Did NBA playoff referees cheat in favor of the home team?

Columnist and Economist Kevin Hassett today looks at the 2007-08 NBA playoffs, and its unusually large home-court advantage. He sees "troubling" evidence to suggest that the NBA manipulated the results, perhaps instructing referrees to favor the team trailing in games, allowing the league to wind up up with longer, more exciting, more remunerative series.

That's a huge accusation, and one that calls for a lot of supporting evidence. And I think Hassett doesn't provide nearly enough evidence.

Here's what he does tell us. First, in terms of game statistics, then, in terms of wins:


Game Stats

First, in the last two playoff seasons, when the home team was leading 3-1 in games, the visiting team actually had a better field-goal percentage than the home team, by 1.1 percentage points. But when the visiting team was leading 3-1, the home team outshot the visitors by 5.4 points:

-1.1: home team ahead in series
+5.4: home team behind in series
Hassett tells us there were 25 games total. Assuming 12.5 games in each group, and 80 field-goal attempts per team per game, the SD of the difference would be 2.2 points, which would make the observed difference very significant.

But if there were only, say, 5 games in one of the two groups, the SD increases to 2.8 points. And if there were only 2 games in one of the two groups, the SD is now 4.1, and the result isn't significant at all.

However: is it possible that success rates aren't independent? A team that falls behind might try higher risk shots, for instance. That would increase the SD, and lower the observed significance.

And isn't it possible that when a team is down 3-1, it plays harder, plays its regulars more, doesn't worry as much about fouling out, and so on?

Bottom line: it looks like this may be significant, but there isn't enough information to tell.

Second, in Game 6s, again over the past two playoffs, the home team was called for 4.1 fewer fouls than the visiting team when it was behind in the series three games to two. In Game 7s, the 4.1 drops to 1.0.

Home team 2-3 in games: 4.1 fewer fouls
Home team 3-3 in games: 1.0 fewer fouls

Again, how many such games were there? There's nothing about sample size, variance, or statistical significance. As it turns out, there have been only four seven-game series in the past two seasons. I'd bet a lot of money that 1.0 is not significantly different from 4.1 over *four games*. And that's even without considering the "desperation" factor.

Third, Hassett shows that in the regular season, home teams get called for 0.8 fewer fouls than visitors. In the playoffs since 2003, the figure jumps to 1.4. The home team advantage in field goal percentage also increases in the playoffs, from 1.3 to 2.3. This year, it was 3.4. This, Hassett writes, "suggests the home team is allowed to play aggressively."

These differences are statistically significant. But it's a long step from showing a difference in home advantage, to showing the league is corrupt. There are lots of possible other explanations:

-- in the playoffs, all teams are fairly strong
-- in the playoffs, teams are more closely-matched than in the regular season
-- in the playoffs, teams may play more aggressively
-- in the playoffs, teams may give their regulars more court time.

Or, how's this? In the playoffs, games are closer because the teams are more evenly matched. Deliberate fouls in the last minute happen only in close games. Therefore, there are more fouls in the playoffs by the team that's behind.. And the team that's behind is more likely to be the visiting team.

So I'm certainly not convinced that this is evidence of anything.



Game Stats

Hassett writes,

"In the 2008 playoffs, the home team won 64 of 86 games -- or 74 percent of the time. If we exclude the first round, where there are bound to be some blowouts, the home team won 34 out of 42, an 81 percent clip ...

"Since the 2002 regular season, home teams won a little more than 60 percent of the time ... If the true odds of the home team winning were six in 10, as in the regular season, then the odds of observing 34 home victories in 42 games simply by chance are close to zero."

There are a few objections to this analysis.

First, why exclude the first round? Isn't that cherry picking, to notice that most of the observed effect is in subsequent rounds, and rationalizing after-the-fact that only those rounds matter?

Second: unlike the regular season, not every team has the same number of home games. In the first three rounds of the NBA playoffs, if the series goes an odd number of games, the better team gets the extra game. So the naive measure of HFA in the playoffs suffers from selection bias – better teams are overrepresented.

In 2008, there were eight series that went exactly 5 or 7 games; five series in the first round, and three in the next two rounds. The home team, obviously, won all those games.

So let's adjust for that.

In the playoffs as a whole, the home team would have been expected to go .613. Let's assume it should have gone .750 in those odd-numbered games (because the better team was at home). That means 78 games at .600, and 8 games at .750. That works out to .613.

Home teams actually went .744. They went 64-22; they should have gone 53-33. In winning percentage terms, they overachieved by .131.

Using the binomial approximation, the SD of winning percentage over 86 games is .052. That means the .744 figure is 2.5 SDs over expected, which is significant at 0.6%.

That means what happened this year should have happened 1 out of 161 years.

Is 1 in 161 strong enough evidence that we should consider the conspiracy allegations? I don't think so. Do you really want to be in a position where you falsely accuse the league of this kind of corruption every 161 years? If you include all four major sports, you'd be making a false accusation every 40 years. If you looked at statistics other than home field advantage, for anything unusual, it would be even more often than that. (I'm sure that if, instead of an abnormal HFA, there were some unusual comebacks, there would be accusations of collusions for those, too.)

If you look at 20 different statistical categories in each league, and they're independent, you'll come up with a corruption charge every two years. The 1 in 161 is certainly noteworthy, but not so noteworthy that it's enough in and of itself.

Also, the 1 in 161 again assumes that there is no other explanation for why home field advantage is higher in the playoffs than in the regular season. If you hypothesize that, because of changes in play, the actual playoff HFA is .629 (which it was in the 2002-03 regular season), you wind up below 2 SD.

Also, why concentrate so hard on home-field advantage? If the NBA were indeed colluding with the referees to make series go longer, why would they tell them to favor the home team? Wouldn't they tell them to favor the team that's behind in the series? If the goal is to extend the series, and the visiting team wins the first two games of the series, why would the NBA tell the refs to go ahead and continue to cheat for the home team, when that would just ensure a 4-0 series finish? That would work against their interests!

I suppose you could argue – albeit implausibly -- that the NBA doesn't want the referees to know it's looking for longer series. It could lie to the refs and tell them they're favoring the visiting team too much. The refs then lean towards the home team.

But would that really make the series longer? I ran a simulation. I assumed one team would be .600 over the other team at a neutral site (which means the other team would be .400). Then, with a .110 home field [dis]advantage, 28.2% of series to go 7 games. Increase the HFA to .160, and the percentage increases to 29.5%. It's a very minor difference -- one series out of 70.

Rather than corruptly increase the HFA, the league would do better to pick one crucial game, and arrange for the referees to favor the team that's behind just that one game. That would be worth more than any reasonable estimate of the value of illicitly increasing home field advantage, and greatly reduce the amount of effort required.

Even then, the benefit is going to be small. Would the NBA risk its existence, government hearings, and criminal charges, for a tiny bit of advantage? Would the NBA be so stupid as to believe that the referees would not only agree, but also be able to keep it all a secret? Would they have thought that if they started favoring the home team, nobody would notice?

The whole idea is just absurd, this weak statistical evidence notwithstanding.

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UPDATE: I originally had written that the true HFA in basketball was .613 rather than .600. Thanks to Justin Kubatko, who provided me with the data, I found out that .600 is indeed accurate for the past few regular seasons (.613 was the figure for 2003-04). I have now updated the calculations and arguments to use .600.