Heavy Topspin: A Tennis Blog

The more gut-wrenching the moment, the more likely it is to stick in memory.  We easily recall our favorite player double-faulting away an important game; we quickly forget the double fault at 30-0 in the middle of the previous set.  Which one is more common? The mega-choke or the irrelevancy?

There are three main factors that contribute to double faults:

1. Aggressiveness on second serve. Go for too much, you’ll hit more double faults.  Go for too little, your opponent will hit better returns.
2. Weakness under pressure. If you miss this one, you lose the point. The bigger the point, the more pressure to deliver.
3. Chance. No server is perfect, and every once in a while, a second serve will go wrong for no good reason.  (Also, wind shifts, distractions, broken strings, and so on.)

In this post, I’ll introduce a method to help us measure how much each of those factors influences double faults on the ATP tour. We’ll soon have some answers.

In-game volatility

At 30-40, there’s more at stake than at 0-0 or 30-0.  If you believe double faults are largely a function of server weakness under pressure, you would expect more double faults at 30-40 than at lower-pressure moments.  To properly address the question, we need to attach some numbers to the concepts of “high pressure” and “low pressure.”

That’s where volatility comes in.  It quantifies how much a point matters by considering several win probabilities.  An average server on the ATP tour starts a game with an 81.2% chance of holding serve.  If he wins the first point, his chances of winning the game increase to 89.4%. If he loses, the odds fall to 66.7%.  The volatility of that first point is defined as the difference between those two outcomes: 89.4% – 66.7% = 22.7%.

(Of course, any number of things can tweak the odds. A big server, a fast surface, or a crappy returner will increase the hold percentages. These are all averages.)

The least volatile point is 40-0, when the volatility is 3.1%. If the server wins, he wins the game (after which, his probability of winning the game is, well, 100%). If he loses, he falls to 40-15, where the heavy server bias of the men’s game means he still has a 96.9% chance of holding serve.

The most volatile point is 30-40 (or ad-out, which is logically equivalent), when the volatility is 76.0%.  If the server wins, he gets back to deuce, which is strongly in his favor. If he loses, he’s been broken.

Mixing in double faults

Using point-by-point data from 2012 Grand Slam tournaments, we can group double faults by game score.  At 40-0, the server double faulted 3.0% of points; at 30-0, 4.2%; at ad-out, 2.8%.

At any of the nine least volatile scores, servers double faulted 3.0% of points. At the nine most volatile scores, the rate was only 2.7%.

(At the end of this post, you can find more complete results.)

To be a little more sophisticated about it, we can measure the correlation between double-fault rate and volatility.  The relationship is obviously negative, with an r-squared of .367.  Given the relative rarity of double faults and the possibility that a player will simply lose concentration for a moment at any time, that’s a reasonably meaningful relationship.

And in fact, we can do better.  Scores like 30-0 and 40-0 are dominated by better servers, while weaker servers are more likely to end up at 30-40. To control for the slightly different populations, we can use “adjusted double faults” by estimating how many DFs we’d expect from these different populations.  For instance, we find that at 30-0, servers double fault 26.7% more than their season average, while at 30-40, they double fault 28.6% less than average.

Running the numbers with adjusted double fault rate instead of actual double faults, we get an r-squared of .444.  To a moderate extent, servers limit their double faults as the pressure builds against them.

More pressure on pressure

At any pivotal moment, one where a single point could decide the game, set, or match, servers double fault less than their seasonal average.  On break point, 19.1% less than average. With set point on their racket, 22.2% less. Facing set point, a whopping 45.2% less.

The numbers are equally dramatic on match point, though the limited sample means we can only read so much into them.  On match point, servers double faulted only 4 times in 296 opportunities (1.4%), while facing match point, they double faulted only 4 times in 191 chances (2.2%).

Better concentration or just backing off?

By now, it’s clear that double faults are less frequent on important points.  Idle psychologizing might lead us to conclude that players lose concentration on unimportant points, leading to double faults at 40-0. Or that they buckle down and focus on the big points.

While there is surely some truth in the psychologizing–after all, Ernests Gulbis is in our sample–it is more likely that players manage their double fault rates by changing their second-serve approach.  With a better than 9-in-10 chance of winning a game, why carefully spin it in when you can hit a flashy topspin gem into the corner?  At break point, there’s no thought of gems, just fighting on to play another point.

And here, the numbers back us up, at least a little bit.  If players are avoiding double faults by hitting more conservative second serves on important points, we would expect them to lose a few more second serve points when the serve lands in play.

It’s a weak relationship, but at least the data suggests that it points in the expected direction.  The correlation between in-game volatility and percentage of second serve points won is negative (r = -0.282, r-squared = 0.08).  Complicating the results may be the returner’s conservative approach on such points, when his initial goal is simply to keep the ball in play, as well.

Clearly, chance plays a substantial role in double faults, as we expected from the beginning.  It’s also clear that there’s more to it.  Some players do succumb to the pressure and double fault some of the time, but those moments represent the minority.  Servers demonstrate the ability to limit double faults, and do so as the importance of the point increases.

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Some players are much better returners than others.  Many players are such good returners that everyone knows it, agrees upon it, and changes their game accordingly.  This much, I suspect, we can all agree on.

How far does that go? When players are altering their service tactics and changing their risk calculations based on the man on the other side of the net, does the effect show up in the numbers? Do players double fault more or less depending on their opponent?

Put it another way: Do some players consistently induce more double faults than others?

The conventional wisdom, to the extent the issue is raised , is yes.  When a server faces a strong returner, like Andy Murray or Gilles Simon, it’s not unusual to hear a commentator explain that the server is under more pressure, and when a second serve misses the box, the returner often gets the credit.

Credit where credit isn’t due

In the last 52 weeks, Jeremy Chardy‘s opponents have hit double faults on 4.3% of their service points, the highest rate of anyone in the top 50.  At the other extreme, Simon’s opponents doubled only 2.8% of the time, with Novak Djokovic and Rafael Nadal just ahead of him at 2.9% and 3.0%, respectively.

The conventional wisdom isn’t off to a good start.

But the simple numbers are misleading–as the simple numbers so often are.  Djokovic and Nadal, going deep into tournaments almost every week, play tougher opponents.  Djokovic’s median opponent over the last year was ranked 21st, while Chardy’s was outside the top 50.  While it isn’t always true that higher-ranked opponents hit fewer double faults, it’s certainly something worth taking into consideration.  So even though Chardy has certainly benefited from some poorly aimed second serves, it may not be accurate to say he has benefited the most–he might have simply faced a schedule full of would-be Fernando Verdascos.

Looking now at the most recent full season, 2012, it turns out that Djokovic did face those players least likely to double fault.  His opponents DF’d on 2.9% of points, while Filippo Volandri‘s did so on 3.9% of points.  While these are minor differences when compared to all points played, they are enormous when attempting to measure the returners impact on DF rate.  While Djokovic “induced” double faults on 3.0% of points and Volandri did so on 3.9% of points, you can see the importance of considering their opponents.  Despite the difference in rates, neither player had much effect on their opponents, as least as far as double faulting is concerned.

This approach allows to express opponent’s DF rate in a more efficient way, relative to “expected” DF rate.  Volandri benefited from 1% more doubles than expected, Chardy enjoyed a whopping 39% more than expected, and–to illustrate the other extreme–Simon received 31% fewer doubles than his opponents would be predicted to suffer through.

You can’t always get what you want

One thing is clear by now. Regardless of your method and its sophistication, some players got a lot more free return points in 2012 than others.  But is it a skill?

If it is a skill, we would expect the same players to top the leaderboard from one year to the next.  Or, at least, the same players would “induce” more double faults than expected from one year to the next.

They don’t.  I found 1405 consecutive pairs of “player-years” since 1991 with at least 30 matches against tour-level regulars in each season. Then I compared their adjusted opponents’ double fault rate in year one with the rate in year two.  The correlation is positive, but very weak: r = 0.13.

Nadal, one player who we would expect to have an effect on his opponents, makes for a good illustration.  In the last nine years, he has had six seasons in which he received fewer doubles than expected, three with more.  In 2011, it was 15% fewer than expected; last year, it was 9% more. Murray has fluctuated between -18% and +25%. Lots of noise, very little signal.

There may be a very small number of players who affect the rate of double faults (positively or negatively) consistently over the course of their career, but a much greater amount of the variation between players is attributable to luck.  Let’s hope Chardy hasn’t built a new game plan around his ability to induce double faults.

The value of negative results

Regular readers of the blog shouldn’t be surprised to plow through 600 words just to reach a conclusion of “nothing to see here.”  Sorry about that. Positive findings are always more fun. Plus, they give you more interesting things to talk about at cocktail parties.

Despite the lack of excitement, there are two reasons to persist in publishing (and, on your end, understanding) negative findings.

First, negative results indicate when journalists and commentators are selling us a bill of goods. We all like stories, and commentators make their living “explaining” causal connections.  Sometimes they’re just making things up as they go along. “That’s bad luck” is a common explanation when a would-be winner clips the net cord, but rarely otherwise.  However, there’s a lot more luck in sport than these obvious instances.  We’re smarter, more rational fans when we understand this.

(Though I don’t know if being smarter or rational helps us enjoy the sport more.  Sorry about that, too.)

Second, negative results can have predictive value. If a player has benefited or suffered from an extreme opponents’ double-fault rate (or tiebreak percentage) and we also know that there is little year-to-year correlation, we can expect that the stat will go back to normal next year. In Chardy’s case, we can predict he won’t get as many free return points, thus he won’t continue to win quite as many return points, thus his overall results might suffer.  Admittedly, in the case of this statistic, regression to the mean would have a tiny effect on something like winning percentage or ATP rank.

So at Heavy Topspin, negative results are here to stay. More importantly, we can all stop trying to figure out how Jeremy Chardy is inducing all those double faults.