Golf and luck

Given a player's talent, I routinely think of the results of a series of basketball free throws as "luck".  That is, if the player has worked to become an 80% talent, whether he makes the current shot (80 percent chance) or misses it (20 percent) is just random, as if he flipped an 80% coin.

Some people don't like that idea ... they feel that because it's all within the player's control, it's wrong to think of it as "luck" or "random".  I don't agree, but I won't argue that here.  What I want to do here is try out a different example, one that I can use instead of free throws, that maybe we can all agree on.

So ... how about golf shots?  Those aren't completely under a player's control, because of wind.

A difference in wind speed of only about 2 m/s (4.5 mph) is said to affect the ball's distance by around 15 meters (49 feet) (.pdf).  That's pretty big.  Pros sink 20-foot putts only 14 percent of the time, as compared to 38 percent for 10-foot putts ... and that's only a 10 foot difference, not 49 feet. 

Now, you could argue that golfers should take the wind into account when swinging.  And they do.  But, wind changes while the ball is in the air, and it's literally impossible, from the ground, to predict how the wind will change.  If 2 m/s wind is 15 meters of distance, we can guess that 0.2 m/s of wind is 1.5 meters of distance.  If there's an unpredictable 0.2 m/s change for half the time the ball is in the air, that's 2 to 3 feet.  That's still a fair bit.  Moving a putt 2-3 feet closer is a big deal, especially when you're already close. 

Or ... suppose a golfer gets a hole in one.  It's reasonable to assume that if the wind had been even slightly different, in any direction, the ball wouldn't have gone in. 

When does the ball go in on a tee shot?  Consider where the ball would have landed if there were no hole.  Let's say that if that spot is, maybe, 12 inches behind where the hole would be (in the line of trajectory), and four inches left to right, it would have gone in.  That's 0.33 square feet.  Let's round it up to 0.5.

If the wind makes an unpredictable difference of, say, 3 feet each direction, that's a circle of radius 3, or about 28 square feet. 

28 divided by 0.5 is 56.  So, because of wind, there'd be only a 2% chance the ball would go in if you did the exact same swing again. 

That is: if you get a hole in one, you hit a 50-to-1 longshot, by luck.  That's even if you're a perfect golfer in every respect.


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Of course, the better a golfer you are, the more holes-in-one you're going to get.  The argument is not that it's *all* luck -- the argument is that there's *some* luck.  

Holing your tee shot is like winning the lottery.  I'm not a very good golfer, but my lottery ticket might still come in someday.  Tiger Woods, because of his skill, holds several thousand tickets, so he'll get lucky much more often.

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"Iron Byron" is a machine that swings a golf club, exactly the same way each time -- or at least, as close to "exactly" as a machine can get.  But the balls it hits don't land in exactly the same place.  This site says that, after multiple swings, the pattern of balls was 15 feet by 8 feet for cavity-back clubs, and "about 1/4 the size" for the club they were developing.

For the purposes of this discussion, that's close enough to the 3-foot radius I guessed at.

You'd think what the machine did would be the limit of human performance.  Of course, you might think humans can be more precise than machines, which seems unlikely -- but feel free to argue it if that's what you think.  Keep in mind, though, that the human is always a different distance from the pin, and has to adjust his swing every shot!  On the other hand, the machine doesn't have to figure out how hard to swing, because it doesn't matter. 

So the human has to be *more* perfect than the machine, to get the same results.

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Both these arguments -- theoretical, and empirical -- seem to imply that non-human forces have an effect on a golf shot, an effect that's significant enough to affect who wins a tournament.  In other words, that there is at least some "external" luck in golf.

For those of you who disagree that there's luck in free throws, does this argument convince you that there's luck in golf shots?  If someone hits a hole in one and wins a PGA tournament by two strokes, would you be comfortable agreeing that luck had a lot to do with it?

If not, why not?

Is Tiger Woods Irrational?

"Loss Aversion" is a kind of irrationality where people care more about avoiding a loss more than about securing a gain of the same amount. For instance, if one person wins $50, while another identical person loses $50, the unlucky gambler will gain more unhappiness than the winner gains happiness. Because of this, people will expend more effort to avoid incurring a loss than they will in pursuit of the identical gain.

This is irrational; if it takes three hours to save a loss of $20, but you can earn $10 an hour, you're worse off if you spend the three hours to save the $20. That's because if you accepted the loss, and instead spent those three hours earning $30, you'd be $10 ahead. But since, as it turns out, humans tend to value losses as about twice the value of the identical gain, there should be a tendency to spend time trying to save the $20 instead of earn the $30.

The irrational fear of losses is a generalization; it's certainly possible that some people don't have this bias, or at least are aware of it and able to compensate. Life is full of decisions and trade-offs, and you'd think the most successful people would have learned how to be more rational in these situations.

Take golf, for instance. In normal PGA tournament play, your score is just the number of strokes you took. A stroke is a stroke, and so it shouldn't matter if it's a stroke to make birdie (which, if missed, will leave your score unchanged), or a stroke to make par (which, if missed, will make your score one stroke worse). Both strokes are of equal importance. However, the stroke for par may count more in the golfer's mind. That's because, if he doesn't make it, it looks like a loss (his score gets worse). But if he misses a birdie putt, it's just a foregone gain (his score stays the same). If losses irrationally count for twice as much as gains in the golfer's mind, then he should try harder to make the par putt than the birdie putt.

And it turns out he does. According to a recent golf study (.pdf, free download) by Devin G. Pope and Maurice E. Schweitzer, PGA golfers are significantly more likely to make a par putt than the identical birdie putt. It's not even close: overall, the difference is about three percentage points. That's huge: the overall conversion rate for putts is 61 percent, and 3 points out of 61 is about five percent. In baseball terms, it's like a 96-66 team suddenly going 102-60 just by changing an irrational strategy.

Pope and Schweitzer did more than just count putt conversion rates, of course. After all, it's likely that birdie putts are different from par putts in many ways. They might be farther from the hole. They might be in less advantageous places on the green. They are less likely to have followed other putts, which means the golfer doesn't have as good a read of the green. Birdie putts might be associated with different kinds of golfers. They might be associated with different difficulties of greens. And so on, and so forth.

The authors took all these things into consideration. Indeed, their database is so extensive (over 1.6 million putts between 2004 and 2008) that the authors were even able to find thousands of "matching" pairs of putts where one was for birdie and one for par, but where both were in almost exactly the same place on the green, in the same round. The results held: par putts were holed significantly more frequently than birdie putts.

It turns out the difference is mostly length: when pros putted for birdie, they were more likely to leave the putt short than when they were putting for par. The idea is that the birdie golfers are playing conservatively: they want to make sure that if they don't sink the putt, they leave it close to the hole for an easy subsequent shot. On the other hand, the par golfers are scared to miss, because that appears to cost them the loss of a stroke, so they make sure they give it enough weight. For putts of 22.5 feet or longer, birdie putters leave the ball about two inches shorter than par putters.

From a strict strategic standpoint, the conservative play makes little sense. The authors find that after a missed (presumably conservative) birdie try, golfers make the subsequent putt only 0.2 percentage points more often than after a missed (presumably aggressive) par try. So the conservatism costs 3 percentage points, but gains only 0.2 percentage points. Moreover, the gain comes only if the putt is missed: assuming a 50% conversion rate on the original putt, the net gain from conservative play is only 0.1 percentage point!

So why are golfers accepting a much worse first putt for a very, very slight chance at avoiding a three-putt? It must be loss aversion. There are still two possible ways that could manifest itself. It could be that golfers are deliberately trying to hit the ball softer on birdie tries in a conscious effort to be more conservative. Or, it could be that golfers are just trying harder in general on par tries. Either way, that's something that you'd think pros would be able to control, if they realized that what they were doing makes no sense.

Has any golfer figured this out? Apparently not. The authors calculated the effect for each of 188 golfers in their data sample; they got a bell-shaped curve centered around 3.5 percentage points. The "most irrational" golfer's difference was 7 percentage points; the "least irrational" golfer was at half a percentage point. That is: every one of the 188 pro golfers exhibited this bias, to one extent or another. Just as surprising was the finding that the size of the bias was barely correlated with the ranking of the player. Better golfers had less bias, but only very slightly less. Tiger Woods, the best golfer in the world, was almost exactly average in this measure. By eliminating the bias, and shooting birdie putts the same as par putts, Pope and Schweitzer calculate that the average PGA pro will take one stroke off his score for a 72-hole tournament. If a top-20 golfer did this (and none of the other 19 did), his earnings would improve by 22% -- over one million dollars per year.

That's hard to believe, but there it is.

Read the whole study; the authors diced the data many different ways to confirm their thesis, and there are many interesting findings. A New York Times article about the study is here.

P.S. I've been getting lots of spam comments lately, so now comments on older posts (28 days old or more) are moderated. Comments on current posts will continue to appear instantly.

Hat Tip: Inside the Book

A new golf handicapping system

The Royal Canadian Golf Association (RCGA), Canada's governing body for golf, has a committee to consider updating the system by which a golfer's handicap is computed. Tim B. Swartz, the committee's statistical guy, has a paper in the most recent JQAS explaining the new proposed system.

I'm going to simplify things a bit and explain the situation as I understand it from the paper. Leaving out the technical adjustments (even at the cost of a bit of inaccuracy), I'll describe the current handicap system like this:

You start with your 20 most recent scores (with a few adjustments that I'll discuss later), relative to par. Then, you drop the 10 worst scores, leaving only the 10 best. You average those 10 best. That's your handicap.

Why change this system? For one thing, as Swartz points out, your handicap isn't a true indication of your expected score; golfers fail to shoot their handicap more often than not. As I see it, you're taking the average of the top half, so you'd expect the handicap to be at about the 25% mark. If you assume that scores are normal, then since the normal curve is fatter near the middle, it's a bit more than 25%. As it turns out, the mean of the right half of the normal curve is about .6367 .798, and the chance of beating a Z-score of +.6367 +.798 is about 26.2% 21.2%. So you'd expect a golfer to beat his handicap about 21% of the time.

Swartz checked, using a database of scores from a golf club in Alberta. As it turns out, golfers actually beat their handicap 36% of the time, not 21%. Maybe I made a mistake in the calculation; maybe golf scores aren't really normal; or maybe the various adjustments are causing the difference.

Another problem with the current system is that in casual head-to-head play, it favors the better golfer. Swartz generated a bunch of random matches from his database, and found that the better golfer won 55 percent of the time, rather than 50 percent.

A third problem, and an important one, is that in multi-player tournaments, the winner is likely to be a golfer with a higher handicap. That's because a bad golfer, with a handicap of (say) 20 (which represents a score of about 92), could reasonably have a very good day and shoot an 80, finishing at -12. But a scratch golfer (0 handicap, 72 average) is much less likely to match the -12 by shooting a 60 on the day.

The more players in the tournament, the more likely someone will have a much better game than normal. And those "much betters" are likely to be from the worst golfers.

In his simulation, Swartz found that the top third of golfers won only 27% of the 99-player tournaments. The middle third won 33%, and the worst third won 40%. So the current system favors the better golfer in tournaments of two players, but favors the worse golfers in tournaments of many players.

So how does Swartz fix the current system? Two ways: he makes the handicap represent the player's average score, instead of his 74th percentile score. Second, he divides by a player's standard deviation (effectively converting a raw score to a Z-score), which neutralizes the luck factor in large tournaments.

Here are the details.

Like the current system, Swartz considers only the 20 most recent scores. But instead of dropping the worst 10, he drops only the worst four, leaving 16 scores. Then, instead of just averaging them, the new system uses mathematical statistical techniques to estimate the best normal curve to fit the data (keeping in mind that the four worst scores are missing). That is, it asks the question: what is the best fit normal curve that takes into account that we're looking at only the best 16 of 20 observations?

Swartz gives linear formulas (like a Linear Weights estimate of the 16 scores) to estimate the mean and SD of that best-fit curve; he says that those formulas are minimum variance linear unbiased estimators, which means you can't do better (by using different weights) unless you go to a non-linear estimator.

Those estimates of mean and SD become the player's stated handicap (so, effectively, there are two numbers for the handicap instead of one). Then, for his next (21st) round, his raw score is converted to a Z-score, and that's what gets compared to the other players' Z-scores to determine the winner.

In the study's simulations, golfers shot their handicap 45% of the time with this new system (fairer than 36% with the old system); one-on-one matchups were won by the better golfer only 48 to 51 percent of the time (fairer than 55%); and in tournaments, the best golfers won 29% of the time (fairer than 27%) while the worst won 32 to 34 percent of the time (fairer than 40%).

I promised some details of the adjustments to player scores that go into the formulas. I'll outline them here, and you can see the paper (which is nicely presented and very easy to read) for the details.

First, under both systems, scores are adjusted twice for the difficulty of the course. There's the course rating, which specifies how hard the course is for excellent (scratch) golfers, and the slope rating, which specifies how hard the course is for worse (bogey) golfers after adjusting for the course rating.

Then, there's something called "equitable stroke control" (ESC). That sets a maximum possible score for each hole, so that (for instance) a bad golfer can't score more than a quadruple bogey. Even if it takes him ten strokes to finish a par-three, he can't put more than 7 down on the scorecard. (In Canada, the stroke limit varies by handicap between bogey and quadruple-bogey; in the US, it seems the limits are fixed and not based on par. Swartz says this is the only difference in the current system between the two countries.)

The idea is that very high scores measure golfer frustration rather than skill, and should be discounted. Also, Swartz says, it discourages "sandbagging," which is deliberately trying to inflate your handicap, and provides a maximum if you forget to write down your score.

In this study, Swartz often gives results both with ESC and without, and the results are fairly similar.

But, after all those adjustments, I think the essence is:

-- the old system adjusts for your score relative to your own 63rd percentile.
-- the new system adjusts any outliers in your worst quintile, and gives you a Z-score relative to your own distribution.

I'm not a serious golfer, so I don't know if the added complexity of the new system is worth the advantages. It does seem to me that the new system is better, though.

Golf numbers: comparing amateurs to pros

Last Monday, the New York Times described some interesting findings in golf sabermetrics.

The analysis, by golf researcher Mark Broadie (who does research for the PGA), describes and quantifies some of the differences between amateur golfers and the pros. So if you're looking to compare Tiger Woods to Phil Mickelson, Broadie's work probably won't help you quite yet. But it's interesting nonetheless.

Broadie had the PGA database with which to analyze pro scores. For amateur scores, he got players at a local course to log all their shots. He wound up with 43,000 amateur strokes, which, I suppose, is about 400-500 rounds.

He then analyzed the entire database to break down some of the score differences. For instance, at what distance is there a 50% chance of sinking a putt? The professionals break even on 8-foot putts, but, as you would imagine, it's shorter for amateurs. The breakdown by handicap:

8 feet: Pros
6 feet: Amateurs with handicap of 0-9 [I'll call these "A" amateurs]
5 feet: Amateurs with handicap of 10-19 ["B"]
4 feet: Amateurs with handicap of 20-36 ["C"].

In yardage off the tee:

279 yards: Pros
248 yards: Group A amateurs
237 yards: Group B amateurs
216 yards: Group C amateurs

And, a very interesting statistic: when hitting from 100 to 150 yards away from the hole, what percentage of that distance is left after the stroke?

5.6%: Pros
8.7%: Group A amateurs
12.0%: Group B amateurs
17.3%: Group C amateurs

And so Broadie suggests a way to see if your short game needs more help than your long game: figure out your own percentage, and compare it to which group you fit into. For instance, suppose your handicap is 3, which puts you in Group A. If your percentage is higher than 8.7, it means your short game is worse than that of your peers (which means your long game is better).

There are more tidbits: check out the entire article.

HT: Bob Timmermann

Guest post: Don Coffin on golf performance measures

(This is Phil. In response to a previous post on golf scores, Don Coffin, an economist and sabermetrician from Indiana University Northwest, did a little extra research. Here's Don:)

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I [Don] have done some research on the relationship between various measures of golfer performance and overall performance (strokes per round; prize money), but have found it difficult to know exactly where to go with it. Here’s how I have approached it.

Overall performance, measured as strokes per round, has three components:

1. Shots off the tee. There is essentially no variation in this measure of performance. Everyone has essentially one tee shot per hole, or 18 per round, and the standard deviation is almost zero.

2. Putts. There is some variation here, but less than one would expect. In 2007, according to data at PGATour.com, the average number of putts per round (averaged across golfers, so this is an average of averages) was 29.30, with a standard deviation on 0.52. The coefficient of variation was 1.77%.

3. All other shots. There’s a very little more variation here; again, using 2007 data, the average was 23.98 “other” shots per round, with a standard deviation of 0.63, and a coefficient of variation of 2.62%.

Overall, golfers in 2007 averaged 71.28 strokes per round, with a standard deviation of 0.59 strokes per round, for a coefficient of variation of only 0.83%. Overall performance was, then, less variable than the components of scoring with some variation.

The PGA reports a number of what it calls “skill statistics;” all of these are reported in Table 1. (Putts per round shows up in the “skill statistics;” “Other Shots” is Strokes per Round, minus Putts per Round, minus 18). If our objective is to explain overall performance, as measured by Strokes per Round, then we have to select explanatory variables from among the available performance measures. For 2007, the PGA reported all the “skill statistics” data for 196 golfers.

I believe it is inappropriate to use Putts per Round as an explanatory variable. If we could control adequately for other performance measures, then the (expected) coefficient on Putts (in a multiple regression) would be 1—each additional putt would raise Strokes per Round by 1. What would be useful, however, would be to find explanatory factors for the components of Strokes per Round—Putts, and Other Shots.

( ... continued)

Long tee shots: how much do they improve PGA golf scores?

PGA golfers are hitting for distance better than ever before. Is that contributing to an improvement in their scores? Or, by going for the long drive, are they losing so much accuracy that the increased distance doesn't help?

An article (fortunately available online) in the new issue of Chance Magazine (published by the American Statistical Association) tries to answer that question. It's Called "Today's PGA Tour Pro: Long but Not so Straight," by Erik L. Heiny.

Heiny starts by showing us that average PGA driving distance has increased substantially in recent years. Between 1992 and 2003 (these seasons are the ones used throughout the paper), Figure 1 shows an increasing trend from 260 yards to 287. Driving accuracy, though, is flatter; Figure 19 shows a relatively stable trend up to 2001, at which point accuracy suddenly drops from 68% in 2001 to 66% in 2003.

Then, Heiny runs some straight regressions between various aspects of golf performance, and gives us the year-to-year correlations. Unfortunately, he doesn't give us the regression equations, which is where most of the knowledge is. For instance, driving distance is positively correlated with score. But what's the size of the effect? If Phil Mickelson increases his distance by 5 yards, what can he expect as an improvement? Half a stroke? One stroke? Two strokes? This is important and useful information, but Heiny doesn't tell us.

I found the correlations for the different variables much more interesting than the year-to-year trends, which are mostly flat, with occasional exceptions in 2003. Given that 2003 is also the year driving accuracy dropped, you've got to wonder if something specific happened that year to make all these things happen at once.

In any case, the most interesting correlations (Figures 2-15) are:

Driving distance vs. scoring
Driving accuracy vs. scoring
Greens in regulation vs. scoring
Putts per round vs. scoring
Sand saves vs. scoring
Scrambling vs. scoring
Bounce back vs. scoring

(I should give you the definitions for some of the less obvious variables. "Driving accuracy" is percentage of drives (excluding par 3s) that landed on the fairway. "Greens in regulation" is percentage of holes in which the green was reached in (par – 2) strokes or fewer. "Sand saves" is percentage of balls in sand traps holed in two shots or fewer (I think). "Scrambling" is how often a par (or better) was made, as a percentage of holes where the green was *not* reached in regulation. And "bounce back" is the percentage of times a player gets a birdie or better after a hole of bogey or worse.)

I was surprised at some of the correlations. Which of the seven factors above would you think had the biggest influence on score? I would have thought "greens in regulation" and "putts per round." I was half right. Here are the approximate numbers, as I eyeballed them from the graphs:

0.2 Driving distance vs. scoring
0.3 Driving accuracy vs. scoring
0.7 Greens in regulation vs. scoring
0.3 Putts per round vs. scoring
0.3 Sand saves vs. scoring
0.6 Scrambling vs. scoring
0.5 Bounce back vs. scoring

(The author also repeats these correlations for money instead of scoring, but, for the most part, the conclusions are the same, so I won't mention them further.)

From these correlations, Heiny draws some tentative conclusions about how driving distance has affected play. For instance, in 2003, the correlation between driving accuracy and scoring decreased from 0.2 to 0.0. Heiny writes that this suggests that

"... with the driver going so far, it just didn’t matter whether the player was in the fairway. With longer drives, players could get close enough to the green to play short irons or wedges. Even from rough, they could control the shot into the green."

I'm not sure I'd agree with that. Driving distance went up only 6 yards that year, and I'd be more inclined to view the big drop in correlation as random. Indeed, in the first ten years of the study, distance increased 20 yards, with little change in accuracy or correlation.

In any case, the author proceeds to multiple regressions, where he predicts a player's score based on the seven variables above. He runs one regression for each year.

Again, we don't get any equations, just signficance levels. Summarizing the 12 regressions, here's what the article shows:

Highly significant: driving distance
Highly significant: greens in regulation (GIR)
Highly significant: putts per round
Highly significant: scrambling
Occasionally significant (3 years out of 12): driving accuracy
Occasionally significant (5/12): bounce back
Not significant (0/12): sand saves

Strangely enough (to me), a couple of results were slightly different when predicting (the logarithm of) money winnings instead of score: drive accuracy went from 3/12 to 7/12, and scrambling went from 12/12 (all of which were < .0001) to 2/12 (many of which were greater than 0.5, which means *negative* correlation!) I can think of a few reasons why this might occur (high money might be correlated with a longer course, which means higher scores; high money might mean better caliber golfers, which might have different characteristics; and so on). In the money regressions, driving distance was extremely signficant (p < .0001) up to 2000, when it started becoming less important, hitting p = .0545 (not significant at 5%) in 2003. Also, driving accuracy seemed to be more signficant in the early years than the later years. This prompts Heiny to say that

"it seems to be evident that accuracy off the tee is becoming less important as driving distances increase."

But it seems to me that you can't really draw any conclusions like that from the regression, because of the choice of variables.

For instance: how would in increase in driving accuracy improve scoring? It would do so by increasing the chances of landing on the green in regulation. But "greens in regulation" is another variable being used in the regression! And so even if driving accuracy doesn't come out as significant, that might be because most of its influence is on the "greens in regulation" variable.

We know that if you're a pitcher, giving up a lot of hits increases your chances of losing the game, right? But if we run a regression on losses, and include hits *and runs*, hits will come out as completely insignificant. Why? Because your chance of winning depends on how many runs you give up – but, if you give up four runs, *it doesn't matter* how many hits you allowed in producing those four runs. Hits and runs don't indpendently cause wins: hits cause runs, and runs cause wins. Hits are irrelevant if you know runs.

Hits -----> Runs -----> Wins

The same kind of thing holds for driving accuracy:

Accuracy -----> Greens in Regulation -----> Score

If Vijay Singh his 72% of greens in regulation, it doesn't matter much how he got to that 72%: his score will be roughly the same if he got it by inaccurate drives with good recoveries from the rough, or accurate drives with average approach shots. And so, just like hits are irrelevant if you know runs, accuracy is irrelevant if you know GIR.

The analogy isn't perfect: accurate tee shots affect more than just GIR. They may lead to a better position on the green, which will affect putts per round (which is also a variable in this regression). In cases where you miss the green, a more accurate tee shot might lead to an easier scramble (also a variable), or a better lie in the sand (again a variable).

Since all those other things are accounted for in the regression, the surprise is that accuracy would ever be significant at all! There must be other ways in which accuracy improves scores. What might those be?

We can start by noting that score can be computed *exactly* from these four variables:

A = Percentage of GIR
B = Percentage of greens in one stroke less than regulation
C = Number of putts taken per hole
D = On missed GIRs, average number of extra strokes taken to get on to the green

Then, the average hole's score relative to par simply equals (C – 2) - B + D(1-A). Simplifying gives

Exact (average) Score = C - B + D - DA - 2

So if you ran a regression on B, C, D, and DA, you would predict score perfectly -- you'd get a correlation of 1, and the above regression equation. (If you ran it on A, B, C and D, you'd come close to 1, but wouldn't hit it exactly, because there's an interaction between D and A that you wouldn't capture.) And adding other variables to the regression – such as accuracy – would do almost nothing, because accuracy "works" by causing a change in one or more of A, B, C, or D.

Now, Heiny didn't actually include all of A, B, C and D in his regression. But he came close. His regression did include:

A
C
Scrambles, which is significantly correlated with D and C.

So what's left that he didn't include? B, and part of D. So when one of the other variables, such as driving accuracy, comes out significant, it must because of its effect on B or D:

-- it increases the chances of getting to the green in less than regulation (B); and
-- it increases the chance that if you miss the green in regulation, you'll get back on in few strokes (D).

Looked at in that light, it's easy to see why driving distance is significant: to get yourself a B, you need to hit a par-5 green in two strokes, so you better be hitting long. And it's easy to see why accuracy is significant – landing on the fairway means you avoided losing your ball or hitting in the water, saving yourself a lot of D.

(By the way, could it be that "sand saves" comes out insignificant because all sand saves are actually scrambles? Or does the definition of "scramble" explicitly exclude bunker shots?)

But those minor results are not what the study was looking for. The point of the study was to see if distance and accuracy caused scores to drop in general, not to see if it caused scores to drop only because of "greens in better than regulation" and "water hazards avoided".

And if want the overall effects of distance and accuracy, you can't include any variables that are *also caused* by distance and accuracy. Run your regression on distance and accuracy only, and see what you get.

P.S. I wrote about this kind of flaw earlier, probably more understandably.

P.S. A PGA study once showed that "driving the green" -- trying for a "B" by going for the green in less than regulation -- was associated with lower scores.

Do golfers give less effort when they're playing against Tiger?

According to this golf study by Jennifer Brown, the best professional golfers don't play as well in tournaments in which they're competing against Tiger Woods. Apparently, they have a pretty good idea that they won't beat Tiger, and so they somehow don't try as hard.

Brown ran a regression to predict a golfer's tournament scores. She considered course length, whether Tiger was playing, whether the golfer was "exempt" (golfers who were high achievers in the past are granted exemptions) or not, whether the event was a major, and so forth. It turned out that when Tiger was playing, the exempt golfers' scores were about 0.8 strokes higher (over 72 holes) than when Tiger sat the week out. Non-exempt golfers, on the other hand, were much less affected by Tiger's participation – only 0.3 strokes. Brown suggests that the difference is that the non-exempt golfers know they can't beat Tiger anyway, so his presence doesn't deter them from playing their best.

Also, when Tiger was on a hot streak – his scores in the previous month were much better than other exempt golfers – the effect increases. Now, instead of being just 0.8 strokes worse than usual, the exempt golfers are 1.8 strokes worse. During a Tiger "slump," the exempt golfers are so pumped by their chance of winning that they're *better* than usual, instead of worse – by 0.4 strokes.

The result seems reasonable – the less chance you have of winning, the less effort you give. But 0.8 strokes seems like a lot. That's especially true when you consider that it's not *every* time that Tiger Woods runs away with the lead. It might be 0 strokes when Tiger is struggling, but 1.6 strokes when it's obvious that it's a lost cause this week. And how does Phil Mickelson lower his scores by 0.8 just by trying harder? Is it more practice? Is it setting up better? Is it spending more time reading the green? What actually is it?

(UPDATE: a couple of commenters noted that with Tiger in the field, certain opponents might change their style of play to take more risks to beat him, and that might cause the scoring difference. However, the study checked for that, by looking at the 72 individual holes, and it found no difference in the variance based on whether or not Tiger was playing.)

There's something else that's confusing me, and that's one of the actual regression results in the paper. Brown has a dummy variable for (among other things) every player, every golf course, and whether or not the tournament is a major. The coefficient for the major is huge: around 17 strokes.

That doesn't make sense to me, because that's after adjusting for the player and the course. It says that if Phil Mickelson plays Pinehurst #2 twice, with the same course length and the same wind and temperature, but one time he's playing a major and the other he's not, *his score will be 17 strokes higher in the major*. That just doesn't sound right to me. Why does calling a tournament a "major" make every golfer 17 strokes worse? (See tables 4 and 5 of the study.)

(Also, how can you divorce the course from the major? As far as I know, the Augusta National Golf Club, which hosts the Masters (a major), doesn't host any other PGA tournaments. So what happens when you run the regression, and the Augusta dummy is always the same as the Masters dummy? I'm not an expert after my one course – but doesn't that cause some kind of matrix problem in the regression?)

I wouldn't even expect a coefficient of +17 only if you didn't adjust for the course at all. Well, maybe if you look at last year's Masters. At Augusta in 2007, every golfer finished over par. But in the 2008 Buick Invitational (which is not a major), 34 players finished at or below par, and the winner (Tiger) was at –19.

But that's an exception. Eyeballing the four majors on Wikipedia, the Masters and PGA Championship usually feature a winner around –8. The British Open looks a little easier, and the US Open a little tougher. Eyeballing the non-majors in 2007, the typical winning score looks to be somewhere in the mid-teens.

So the difference is probably around 7 strokes or so. After correcting for the higher caliber of golfer in the majors, you might get to, what, 12 or 13? You're still short of the 17 the regression found. And, again – that's BEFORE adjusting for the difficulty of the course! So I just don't understand.

What's going on here? Am I figuring something wrong?

Hat Tip: The Sports Economist

An inconsistency in Tiger Woods betting markets?

There are many PGA golf tournaments throughout the year. A specific four of those tournaments are called "majors."

What are the chances that Tiger Woods will win all four majors?

You can start by assuming that Tiger has the same chance of winning any of the four. This probably isn't true, but it's a good start. A friend of mine notes that at his online bookmaker, the current odds (expressed as probabilities) of Tiger winning are:

42% Masters
44% US Open
37% Open Championship
37% PGA Championship

Taking an average of about 40%, we can calculate that the chance of Tiger winning all four events is 0.4 to the fourth power, which is 2.56%. Without showing my work, here are the binomial odds of Tiger winning various numbers of majors:

4 majors: 2.6%
3 majors: 15.4%
2 majors: 34.6%
1 majors: 34.6%
0 majors: 13.0%

However: at TradeSports.com (choose "Golf" from the left menu, then "Tiger Woods Props"), the current odds are very different. Here they are, taking the halfway point between the bid and ask. (My friend's bookie's odds for these bets are almost the same.)

4 majors: 8.5% (-not the theoretical 2.6%)
3 majors: 12.7% (not the theoretical 15.4%)
2 majors: 24.5% (not the theoretical 34.6%)
1 majors: 35.0% (yes the theoretical 34.6%)
0 majors: 22.5% (not the theoretical 13.0%)

There appears to be a mismatch between these odds and the individual tournament odds. It could just be that TradeSports bettors are assuming a different win probability than 40%.

What would it take to give Tiger an 8.5% chance of winning all four majors, as Tradesports estimates? He'd have to have a 54% chance of winning each major (.54 to the fourth power is about .085). But if he DID have a 54% chance, the other odds should be different. Here's the full table assuming 54%:

4 majors: 8.5%
3 majors: 29%
2 majors: 37%
1 majors: 21%
0 majors: 4%

These are very, very different from the posted odds. If you really thought Tiger had a 54% chance of winning each major, you should also believe he has only a 4% chance of losing them all. You should sell the 22.5% contract for $2.25, knowing it's only worth 40 cents, and make lots of money.

But what if you don't believe the "4 majors" odds of 8.5%? What if you believe that the "0 majors" odds of 22.5% is the correct number?

In that case, you'd have to assume Tiger had a 31% chance of winning each tournament and a 69% chance of losing, since .69 to the fourth power equals 22.5%. Based on that assumption, the true odds are:

4 majors: 0.9% (TradeSports: 8.5%)
3 majors: 8.2% (TradeSports: 12.7%)
2 majors: 27 % (TradeSports: 24.5%)
1 majors: 41 % (TradeSports: 35.0%)
0 majors: 31 % (TradeSports: 22.5%)

In this case, you'd believe that Tradesports is hugely overestimating the "4-majors" odds. You'd be happy to sell a "4 majors" contract at 8.5%, knowing the true odds are really only about 1%. You'd be making, on average, a 700% markup!

No matter what probability you assume for a Tiger win, the odds just don't seem to make sense. It looks like there are some serious opportunities here to make money. If you believe Tiger is a 54% winner, lay odds on "0 majors." If you believe Tiger is a 31% winner, lay odds on "4 winners". And so on. Whatever you believe Tiger's correct odds are, there's an advantageous bet for you.

And not just a *slight* advantage – a HUGE one! If you go with the 54% estimate, and lay odds on "0 majors," you're getting a 500% markup. If you go with the 31% estimate, and lay odds on "4 majors," you're getting that 700% markup.

Indeed, there is no single-tournament probability that causes the posted odds to make enough sense that there isn't a huge opportunity one way or another.

That confused me. Aren't markets supposed to be efficient? How can these probabilities be so far out of line?

Then it occurred to me: in the arguments above, which bet was advantageous depended on Tiger's true odds of winning. But what if that itself is unknown? Not just unknown, but REALLY unknown – random, in the sense that no amount of study can figure it out. Maybe Tiger has off-years and on-years, and there is absolutely no way of knowing which it's going to be. In fact, Tiger's skill level might not yet be set – it might depend on his practice level, or his health, or his personal life. Suppose it would be randomly decided just before the Masters?

More specifically, let's suppose that Tiger's single-tournament probability might be 15%, or 20%, or 25%, or so on, up to a maximum of 70%. And suppose each of those 12 probabilities is equally likely. If we do the math based on that assumption, we come up with:

4 majors: 08.0% (market odds 8.5%)
3 majors: 19.6% (market odds 12.7%)
2 majors: 26.2% (market odds 24.5%)
1 majors: 26.9% (market odds 35.0%)
0 majors: 19.2% (market odds 22.5%)


On a more practical level, the theoretical odds now come reasonably close to the market odds. Not perfect, but much better than any of the situations based on a single, fixed probability. Indeed, maybe there's a model that might come even closer to market odds: maybe, for instance, if Tiger has a 50% chance of being way off, with a 10% probability, but a 50% chance of being on, with a 70% probability, we get the TradeSports odds exactly. I haven't tried that, or any of the infinity of other possibilities. The closest I came, though trial and error, is the chart above. But there might be a way to solve this mathematically, instead of by hacking away like I did.

(Another interesting thing about this method is that the assumptions have Tiger's expected single-tournament odds at 42.5% (the sliding scale from 15 - 70% averages 42.5%). The chance of winning four straight tournaments at 42.5% is about 3.3%. At first glance, that might lead you to believe that the odds of "4 majors" should be 3.3%. But, because the probability of a sweep isn't linear on the probability of a single tournament, it doesn't work that way. The actual chance is 8%, not 3.3%. This could be the explanation of why my friend's bookie has each tournament at 40%, but the parlay of all four tournaments at much more than 40% to the fourth power.)

In any case, I originally thought there was a seriously advantageous betting opportunity. There's less than I thought. However: in everything I tried, the "3 majors" number was significantly higher than the "4 majors" number. It's only 50% higher at TradeSports. I think money can be made by selling the "4 majors" contract at 8.5%, and buying the "3 majors" contract at 12.7%. It's not a sure thing, but I think the expectation has got to be positive.

Here's a challenge: can you come up with some plausible set of assumptions under which the "3 majors" probability is only 50% higher than the "4 majors" probability? My gut says you can't – that this is one of those examples of "favorite-longshot bias," and one that you can easily take advantage of right now.

But I'm not certain of this, and markets are a lot smarter than I am. There might be something I'm missing. Don’t blame me if Tiger hits the grand slam and you lose your shirt.

"Win Zone" gives golfer win probabilities for tournaments in progress

Suppose that after the first round of a four-round PGA golf tournament, Tiger Woods is at 67 while some guy you never heard of is leading at 66. Who has the better chance of being the eventual winner?

It's obviously Tiger: he's capable of putting a good score together every round, while the leader probably just had a lucky Thursday. Tiger is likely to wind up around 67-67-67-67, while the other guy will probably go something like 66-73-78-74.

A new statistic from the Golf Channel, called "Win Zone," follows this logic to try to estimate players' chances of winding up the winner of the tournament. It looks at the players' histories, their scores, what hole they're on, how hard the course is, and so on, and comes up with a number representing the player's actual chances.

The Golf Channel website describes the system via a text summary and a video (follow above link), but doesn't give a lot of details on how it works. They do say that it runs "over 2 million calculations every minute," which suggests a simulation, or perhaps a Markov Chain analysis (but a simulation seems much more likely).

As I write this, the second round of the "2007 AT&T National Pebble Beach Pro-Am" is complete. Here are the Golf Channel's top 10 in "Win Zone," along with those players' leaderboard stats. (This link is probably good until tomorrow when the next round starts.)

01. 35.7% Jim Furyk .......... –12 (tied for 1st)
02. 33.0% Phil Mickelson ..... –12 (tied for 1st)
03. 11.1% John Mallinger ..... – 9 (tied for 3rd)
04. 06.5% Kevin Sutherland ... – 9 (tied for 3rd)
05. 03.4% Craig Kanada ....... – 7 (tied for 5th)
06. 03.0% Davis Love III ..... – 7 (tied for 5th)
07. 02.5% Mark Hensby ........ – 7 (tied for 5th)
08. 01.8% Vijay Singh ........ – 5 (tied for 11th)
09. 01.5% Aaron Baddeley ..... – 5 (tied for 11th)
10. 01.0% Jesper Parnevik .... – 4 (tied for 18th)
10. 01.0% Atwal Arjun ........ – 2 (tied for 40th)
10. 01.0% Kirk Triplett ...... – 2 (tied for 40th)
10. 01.0% Justin Leonard ..... – 1 (tied for 57th)

This is pretty much what you'd expect – but I'm surprised at how Justin Leonard is seen to have as good a chance to win as Jesper Parnevik. Parnevik is three strokes ahead of Leonard, and has only 17 players to pass. Leonard has to make up 11 strokes in two rounds, and jump over 56 other players on the way up.

You'd expect Leonard's high probability of winning, despite his mediocre score so far, would mean he's a better golfer than Parnevik. It doesn't seem that way. I don't follow golf much, but Leonard has a "World Golf Rating" (whatever that is) of 176th, while Parnevik is 107th. So that can't be it.

What is it, then? According to the Golf Channel article and video, the ranking takes into account such other factors as:

-- who played well on the course in previous rounds of this tournament
-- how players have done on this course in the past
-- how players have done on *specific holes* of this course in the past.

So I'm wondering whether Win Zone isn't reading too much into the small samples of previous player/course/hole results. There's no way to tell, because they don't give details of their calculations or weightings. But I'd be wary of any statistic that considers whether a player is on a "hot hand," in light of the fact that studies have generally been unable to find such an effect in other sports.

Aside from that, does Win Zone work?

Well, there's really no way to know; Win Zone is new, and there's not enough data to analyze. The Golf Channel's arguments in its favor aren't really all that relevant. For instance, they tell us that Win Zone gives a better chance of picking the winner than just the current leaderboard. But that's not much of an achievement – in the example in the first paragraph, it would be obvious to any fan that Tiger had a better chance of winning than his no-name opponent, and we wouldn't need a fancy methodology to tell us that.

(Also, I disagree with some of the explanations in the video – for instance, they argue that a Win Zone probability of 50% is a "milestone," because at that point "the odds are with you." To which I say, "so?" Why should 50.1% be that much more important than 49.9%?)

But one reasonable way to check the system is to compare it to "market odds" from tradesports.com:

Furyk ...... 31.8 to 34.3% (Win Zone says 35.7%)
Mickelson .. 33.2 to 35.5% (Win Zone says 33.0%)
Love III .... 3.8 to 6.7 % (Win Zone says 3.0%)
Singh ....... 1.9 to 4.3 % (Win Zone says 1.8%)

Of course, the bettors' actions could be influenced somewhat by Win Zone – but betting markets tend to be pretty smart, and I trust their estimates to be hard to beat.

So it seems to me that Win Zone would give you a reasonably accurate rundown of the probabilities, at least for the favorites. I do wish they had given more details of their system, but even without those details, their win probabilities are a useful addition to the regular leaderboard stats.

(Thanks to John Matthew IV for the pointer.)