%


So, I just don't see it. If umpires DO call more strikes for visiting teams in low-leverage situations, maybe that's compensated for by those pitches actually being strikes ... but being worse pitches in location and movement and velocity. That is, maybe HFA comes from pitchers throwing more accurately, but more hittably.

In any case, if my data are correct, and the authors' data are also correct, it can't be the case that the authors' findings are an explanation of HFA.

------

Now, let's look at number 18, the hockey case. The authors argue that HFA is caused almost entirely by penalties. If that's the case, then you'd expect home and visiting teams to have similar numbers at equal strength.

They do not. The NHL.com website has home/road goal breakdowns. Here they are for the 2008-09 season, averaged by team:

Even strength... 124-110 (home advantage 12.5%)
Power play...... 35-30 (home advantage 15.1%)
Shorthanded..... 4-4 (home advantage 1.0%)

There's almost as large an advantage at even strength as there is on the power play. Admittedly, the extra power play boost is probably caused by more penalties, as the authors say, but the overall contribution of the extra penalties seems to be pretty small.

Just to make sure it wasn't a fluke, I ran the same numbers for 2009-10:

Even strength... 121-106 (home advantage, 13.9%)
Power play...... 30-25 (home advantage, 21.0%)
Shorthanded..... 4-3 (home advantage, 32.9%)

A bit more extreme in favor of power play. But how do you explain the sizeable advantage for home teams at even strength? One possible explanation is that visiting teams have to play an overcautious game, to avoid being penalized by biased referees. But for a 13.9% disadvantage, that caution would have to be way out of line, wouldn't it?


------

Both of these examples -- and, by the way, they're the only two I checked -- cast doubt on the authors' hypothesis that HFA is almost all refereeing. I have never disagreed that *some* of it might be refereeing, but there's obviously a lot more going on.

And I have to say that the authors have indeed provided a blueprint for how this kind of research should go -- try to break down performance into its constituent parts, and check those.

If there's no home advantage in foul shooting, why not? If there's no HFA in hockey shootouts, why not? If we get a list of areas with high HFA, and a list of areas with low HFA, we can maybe start narrowing down what the causes might be.

But the authors have amassed a lot of evidence, and the must be something to at least some of it, no? For instance, I can't think of any explanation for the injury time phenomenon (maybe I should look up the relevant study). And it seems reasonable that referees will call more fouls on visitors, even if they're unbiased. Why? Because they might be using crowd noise as a guide ot what is and what isn't a foul. If the fans scream when a visitor trips an opponent, but not when a home player trips an opponent, that will simply make it more likely that an unbiased referee will have enough evidence to correctly "convict" the visiting player.

But the question is not just whether referee bias exists, but *how much* of it there is, and how much of HFA it's responsible for. The authors of "Scorecasting" seem more focused on "existence" evidence, and it seems to me they've made only a small dent in terms of explaining the real-life observed HFA. I wish the authors had provided more details of some of their findings, so we can figure out what's going on and maybe quantify it a bit more ... but I guess it is what it is.

I know there are a lot of working sabermetricians reading this ... if you have expertise or evidence on any of the authors' points, please weigh in.

-------

UPDATE: I have a full review of the book here.