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Research on Pyschological Impacts on Performance

I have written a lot on the consistency of performance and using past performance to predict future performance. Once you have the data, those studies are straightforward to conduct and produce intuitive results. I’ve neglected much discussion of the mental side of the game because, on the whole, there isn’t any data out there that directly measures whether a player is confident, nervous, distracted, overwhelmed, able to cope with pressure, etc. I’ve just read two papers – Confidence Enhanced Performance by Rosenqvist & Skans and The Impact of Pressure on Performance by Hickman & Metz – that attempt to measure the psychological impacts on performance inherent in golf using performance data.

Confidence Enhanced Performance:

Rosenqvist & Skans use European Tour data from the past decade to measure the impact of confidence on performance. Because of the existence of the cut in most tournaments and the natural division of the field into successes and failures by the cut, it’s possible to look at how making or missing the cut affects performance in the subsequent tournament. Players who make or miss the cut are separated by very small differences in performance (as little as a single stroke for those directly on either side of the cut line) and are also nearly identical in terms of long-term talent. That means we should expect their performances to be similar in subsequent weeks – assuming that there isn’t any impact from prior weeks.

What Rosenqvist & Skans find is that there is a difference in performance between those who barely made the cut and those who barely missed the cut (they create these groups using players within six strokes of the cut in either direction, though they also compared smaller ranges). Players who just made the cut in the treatment tournament are ~3% more likely to make it in the outcome tournament. Players who make the cut also play ~0.125 strokes better per round in the first two rounds of the following tournament. The authors explain this outcome as a product of enhanced or diminished confidence effecting the players’s performance.

I’ve found similar impacts on performance in my own work. Players who exceed their normal or expected performance one week retain a small portion of that over-performance the following week; that is, they continue to perform slightly better the following week. The same is true for those who play worse than expected. It’s very difficult to say precisely why this occurs.

The authors say it’s because the player in question is more or less confident than normal. They designed their study to directly compare players with similar performance who were clearly separated into successes and failures. However, the difference in strokes between making and missing the cut is still large – even comparing only those players on either side of the cut line. In their study, the smallest range they examined was those players within four strokes to either side of the cut line. There is still a substantial difference in performance between the two groups; the missed cut group on average played one stroke per round worse than the cut line and the made cut group on average played one stroke per round better than the cut line. That shows a two stroke difference between the two groups of players. The authors showed that both groups were comprised of fairly similar players, so the made cut group slightly overachieved by roughly one stroke per round and the missed cut group slightly underachieved by roughly one stroke per round.

So the authors were not comparing players who were separated only into a successful and an unsuccessful group. They were comparing players who were successful/had overachieved versus a group who were unsuccessful/had underachieved. It’s impossible, with this data, to state that the observed differences in making or missing the next cut can be explained by confidence versus other factors. The slightly higher probability of making the cut in subsequent tournaments can be just as easily explained by saying the player’s swing was slightly better than normal or that his mind/body were in better physical condition. Teasing apart the impact of psychological vs. physical is difficult – perhaps impossible without administering a psychological evaluation and analyzing Trackman launch monitor data.

The authors’s finding of a small carry-over in performance is the important discovery, though. They show that players who perform well the previous week perform slightly better the following week than those who did not perform well the previous week – by around 0.125 strokes per round. However, this impact is extremely small and is surely overstated by those writing about and discussing golf. Good play the previous week only barely increases the probability that you will play well the following week. This reinforces the importance of looking at long-term performance when attempting to predict future performance.

The Impact of Pressure on Performance:

Hickman & Metz use Shot Link data to examine the probability of making a putt on the last hole of a tournament, considering the amount of money riding on the putt, the distance of the putt, the experience of the player, the amount of money the player won in the previous season, and the putting performance of the player so far in the tournament. They find that for every ~$30,000 that is riding on a putt (ie, if someone makes they win $100,000 and if they miss they win $70,000) the probability of making the putt drops by 1% all else being equal. They also find that this most impacts short putts of between 3 to 12 feet.

I don’t have much to say about this study except that I wish the authors had included a better control for putting ability. They controlled for ability using Total Putts Gained (basically Strokes Gained Putting throughout the tournament). Unfortunately, putting performance in small samples like one tournament is essentially noise. If you know how well a player putts in general, how well they are putting in a tournament isn’t predictive at all. In their study you’ll see that TPG is a significant factor in predicting the probability of making a putt, but if you look at the coefficients you can see that for every Total Putt Gained in the tournament (or for every 0.25 TPG in each round) the probability of making a putt increases by 0.9%. Over 18 holes this would translate to 0.16 putts gained. So players who putt well, in general, will tend to putt well in a tournament, so they’ll tend to make slightly more putts on the 18th hole. I don’t think this effects their results, but I do wish they’d controlled for ability in a better way. It’s possible that good putters block out the pressure better, or that bad putters are less affected by pressure.

Edit: It’s also unclear whether they take putts gained on the final hole out of their TPG measure.

However, the effect they’ve found is significant in terms of examining the impact of pressure. It indicates that for putts with very large differences in prize money (for example the Bubba Watson example they quoted on page 9 where $300,000 was on the line) the difference in probability of making the putt compared to a non-pressure situation could be up to 10%.

The Impact of Tournament Position on Performance:

I’ve dug into my database of results for the last half decade and examined the impact on performance of starting position in terms of strokes behind the leader. I’ve found two interesting results: 1. players who begin the second and fourth rounds a large number of strokes behind the leader perform worse than expected (I have compared all performance to my expected z-score performance) and 2. players who begin the fourth round in the lead or one stroke back perform worse than those who begin the round near the leader, but further back. For #1, it’s possible that these players are out-of-form (whether because of injury, swing, fatigue, etc.) or that they’re “giving-up” – focusing less because there’s less on the line for them. For #2, I suggest that players near the lead “choke” or play slightly worse than normal because of pressure.

It’s the nature of a golf tournament that after the second round a cut is taken that eliminates the worse half of the field. After the fourth round, prize money is awarded based on finish –  with those at the top earning as much as 17% of the purse and most of those near the bottom earning prizes of around 0.5-1% of the purse. In other words, players who begin the second round far behind the leaders normally will not be able to make the cut, while those who begin the fourth round far behind the leaders are normally locked into a very small prize. This means that players who are near the bottom beginning the second and fourth rounds aren’t playing for much; the first group is likely to miss the cut and earn nothing while the second group is largely locked into a small prize. I’ve found the negative impact on performance to be as much as a quarter to half a stroke for those near the bottom of a leaderboard (the green and blue lines on the below graph). It’s impossible, with this data, to attribute this effect to “giving-up” or to physical factors – out-of-form swing, injury, fatigue, etc.

impact of starting position

Similarly, if you focus on the fourth round results in blue, you can see that players in positions zero and one strokes behind the leader performed approximately 0.1 strokes worse than expected while those in positions two to seven strokes behind the leader performed approximately 0.06 strokes better than expected. All of these players had a reason to remain fully engaged mentally with the tournament. Those finishing in the positions they started the round in stand to earn the large prizes. This shows that players who begin the round in or near the lead typically play slightly worse than would be expected by their prior performance, and more importantly that the same is not true for players who begin the round in close positions, but not in or right behind the leader. This reinforces the idea that pressure exerts a negative effect on those in the lead.

I also should address the red line for the third round. Players in the third round begin in opposite order of performance so far, meaning those furthest back of the leader tee of early, while those closest to the leader tee off late. PGA Tour courses play more difficult, in general, in the afternoon than the morning. Steven Rachesky found a difference of roughly 0.15 strokes between early and late tee-times – similar to the results I’ve observed from my data. That is the major reason why the data for the third round looks drastically different.

 

Predicting Putting Performance by Distance

Mark Broadie’s research of the Shot Link data established a clear relationship between putt distance and % of putts made. PGA Tour pros make a very high percentage of their close putts, but only about half of their putts around 10 feet and only around one in six around 20 feet. Pros hole very few (~5%) of their longest efforts from 25 feet and beyond. That data on % of putts made for each distance now forms the backbone of the PGA Tour’s Strokes Gained Putting statistic where players are credited and debited for making or missing every putt from every distance. Over a single season Strokes Gained Putting is often an unreliable indicator of putting performance, particularly at the extremes and also for players who have putted much worse or much better than in previous seasons.

Putting performance is polluted by randomness; Tour players just don’t attempt enough putts over the course of the season to get an accurate picture of their underlying putting ability. However, to make accurate projections of putting ability, you need to know whether Graeme McDowell’s 0.9 putts gained this season represents more talent or more luck. I’ve broken down putting performance into four different distance buckets from the PGA Tour data: putts inside 5 feet, 5-15 footers, 15-25 footers, and putts outside 25 feet. The results show that putting performance is far more predictable and consistent at the short distances. Long putting is so noisy that it’s difficult to say anyone gains much of an advantage from their long putting over the long-term.

Inside 5 Feet:

These putts are almost always converted (average 96%). The spread in performance between 2011-14 was 93% to 99%. The spread in expected performance derived from weighting the previous four seasons is 94.3% to 97.8%. This indicates that we should expect every regular Tour player’s true talent from inside 5 feet to fall somewhere inside that 3.5% range. Based on an average of over 900 putts attempted inside 5 feet over a season, we should expect every regular Tour player’s talent in terms of putts gained or lost to fall between +0.2/round and -0.3/round.

The graph below shows the correlation between a three year average (2011-13) and 2014 performance for all players with qualifying rounds in all four seasons. The correlation (R=0.56) between prior performance and 2014 performance is strongest in this distance range.

inside5feet

5-15 foot Putts:

This length is either short birdie putts or par putts after a scrambling shot that are converted approximately half the time. The spread in performance between 2011-14 was 36% to 54%. The spread in expected performance derived from weighting the previous four seasons is 40% to 52%. Based on around 450 putts attempted from 5-15 feet over a season, we should expect every regular Tour player’s talent in terms of putts gained or lost to fall between +0.4/round and and -0.5/round. Compare that to the best putters on Tour gaining about 0.75 putts/round.

The correlation between three year average and 2014 performance is below. The correlation (R=0.53) is similar to that for the short <5 foot putts.

5-15 footers

15-25 foot Putts:

These length are normally longer birdies putts and are converted about 16% of the time. The spread in performance between 2011-14 was 8% to 26%. The spread in expected performance derived from weighting the previous four seasons is 12% to 20%. Based on around 225 putts attempted from 15-25 feet over a season, we should expect every regular Tour player’s talent in terms of putts gained or lost to fall between +0.15/round and and -0.15/round. There’s much less at stake from this range than the previous two, just because so few putts are attempted from 15-25 feet.

The correlation between three year average and 2014 performance is below. There’s not much of a relationship (R=0.28), showing that putting performance from this range is much more affected by random chance over a full season than the shorter length putts.

15-25 footers

Putts outside 25 feet:

These length are the longest birdie putts, often really lag putts just to get it close for par. The spread in performance between 2011-14 was 2% to 13%. The spread in expected performance derived from weighting the previous four seasons is 4% to 9%. Based on around 300 putts attempted from beyond 25 feet over a season, we should expect every regular Tour player’s talent in terms of putts gained or lost to fall between +0.1/round and and -0.1/round. Again, there’s very little difference in expected performance from this distance. Even the very best long putter on Tour will gain little from these putts – over the long term.

The correlation between three year average and 2014 performance is below. There’s almost no relationship (R=0.10), which means it’s almost impossible to predict how well a player will putt on these long putts. The top ten long putters from 2011-13 average hitting 7.6% of their putts (versus 5.5% average). They only hit 6.7% of their putts in 2014 – a regression of almost 50% to the mean.

outside25ft

The Big Picture:

This graph shows performance in all four ranges. The longer putts show little relationship to future performance, while the shorter putts do show a more consistent relationship. This means that players who gained a lot of putts last season based off their longer putts will start making putts at a lower rate, while those who gained a lot of putts based on shorter putts are better bets to retain that putting ability.

bigpicture

Most Improved Putters from 5-15 feet in 2014:

1. Graeme McDowell

2. Charley Hoffman

3. Billy Horschel

4. Justin Leonard

5. Michael Thompson

These guys have a better chance of retaining their putting performance into 2015.

Most Improved Putters from > 25 feet in 2014:

1. Rory McIlroy

2. Y.E. Yang

3. David Toms

4. Brendan Steele

5. Brian Gay

These guys look likely to regress in terms of putting performance, especially McIlroy who performed to career average on all other putts, but hit 8% more of his long putts – gaining almost a third of a putt per round over his career average.

Measuring the Signal in First Round Performance

After the 1st Round of the Deutsche Bank Championship a month ago, Keegan Bradley sat two strokes off the lead. Playing in front of the home fans, Bradley fired a six under 65 fueled by great putting (4.2 strokes gained) and a solid long game (2.3 strokes gained on tee shots and approach shots). At that point he looked in great shape keep it going and capture his first win of the season. However, he came out the next three rounds and shot 71-69-71 to finish T16. The culprit wasn’t his long game either; he gained 1.6 strokes on the field per round in the second, third, and fourth rounds, good enough to finish in the top ten for the event in strokes gained off tee shots and approach shots. No, it was the putter that let him down. After being hot in the opening round, he actually lost 0.4 strokes per round from his putting.

My question is: how common is Bradley’s experience? When golfers come out in the 1st round and play/putt very well, how often do they keep playing/putting well? What about when they come out hitting the tee shots and approach shots well? Does that carry over to the next day? Many around the game act like one round of performance is really meaningful (just look at everyone who advocated for playing Jordan Spieth and Patrick Reed after their Friday morning 5&4 win at the Ryder Cup), but does first round performance tell us anything about how a player will perform in the following round?

Looking at Putting:

I gathered round by round Strokes Gained Putting data from the twelve most recent PGA Tour tournaments (Travelers Championship through the Tour Championship). First, I checked how 1st round putting performance predicted 2nd round putting performance. That’s the first graph below, and the results show how player putted in the 1st round hardly sheds any light on how they will putt in the 2nd round (R^2 of 0.001). In fact, someone who putted as well as Keegan Bradley did in the above mentioned round would be predicted only to putt 0.2 strokes above average the following round.

rd1SGP v rd2SGP

Next I generated prior expectations of Strokes Gained Putting performance from the past several years of data. I’ve shown before that putting performance isn’t very consistent season-to-season, so I’m using performance from 2011 to 2014 to generate the prior. The below graph shows how well the prior expectation predicted 2nd round putting. The results still were not highly predictive – R^2 of 0.01 (performance round to round is highly variable in golf) – but the regression line produced tracks pretty closely with results. Players predicted by the prior to putt well generally putted well and those predicted to putt poorly generally putted poorly.

priorSGP v Rd2SGP

Finally, I tied both pieces of information together. The prior estimate proved way more predictive than just 1st round performance, but does 1st round performance have any information to add? I set-up a linear regression with the prior estimate as x1 and the 1st round performance as x2. The results indicated 1st round putting performance provides no extra information to predict 2nd round putting performance (the coefficient was indistinguishable from zero). If you have a good guess of how well a player will putt, you can safely ignore first round putting performance.

Looking at Long Game Performance:

The long game is tee shots and approach shots (drivers/woods/irons essentially). I gathered long game performance data from the same twelve PGA Tour tournaments for the first and second rounds. I then ran the exact studies as above just substituting long game data for putting data. The correlation between 1st round long game performance and 2nd round long game performance was higher than with putting, but still didn’t contain a lot of information (R^2 of 0.03). If a player plays four strokes above field average in long game strokes gained, they’re expected to play 0.6 strokes better in the long game in the 2nd round.

rd1LONG vs rd2LONG

There was also a higher correlation between my prior estimate for long game ability and 2nd round long game performance (R^2 = 0.10). Again though, the regression line tracks closely with the results. Top ten long game players (around +1.2 strokes or above) generally performed to that level in the 2nd round.

priorLONG vs. rd2LONG

Tying both pieces together indicated that there is a small amount of signal in 1st round long game performance. Combining the prior estimate with 1st round performance slightly increases the fit of the model. The regression equation suggests that you should weight your prior estimate at twelve times the strength of first round performance. This indicates that someone who is PGA Tour average in long game shots, but produces an elite round of 4.0 long game strokes gained, should be expected to play about 0.3 strokes above average in long game shots. That seems like a small difference, but it’s enough of a shift in talent to move a player from around 50th best in the world to about 30th best in the world.

The Takeaway:

Based on these results, it looks like 1. a single round of performance is much less predictive than an estimate built on past-performance and 2. the small amount of signal contained in single rounds is from performance on tee shots and approach shots. Putting results from one round provide no more information than was available before the round. On the other hand, golfers who play particularly well on tee shots and approach shots in a round should perform slightly better than expected the following round.

 

How Real are Hot Streaks in Golf?

Whenever a golfer goes on a high profile hot streak – think Rickie Fowler since June, Billy Horschel for the FedEx Playoffs, Henrik Stenson at the end of last year – there’s always a ton of talk in the media and among fans about how their new swing/putting stroke finally clicked, or that player is returning to form, or they’re finally mature enough to win, etc. Humans love writing narratives to explain why things happen. The end result of all that talk is that a guy in the middle of a hot streak is considered to be much better than they would’ve been considered before the hot streak. No one thought Billy Horschel was deserving of a Ryder Cup pick a month ago, but now everyone thinks we should toss Webb/Mahan off to make room for him. No one thought Rickie Fowler was one of the 1-2 best American players in May, but now that’s almost assumed. Everyone around golf seems to think that hot streaks are real – that they actually predict who’s going to continue to play well. In this post I’ll provide evidence that shows that hot streaks are retained to a small degree – even months later – but that extreme performances still regress strongly to prior expectations.

Methodology:

I settled on using five week periods to measure performance. My sample was everyone who had recorded at least 8 rounds in a five week period and then recorded at least 8 rounds in the next five weeks. All my data is from the 2011-2014 seasons. The actual metric I used to measure performance was my z-score ratings, which are basically strokes better or worse than the field adjusted for the strength of the field. I compared each player’s z-score over that five week sample to my prior z-score rating. I have a prior rating for every player in my sample generated each week which mostly uses prior performance and very recent play to predict how well a player will play that week. They’re designed to be the most accurate prediction of performance. I subtract the prior expectation from the sample performance to get the change in performance which I’ll call the Initial Delta.

So my metric looks like this:

(Sample performance over 5 weeks) – (Prior expectation) = (Initial Delta)

I generated an Initial Delta for every player who qualified for my sample, generating over 27000 separate data points.

I then calculated a Subsequent Delta for every player using the same method only using the next five weeks as my Sample performance and the same Prior expectation used above (meaning I don’t consider any recent results). I then compare the Initial Delta to the Subsequent Delta. If players get hot and stay hot, the two should be strongly correlated. If whether a player has been hot or cold does not predict their subsequent performance, the two will not be correlated.

tl;dr of the above is I’m comparing how much better/worse a guy played over the first 5 or 10 weeks to how much better/worse he played over the next 5 weeks.

Results:

The results show that in general players retain only a small portion of their over or under-performance. Overall, about 20% of the Initial Delta is retained over the next five weeks. This means that if Billy Horschel played 1.8 strokes better than expected over the last five weeks, he should be expected to play about 0.36 strokes better than previously expected in the next five weeks. Now, 0.36 strokes is a large amount, but it’s not enough to bring him up to Bubba/Fowler/Keegan’s level (here is an example of the distribution of talent among the top 50 in the world). Right now, he should be considered slightly better than Mahan or Webb, but not to some ridiculous amount and certainly not to any degree that’s going to effect the outcome of the Cup.

5weekNOPRIOR

Looking Further Ahead:

The above shows that hot streaks can be retained to some degree over a short period of time, but how much is retained further down the road? Is Billy Horschel going to be able to retain any of that ability he showed to win the FedEx Cup going into next season? I set-up the same study as above, only instead of looking at performance in the next five weeks, I looked over the next four months (16 weeks to be precise). Everything is calculated the same, though I only included players with at least 20 rounds over that four month span.

The results here showed that about 18% of the Initial Delta is retained over the next four months, a similar amount to what is retained over the next five weeks. Golfers who play significantly better than expected over five weeks should perform better than previously expected, but only to a small degree. To give you a sense of when recent performance becomes mostly insignificant, if a player performs 0.5 strokes better than expected over five weeks (basically what Chris Kirk has done in the FedEx Cup Playoffs), he is expected to retain only around 0.1 strokes (which is insignificant, basically a rounding error in predictive terms).

4monthNOPRIOR

Adjusting Expectations:

I’ve attached a list of the top and bottom ten guys who have most over or under-performed over the last five weeks (PGA/European Tour only).

Movers9182014

Obviously Horschel is at the top along with some FedEx Playoff stalwarts like Palmer/Fowler/Day. Ryder Cupper Jamie Donaldson has been killing it over in Europe as well. Among the trailers, Phil’s name sticks out like a sore thumb. The US team has to hope his multiple weeks off can help him rediscover his game before the next week. Probably the most terrifying thing is how close Ryan Moore came to making this team – he finished 11th in points, but was only a stroke away from jumping Zach Johnson in points at the PGA Championship. Moore is dead last of 339 pro golfers in terms of his performance relative to expectation.

Putting Driven Performance Changes are Illusory

Last week I posted about how repeatable performance on different shot types was from season to season. Tee to green play is more repeatable than putting which is more repeatable than scrambling. That makes sense once you realize that golfers play 2-3 times more tee to green shots than meaningful putts in a round; there’s just more inherent randomness in a season’s worth of putts than in a season’s worth of tee to green shots. Golfers play even fewer scrambling shots resulting in even more randomness in a season’s worth of scrambling.

Last month I also examined how repeatable small samples (4-8 tournaments) of putting performances are, in the context of discussing why I expected Jimmy Walker’s performance to regress to the mean. That micro-study indicated that there was very little correlation between a golfer’s performance in a 4-8 tournament sample of putts and the following 4-8 tournament sample of putts. In the whole, performances in such short samples regress almost entirely to the mean.

Those two lines of inquiry led me to examine whether putting was more random than tee to green performance. I have always believed that improvements/declines that were driven by over-performance in putting were less real than those driven by tee to green over-performance, but I had never actually tested that hypothesis. The key question is whether changes in performance driven by putting are less persistent than those driven by tee to green play. That is when a golfer performs better over the first half of a season, and much of the improvement can be traced back to an improvement in his putting stats, will that golfer continue to perform better in the second half of the season? The evidence says changes in performance driven by putting are more illusory than changes in performance driven by tee to green play.

Design:

I gathered the tournament by tournament overall, tee to green, and putting performances of all PGA Tour golfers in rounds measured by the ShotLink system for 2011-Present. I divided those rounds into roughly half-season chunks (January-May 2011, May-November 2011, January-May 2012, May-November 2012, January-May 2013, May-September 2013, October 2013-Present). Each chunk included around 15-18 tournaments. I considered all golfers who recorded at least 20 rounds in consecutive half-season chunks.

To measure putting performance I used the PGA Tour’s Strokes Gained Putting stat and to measure tee to green performance I used my own overall ratings with putting performance subtracted out. This methodology is consistent with my measurement of tee to green performance in numerous recent work.

Half-Season Correlations by Shot Type:

First, I measured how repeatable putting and tee to green performance was between half-season samples, much like the full-season samples used in this study. I included all golfers with at least 20 rounds in consecutive half-season samples and compared each half-season to the half-season that directly followed, including 2nd halves to 1st halves of following calendar years. This yielded samples of ~800 golfers for both tee to green and putting. Graphs are below.

half tee to green

half putting

Tee to green performance was again more repeatable than putting performance. In the study linked above consecutive full-seasons of tee to green performance were correlated at a R=0.69 level. I found a correlation of R=0.62 between consecutive half-seasons, understandably less given the smaller number of rounds/shots played. The full-season correlation for putting was R=0.55. Half-season putting performances were similarly less correlated than full-seasons at R=0.40. Both these findings are consistent with the understanding that randomness between samples increases when fewer rounds/shots are compared. Most importantly, putting is less repeatable than tee to green play.

Persistence of Changes in Performance by Shot Type:

Next, I measured how persistent changes in performance are when considering putting and tee to green play. That is, when a golfer improves their putting over a half-season sample, how much of that performance is retained in the following half-season? If 100% of the performance is retained, changes in putting performance over a half-season entirely represent a change in true talent. If 0% of the performance is retained, changes in putting performance over a half-season entirely represent randomness. The same for tee to green play. My assumption was that a larger percent of performance would be retained for tee to green play than putting, meaning that half-season samples of putting are more affected by randomness than half-seasons of tee to green play.

To measure the effect, I first established prior expectations of performance for every golfer in my sample. I simply averaged performance in tee to green play and putting for the three years prior to the beginning of each half-season sample. For example, for the May-November 2011 sample, I averaged play between May 2008 and May 2011. This is not an ideal measure of performance, but it provides a consistent baseline for comparisons to be made.

I removed all golfers from the sample who had no prior performances. This reduced my sample to around 750 consecutive half-seasons.

The values I compared were the initial delta (Prior minus 1st Half-season) and the subsequent delta (Prior minus 2nd Half-season). Using this method I can find how persistent a change in performance is between to half-seasons. I did this considering putting and tee to green play. Graphs are below.

persist tee to green

persist putting

Changes in tee to green play were twice as persistent as changes in putting play, meaning golfers who improved their tee to green play retained twice as much of those improvements as golfers who improved a similar amount in putting. Golfers maintained around 60% of their tee to green improvements, but only 30% of their putting improvements. This indicates that putting performances regress more sharply to prior expectations than tee to green performances.

Are Putting Performances More Illusory?

Finally, I gathered the data from above to measure whether changes in performance driven by putting less real than changes in performance driven by tee to green play. I ran a linear regression using the initial delta for overall performance and the initial delta for putting performance as independent variables and the subsequent delta for overall performance as the dependent variable. In short, given a certain overall change in performance and a certain change in putting performance over the first half-season, how much of that overall change in performance is retained over the second half-season?

As the following table shows golfers retain much more of their improvement or decline when that improvement or decline occurred in tee to green shots than if it occurred in putting. The columns show improvements/declines in overall play (considering all shots) and the rows show improvements/declines solely in putting. The table shows that a golfer who improves overall by 0.50 strokes will retain only a quarter of their improvement if all of the improvement was due to putting (0.50), while they will retain over half of their improvement if none of the improvement was due to putting (0.00). The equation used to produce this chart is Subsequent Delta = (0.56 * Initial Overall Delta) – (0.28 * Initial Putting Delta).

delta comparisons

Discussion:

These findings should fundamentally alter how we discuss short-term changes in performance. I’ve already shown repeatedly that performances better than prior expectation will regress to the mean over larger samples. That idea is consistent across sports analytics. However, these findings indicate that the amount of regression depends on which part of a golfer’s game is improving or declining. Golfers who improve on the basis of putting are largely getting lucky and will regress more strongly to the mean than golfers who are improve on the basis of the tee to green game. Those who improve using the tee to green game are showing more robust improvements which should be expected to be more strongly retained.

The golfers who represent either side of this for the 2014 season are Jimmy Walker and Patrick Reed. I’ve discussed both in the past month, alluding to how Walker’s improvements were almost entirely driven by putting and how Reed’s were mostly driven by tee to green play. Based off these findings, Reed is more likely to retain his improvements over the rest of the season, all else being equal, than Walker.

 

All graphs/charts are denominated in strokes better or worse than PGA Tour average. Negative numbers indicate performances better than PGA Tour average.

Repeatability of Golf Performance by Shot Type

My main interest in analyzing golf is using past data to most accurately predict future golf performance. Inherent in that are the questions of how to figure out how much randomness is affecting the data and how to remove the effects of randomness from the data. The easiest way to find how much randomness is involved with data is to find the correlation between subsequent measures of performance. For example, in this post I found the correlation between a golfers performance in various samples of years from 2009-12 and their performance in 2013. Based on that, I concluded that using such basic methods produced a correlation of around 0.70 – meaning that in a subsequent season we can expect a golfer to repeat about 70% of their prior performance above or below the mean. I’ve achieved correlations slightly higher than that using more sophisticated methods and more detailed data, but an R of 0.70 should be viewed as the typical baseline for judging the repeatability of golf performance.

In this post, I attempt to find the repeatability of performance on different shot types using the same methodology as above. I will use Strokes Gained Putting to measure putting performance relative to the field, my own Z-Score measure minus Strokes Gained Putting to measure tee to green (driving, approach shots, short game) performance relative to the field, and I’ll use my own Scrambling metric (methodology here) to measure performance on only scrambling shots (a scrambling shot is the first shot following a missed green). I have not stripped these scrambling shots out from the tee to green measure; tee to green measures all non-putting strokes.

I gathered data for all PGA Tour golfers for 2008-2013 who had a qualifying number of rounds played (50). I then paired consecutive seasons for each of my three performance measures and graphed the correlations below.

repeat teetogreen

repeat putting

repeat scram

For all three graphs negative values represent performances better than the field. Values are expressed in standard deviations above or below PGA Tour average.

The measure that was most strongly correlated from season to season was tee to green performance. That makes intuitive sense as tee to green performance includes the most shots/round of any of my measures (roughly 40/round). In fact, tee to green performance is almost as repeatable as overall performance (R = 0.69 compared to 0.69 to 0.72 in the study linked above). This indicates that a golfer who performs well above or below average in a season should be expected to sustain most of that over or under-performance in the following season.

Putting was less repeatable than the tee to green measure, but skill still shows strongly through the noise. An R of 0.54 indicates that a golfer’s putting performance should be regressed by nearly 50% toward the mean in the following season (provided you only know their performance in that one season). I would mainly explain the lower correlation on sample size; golfers normally hit 25-30 putts/round, but many of these putts are short gimmes that are converted upwards of 95% of the time. The number of meaningful putts struck in a round is more like 15. This indicates that it takes a golfer over 2.5 seasons of putting to reach a season’s worth of tee to green shots. This suggests that putting is less repeatable from season-to-season than tee to green strokes, which indicates we should be wary of golfers who build their success in a season largely off of very good putting.

Off the three measures tested, Scrambling was the least able to be repeated (R =0.30). This indicates that performance on these short shots around the green is very random. It’s not atypical for a golfer to perform as one of the best 10% on Tour one season and average the next. Again, this is likely a function of sample size. A golfer hits only 6-8 scrambling shots/round (every time they miss a green). It takes a golfer around five seasons of scrambling shots to reach one season’s worth of drives/approaches.

There are two important caveats with this approach. These correlations explicitly measure only one season compared to the following season. Measures like scrambling and putting likely are more strongly correlated when several seasons are used as the initial sample. I intend to test this in a future post. In addition, this study only considers golfers who played consecutive seasons with >50 rounds/season. This leaves out a certain sample of golfers who played so poorly in the first year that they were not allowed to play >50 rounds the following season. These stats may be slightly more repeatable if those golfers were included.

Thoughts on Patrick Reed (Without Using “Confident” or “Cocky”)

Patrick Reed won a tournament yesterday – his third win since August – and in the process delivered pre-round and post-round interviews where he said he thought he was a top five player in the world. There’s been a lot of bullshit spewed already about his comments so I’m going to try to avoid any of that. I am going to lay out some reasons for and against the idea of Patrick Reed being an elite golfer, with the knowledge that anyone who thinks they know for sure is full of it.

Reasons to Doubt

The main argument against Reed being elite is his aggregate play up to this point in his career. Going beyond his three wins in 51 starts, when you consider all of his rounds (not just the ones since August), Patrick Reed’s performance hasn’t been much different than an average PGA Tour cardholder. I have 180 rounds for him between the Web.com Tour and PGA Tour going back to before he turned pro in 2011. In those rounds he’s played to just barely above the level of an average cardholder (-0.17). 180 rounds isn’t the definitive picture of a golfer, but it tells us that in general he’s been essentially average over a fairly large sample of results over mostly the last three seasons.

As I just wrote in a piece last week, the first two months of the season, when considered alongside the last two years of data, provide little extra information about how a golfer will perform going forward. Reed played to a rating of -0.08 in all rounds prior to January 1st 2014 and he’s played to a rating of -0.81 (over 2 strokes better) in 24 rounds since then. In general, past performances have shown that we should place about 3.5 times as much weight on those prior rounds compared to the rounds from the beginning of the season. Using this line of thinking, Patrick Reed should be considered an above-average PGA Tour player, but no better than Kevin Chappell or Russell Knox or other young guys who no one pays an extra second of attention.

Now some might point to his age saying that plenty of young players break-out in their early to mid 20s. I re-ran the study from the piece linked above to factor in age. The methodology is outlined in the piece, but basically I used a regression analysis to predict performance from March to December of a season using the January/February performance as one variable and the previous two full seasons as another variable. I ran the analysis this time using all seasons from age 27 and younger, age 28 to 38, and age 39 and older. I used the age 27 cut-off because that is where my prior aging studies have shown general age-related improvement halts.

In fact, the age of the player does affect how strongly we should believe in early season improvements/declines, though the evidence still favors the prior two seasons. For the age 27 and under group, the weight was about 2.4 times stronger for the prior seasons than the early season form. The weight on the prior seasons was around 5 times stronger for the age 28 to 38 group and nearly 4 times stronger for the age 39 and older group. Consider that all seasons produced a weight of 3.5 times for the prior seasons and it’s clear there’s an effect for younger golfers. So that indicates that we should believe more in early season improvements for young players, but that we should still defer heavily to the prior performance data. Using this method to project Patrick Reed, I’d compare his abilities now to Billy Horschel or Harris English. Plenty of folks think they’re very good players, but no one (bookmakers included) considers them elite by any stretch.

I then set-up a regression which attempts to predict the delta of the remaining ten months of performance with age and the delta between the prior two seasons and the first two months as dependent variables. The top row of the graph below is the delta between the prior two seasons and the first two months (negative means improvement/positive means decline), while the first column is age. Each cell represents the expected delta between the prior two seasons and the remaining ten months of the season based on a golfer’s first two month delta and their age. You can see that younger golfers that outperform their prior two years are expected to retain more of their improvements over the rest of the season than peak aged or past-peak golfers.

ValueHotStartGraphByAge

Reasons to Believe

Now that I’ve laid out the reasons to doubt Reed, here are a few reasons to think that this may be more real than the general model predicts.

1. Reed was an outstanding amateur golfer, especially during his final two seasons in college. The gold-standard for measuring collegiate golf performance is Jeff Sagarin’s rankings. Sagarin uses a method that compares who you beat/lose to in the same tournament and how much you beat/lose to them. College golf doesn’t provide a huge sample of results – a golfer might complete 40 rounds during a season – but it works in general. During Reed’s two seasons at Augusta State, he played 20 tournaments and finished 4th and 9th in the nation in Sagarin’ rankings (and led Augusta State to two straight NCAA Championships). Less than ten others have finished with a better average rank in college than Reed (including Bill Haas, Ryan Moore, and Dustin Johnson). Elite college performance at least establishes that Reed isn’t coming out of nowhere; this guy was lighting tournaments up in college.

2. I don’t consider Monday qualifier results in my database. The data is provided by local PGA chapters using multiple spellings of names and it’s generally a hassle to collect. For most guys that wouldn’t be a huge issue, but Reed was 6/8 in Monday Qualifiers in 2012, earning his way into six tournaments when he had no Tour status. Monday qualifiers are held at a nearby courses with around 100 golfers participating (mainly PGA Tour members without the status to enter the tournament directly, Web.com golfers, or minor tour pros). Of those ~100, the best four scores over a single round qualify to enter the tournament. Because only the top four advance, these qualifiers require a golfer to play around the level of peak Tiger Woods for a round to qualify. In short, Reed playing that well in 6/8 qualifiers should inflate his overall rating by a small amount.

3. Most importantly, Reed isn’t getting terribly lucky putting so far this season. Putting drives a lot of luck on Tour – largely because it’s easier to sink an extra eight footer every round for two months than it is to randomly pick up an extra couple yards of driving distance on every hole. When I examined Jimmy Walker’s game a few weeks ago, all of Walker’s improvement in 2014 could be attributed to a strokes gained putting that was inflated nearly a full stroke above his career average. In Reed’s case his putting numbers are slightly higher than his career average, but nothing similar to Walker’s stats.

Entering 2014 he had gained 0.27 strokes on the field through his putting and was basically Tour average in driving/scrambling/approach shots/etc. in his career. So far this season, he’s gained 0.52 strokes from putting and 1.91 strokes from driving/scrambling/approach shots/etc. About 10-15% of his improvement can be traced to his putting and the rest to his driving, iron play, and short game. In short, he’s not relying on a lucky putter like Walker, instead he’s hitting his driver and irons more consistently – leading to more distance, more greens hit, and more birdie opportunities.

This is where I should sum up all the evidence and declare a winner. Is Patrick Reed going to keep winning tournaments, maybe a Major this season? Or is he going to regress to just being another guy grinding for his card? But I don’t really have any idea. I do hope we start getting more post-round interviews that are heavier on bravado than modesty.

What’s a Hot Start Worth?

This weekend marks the eighth weekend of professional golf in 2014 and the beginning of the Florida Swing of the PGA Tour schedule. So far this season, guys like Jimmy Walker, Patrick Reed, and Harris English have started off playing like top 20 players in the world. With almost two months of the year (not to mention five months of the Tour season) in the books, it feels like we’re reaching the point where we can start to tell who’s struggling, who’s excelling, who’s going to contend for a Major, who’s the next Big Thing, etc. This post is designed to throw some water on those ideas. Two months doesn’t tell us very much about how the rest of the season will play out, at least when compared to a much larger sample of past tournaments.

To test how predictive the first two months of the golf season are of the rest of the season, I gathered all players 2010 to 2013 who played at least 50 rounds in the two seasons prior to the season in question (ie, 2008-09 for 2010, 2009-10 for 2011, etc.) and who played any rounds in the first two months of the season in question and in the remaining calendar year of the season in question. I kept these requirements fairly loose, but tested other combinations. In total, I found 1984 seasons from players on the PGA Tour, European Tour, Web.com Tour, and Challenge Tour. I found the average performance in z-score for the two years prior to the season in question, the first two months of the season in question, and the remainder of the season in question.

To test whether the first two months were predictive of the remainder of the season I first simply found the correlation between the first two months and the remainder of the season. I found a strong correlation (R=0.57) between the two, indicating that the first two months were highly predictive of the rest of the season. However when I examined the correlation between the prior two seasons and the remainder of the season in question (while ignoring the most recent two months), the correlation grew even larger (R=0.68). Note again that this ignores the first two months of the season. That is, if you lock me in a room from New Years until March without access to any information about professional golf I will do a better job predicting the season than someone who only relies on who’s playing well in January and February.

Now, obviously you don’t have to ignore one set of data (two year average) in favor of another (two month average). I ran a simple linear regression of the two variables (two year average and two month average) on the rest-of-season average to see if the accuracy would improve. Indeed, including both variables increased the correlation slightly to 0.72, meaning that this model explains over half of the variance in the remainder of the season (this again shows how random golf is). More interesting are the coefficients the regression spits out: Y = (0.72*Two Year)+(0.20*Two Month)+0.04. That is, the two year average is 3.5 times more important than the two month average.

I followed this study up with another that repeated the methodology, but winnowed the sample down by restricting it to players with > 100 rounds in the previous two season, > 5 rounds in the first two months, and > 19 rounds in the remainder of the season. The results were consistent with what was earlier observed. Using the smaller sample (N=1300 now) slightly improved the predictive strength and also slightly increased the importance of the two year average relative to the two month average.

However, I conducted a further study that showed that drastic improvements in z-score in the first two months were much less predictive of the remainder of the season than the general sample. Using the stricter sampling method above, I split the seasons into those where the two month average was 0.30 standard deviations better than the two year average (basically the sixth of the sample that improved the most), where the two month average was 0.30 standard deviations worse than the two year average (basically the sixth of the sample that declined the most), and the remaining 2/3rds of the sample. I ran the regression using only the data from the +0.30, -0.30, and middle groups.

The results showed that when considering only those who improved the most, you should almost completely ignore what happened in the first two months and rely on the two year average to predict going forward. For the other two groups, the results were largely consistent with what was observed in the previous studies – two year average is roughly 3 times more important than two month average.

Now, I have to stress that the sample of those who improved the most is only 192 seasons and that the standard errors of the coefficients are large (0.11). The confidence interval for the two year coefficient is 0.66 – 1.13, centered on 0.90 while the confidence interval for the two month coefficient is -0.19 to 0.24, centered on 0.03. The standard errors for the previous studies were much smaller (0.02 to 0.03). The finding that two month average should be largely ignored for those who showed the most improvement certainly needs to be tested further with more data.

I am much more confident in the main conclusions, however. When attempting to predict performance over the rest of the season – like who will contend for Majors, Ryder Cup berths, and the FedEx Cup – weigh more heavily how a golfer has played in the prior few seasons than how they’ve started off the calendar year. If that means we pump the brakes a little on Walker, Reed, and English, so be it. And don’t write off Tiger, Kuchar, Poulter, and Luke Donald for a poor couple months. Those guys have shown for years that they belong in the world’s elite; that’s worth more than a cold start.

Will Jimmy Walker Continue to Putt at an Elite Level?

I got some push-back from Chris Catena on twitter today about my contention that Jimmy Walker’s recent run of great play was driven by lucky putting. In that post, I showed that Walker had established himself recently as an above-average, but not elite putter (a strokes gained putting of around +0.25-0.30/round for the last five years). During Walker’s recent run (Frys.com Open through Northern Trust Open), he’s putted at a +1.20 level. That +0.9 strokes/round improvement is entirely what carried him to three wins in the last four months. I also contended that Walker continuing to putt at this level is very unlikely, simply because no one ever has for a full-season. Moreover, Walker’s best putting season (+0.46) and average putting season (+0.26 from 2009-2013) are far short of the kind of elite, sustained level of play we often see out of the golfers who lead the Tour in strokes gained putting. This post is to defend those claims in more depth and show why I think it’s very unlikely that Jimmy Walker will continue putting and playing as well as he has in the last four months.

JimmyWalkerSGP2012-14

Above is a graph of Walker’s strokes gained putting performance per tournament in every tournament the PGA Tour has data for since the start of 2012. The red dashed line is a linear trendline of his performance. It has basically zero (R=0.03) relationship with the passage of time, indicating that on the whole, Walker’s performance hasn’t improved over time. This is important to note because if we hypothesize that Walker changed something in his ability to putt, it clicked in only weeks after his worst putting stretch of the past 2+ years. Now, poor performance is certainly a motivator to change and try to improve, but a simpler explanation is that Walker got unlucky during the summer, and has been riding a combination of luck and improved putting since.

What Walker has done in the past 23 rounds on Tour isn’t unprecedented even during the 2013 season. I divided the tournaments in 2013 (Hyundai ToC to Tour Championship) into four quartiles with 7-8 tournaments in each quartile. I then found the golfers who had participated in 4+ tournaments in each bucket and averaged their SGP for each quartile. I gathered all golfers who had qualifying consecutive quartiles and compared them using Q1->Q2, Q2-Q3, etc. For Q4, I compared it to performance so far in 2013-14 from the Frys.com Open to the Northern Trust Open. From all that, I had 365 pairs of quartiles where a golfer had played at least four tournaments during each quartile. A graph of of those pairs is follows.

pairs of SGP quartiles

There was very little relationship between a golfer’s performance in one set of tournaments and their performance in the following set of tournaments (R=0.04, indicating a tenuous at best relationship). I had 61 quartiles with a performance > +0.50, averaging 0.72. Those quartiles played to only +0.12 in the next set of tournaments. In fact, in only 12 of those samples of > +0.50 performance did a golfer again average > +0.50 the next quartile. None of the six samples of > +1.00 SGP had > +0.52 SGP in the following quartile. In short, we should be very skeptical of elite putting performances over fairly short periods of time.

Now, when I said that Jimmy Walker’s performance was largely driven by luck I meant the “largely” part. I think it’s extremely unlikely that all of his putting performance can be explained by variance alone. Jimmy Walker has +1.20 strokes gained putting/round in 23 measured rounds so far this season. The observed standard deviation between 23 round samples for PGA Tour players is around 0.35 strokes. That means if an average (+0.00) putter plays an infinite number of 23 round samples, 68% of them will yield an SGP average of -0.35 to +0.35, while 95% of them will yield an SGP average of -0.70 to +0.70. In short, there’s a ton of variation between 23 round samples. For an average golfer, it wouldn’t be shocking for them to putt extremely poorly or very well over 23 rounds. Plugging that standard deviation (0.35), Walker’s 2013-14 SGP (+1.20) and Walker’s five year SGP average (+0.26) into a Z-score equation yields a Z of 2.7 which indicates <1% chance that Walker’s SGP is entirely due to chance. That means there is some signal in all that noise.

But how much? I consider myself a Bayesian in that I think it’s very important to compare any observed performance to our prior expectation for that performance. Up until October 2013, Jimmy Walker was an above-average, but not elite putter. Since then, in 23 rounds, Walker has putted out of his mind. Surely we should consider Walker a better putter than we did in October, but how much better? Fortunately, there’s a simple equation we can use to estimate how the 23 round sample should change our expectation for him. It’s ((Prior Performance)/(Prior variance) + (Sample performance)/(Sample variance))/((1/Prior variance)+(1/Sample variance)). Basically, this equation tells us how confident, statistically, we should be about a golfer establishing a new level of performance based on how far his performance is from the prior expectation and how large of a sample we’re dealing with.

We know the prior performance and sample performance from the previous paragraph. The sample variance is simply the 23 round standard deviation from above (0.35) squared (0.12). To find the prior variance, I was forced to run some simulations as my data was limited. I know the variance for 100 round sample is around 0.025, so the prior variance for Walker over his >300 rounds in 2009-2013 must be no greater than that. Simulations indicated to use a figure around 0.02.

Plugging those values into the equation yielded a new expectation for Walker of around +0.40. That’s significantly higher than his five year average, but also much less than what he’s done recently. The equation is saying that Walker’s been much better, but that 23 rounds isn’t nearly enough to say that he should be expected to continue to putt at an elite level. If we had just seen Walker putt at a +1.20 SGP level for 80 rounds, we’d be much more confident in him continuing to putt at an elite level.

The tl;dr here is that extremely good SGP performances over small samples (~4-8 tournaments) sharply regress to the mean in the following 4-8 tournaments. Sustaining the kind of putting Walker has shown recently is unprecedented over a large sample of rounds from 2013-14. Moreover, the expected level of variance of 23 rounds is very large. It would not be abnormal for an average putter to putt at a top 20 or bottom 20 level over 23 rounds. Considering all that, we should expect Walker to putt better over the rest of the season than he did in 2009-2013, but not nearly as well as he has since October.

Bayesian Prediction of Golfer Performance (Individual Tournament)

I’ve posted several studies attempting to predict golfer performance. This attempted to find the importance of the previous week when predicting the following week. The study was not particularly sophisticated (simple linear regression), but the results indicated that the previous week’s performance should be valued at around 10% of the projection for the golfer the following week (90% would be the two-year performance). This other study attempted to predict golfer performance for an entire season using prior season data. That study found that no matter how many years are used or whether those years are weighted for recency, the resulting correlation is ~70%. Doing better than that for full-season prediction would indicate an additional level of sophistication beyond aggregating prior seasons or weighted data for recency.

This post, however, concerns predicting individual tournament performance using my Bayesian rankings. These rankings are generated each week by combining prior performance and sample performance using the equation ((prior mean/prior variance)+(observed mean/observed variance))/((1/prior variance)+(1/observed variance)). In this way, each golfer’s prediction for a week is updated when new information is encountered. The prior mean for a week is the Bayesian mean generated the prior week. My rankings also slowly regress to a golfer’s two-year performance if they are inactive for a period of weeks. For each week, the prior mean is calculated using the equation  (((Divisor – (Weeks since competed)) / Divisor) * (Prior Mean)) + ((1 – ((Divisor – (Weeks since competed)) / Divisor)) * (Two-year Z-Score)). I use 50 as the Divisor, which weights two-year performance at 2% for 1 week off, 27% for 5 weeks off, and 69% for 10 weeks off.

To measure how predictive these rankings were, I gathered data for all golfers who had accumulated 100 rounds on the PGA, European, Web.com, or Challenge Tour between 1-2010 and 7-2013. My sample was 643 golfers. I then examined performance in all tournaments between the 3-28-2013 and 8-8-2013. My sample was 6246 tournaments played. I then generated Bayesian rankings predicting performance before each of these tournaments played. The mean of my predictions was +0.08, indicating I expected the sample to be slightly worse than PGA average. I then compared each prediction to the golfer’s actual performance.

The table below shows the performance of Bayesian and pure Two-year predictions by including all predictions within +/- 0.05 from the displayed prediction (ie, -0.50 includes all predictions between -0.45 and -0.55). The accompanying graph shows the same information with best-fit lines.

BayesianPredictions

BayesianPredictionsGraph

Obviously, the Bayesian and Two-year predictions perform similarly. To test which is better I tested the mean square error. This shows how closely the prediction matched actual performance. I also included “dumb” predictions which simply predict all rounds will perform to the mean of all predictions (+0.08 for Bayesian, +0.055 for Two-year). The “dumb” predictions are the baseline for judging any predictions. If a prediction can’t beat it, it’s worthless.

The mean square error for the Bayesian predictions was 0.381 and 0.446 for the “dumb” predictions. The mean square error for the Two-year predictions was 0.389 and 0.452 for the “dumb” predictions. So both sets of predictions provide value over the “dumb” predictions, but both perform fairly similarly when compared to the “dumb” predictions (-0.065 for Bayesian and -0.063 for Two-year).

This study indicates two things; first, using Bayesian methods to predict golfer performance doesn’t substantially improve accuracy relative to unweighted aggregation of the last two years of performance, and second, that predicting golfer performance in individual tournaments is very difficult. A mean square error of 0.38 indicates an average miss of 3.5 strokes for golfers playing four rounds and 2.5 strokes for golfers playing two rounds.