• Analytics Blog
Nov 2nd, 2018
Part 1: Age profiles and adjusting strokes-gained from 1983-present
There is a lot of interesting stuff to get to in this article, so we’ll cut the fluff and get right into the details. The goal of this statistical exercise is to project the performance level of golfers several years into the future. The metric of performance we focus on is adjusted strokes-gained. We use this because we believe it is the least noisy measure of golfer performance. Recall that, in a given round of golf, a golfer’s adjusted strokes-gained is the number of strokes better or worse their score is than some benchmark golfer (e.g. the average player on the PGA Tour). You can roughly think of this as being calculated by taking a golfer’s raw score and first subtracting the mean score of the field on the given day, and then making a further adjustment to correct for the average skill level of the field.

We are going to be predicting a golfer’s average adjusted strokes-gained in future seasons. Critical to this are two things: 1) properly adjusting scores over long periods of time, and 2) understanding age profiles of golfers. Let’s tackle these two things in turn.

Adjusting scores over many seasons is fundamentally no different than adjusting them across tournaments in a given season. Therefore, we adjust scores from 1983 to 2018 using the same method we’ve used before to adjust scores across tournaments and tours within a given season. To see some details and a link to the academic paper we roughly follow, see the first endnote of this article. The core idea when adjusting scores over time is that by estimating each golfer's ability at each point in time, we can properly assess the difficulty of a course on a given day. A key intuition is the following: to compare the 2009 version of Tiger Woods to the 1992 version of Fred Couples, we are comparing Woods’ and Couples’ performance against common opponents (e.g. they both played against Phil Mickelson). Additionally, every player’s skill level is allowed to vary over time (e.g. Mickelson was likely at a different skill level when he played against Woods than when he played against Couples). The end result is a strokes-gained measure for every round played from 1983-present on the major professional tours (data for European Tour is from 1996-present, and Web.com data is from 1990-present). For a baseline (i.e. a strokes-gained value of 0) we use the average player on the PGA Tour in the 2000 season. Here is the average value for this strokes-gained measure on some of the major tours since 1983 (subject to data availability):

Notes: Plotted is the average player quality on each of 3 major professional tours since 1983. Each data point is the average adjusted strokes-gained on that tour in a given season. The strokes-gained measure is normalized to equal 0 for the PGA Tour in the year 2000.
The quality of the average golfer on each of the 3 tours we analyze has steadily gotten better over time. From 1983 to 2018, we estimate that the average golfer on the PGA Tour improved by about 1.6 strokes. (In an old article, we used a different (and likely much less reliable) method and found larger differences - we place much more faith in the estimates here).

It is always controversial in sports analytics work to present statistical comparisons of athletes across generations. This is likely because there are always assumptions involved and also because different camps of observers often have strong priors about performance across generations. We won’t spend much time here arguing about the validity of these estimates; as you will see later, the prediction exercise does not hinge on belief in our estimates of changing average skill level over time. That being said, they are important to understanding age profiles of golfers, so that’s why we’re covering it.

Now, let’s talk about age profiles. Like most skills in life, we would expect a professional golfer’s ability to increase as they age before flattening out and eventually declining. It should be clear why the previous analysis and discussion is critical to understanding aging curves: if the quality of the average professional golfer is improving over time, then we may underestimate changes in a golfer’s ability as they age. For example, suppose that your metric for performance is strokes-gained relative to the average professional in the current year. Then, given the results in the first figure, we would wrongly conclude that a golfer who gains 1 stroke over PGA Tour fields in 1990 and does the same in 1995 has not improved. However, after adjusting for the improvements in the average field quality on the PGA Tour from 1990 to 1995, the correct conclusion would be that this golfer improved by about 0.25 strokes per round.

Regardless of which measure you use, either time-adjusted strokes-gained or current-year-adjusted strokes-gained, it is a bit complicated to properly construct age profiles. The reason it is complicated is that not all golfers should be expected to have the same age profiles. In our view, there are two main variables that will affect a golfer’s expected aging curve: 1) their baseline ability (e.g. are they a +1 strokes-gained player at age 22, or a -1 strokes-gained player at age 22?), and 2) how many years they’ve spent in professional golf. The first point is important because the higher is your strokes-gained baseline, the less room for improvement you have (this follows if we assume that there is some natural upper bound to golfer performance). The second point is important if there is a learning curve in professional golf; a 25-year-old that is in his rookie season on one of the professional tours may not be the same as a 25-year-old who has already spent 5 seasons in professional golf.
Below we first present the aging curve using all golfers in our sample, followed by a few aging curves for different samples of golfers. Each data point on these curves is interpreted as a golfer’s expected strokes-gained relative to their performance at age 21 (or whatever the youngest age is in the sample). See endnote [1] for a discussion of how these curves are constructed. In all of the figures that follow, there are two age profiles: the blue line is the “time-adjusted” age profile and the red line is the “relative” age profile. The difference between them is that the time-adjusted profile uses a strokes-gained measure that has corrected for the fact that the skill level of the average professional has changed over time, while the relative profile uses a strokes-gained measure that is relative to the average PGA Tour golfer in the current year.

Notes: Plotted are a set of age coefficients from a regression of annual strokes-gained on a set of age dummies (for ages 21-48) and player fixed effects. Data includes all seasons that were comprised of 25 rounds or more on the PGA Tour, European Tour, or Web.com Tour since 1983 (Euro data starts in 1996, Web data in 1990). The omitted group in all plots is the youngest age; therefore you interpret the data points as the average strokes-gained at that age relative to this base age. The "relative SG" data is strokes-gained relative to the average PGA Tour player in that year; "time-adjusted SG" is a strokes-gained measure adjusted for differences in field quality over time.
Notes: Plotted are the set of age coefficients from the relevant regression. a) Includes all players who averaged 0 strokes-gained or better (using the "relative SG" measure) in their first season. b) Includes all players who averaged worse than 0 strokes-gained in their first season. c) and d) Self-explanatory - includes all players who began their careers in the specified age range. In all cases, data includes seasons comprised of 25 or more rounds.
The age profile constructed using all of our data indicates that the typical professional golfer’s “true” performance improves steadily up to about age 32, flattens from ages 32-37, and then declines steadily up to age 48. In contrast, the relative age profile, which is confounded by the fact that the quality of the average PGA Tour professional has increased steadily over time, indicates that performance starts declining at age 33 and continues to decrease to age 48, ultimately reaching a much lower skill level than the starting performance level at age 21.

There shouldn’t be too much debate around the validity of the relative age profile. While we have adjusted scores within each year for this (across tournaments and tours), there are not any real statistical tricks going on. If you performed the same exercise using average strokes-gained on the PGA Tour (unadjusted for field strength - so, just subtracting off the mean score in each round) you would likely obtain a similar profile. The time-adjusted SG profile, on the other hand, requires believing that we have correctly adjusted scores over this time period. It would be interesting to see if professional golfers agree with the true age profile (i.e. the time-adjusted SG profile) here, given their own experiences. They would have to be able to assess how their performance has evolved over time, irrespective of how their performance has evolved relative to the tour average - a pretty hard thing to do, especially with technological change occurring at the same time.

We provide the other graphs which use specific subsets of golfers to highlight the fact that age profiles are likely to differ depending on several factors. For example, it matters how high the baseline ability of the golfer is. This is due to the fact that we always expect some regression to the mean for any player who has a great season (i.e. Spieth may have been performing above his true ability as a young professional), and it is also due to the fact that players who truly have high abilities at young ages have less room to improve (i.e. even if 22-year-old Spieth truly is a +2 SG player, we know from historical data that he is very unlikely to improve beyond +2.5 SG, for example). In looking at plot (a), it doesn’t seem like selecting for players who had unusually great starting seasons is much of an issue, as the aging profile still slopes up beginning at age 21. But, we do see much less of an improvement leading up to peak performance at age 32 than when we examine players who started their careers performing below the PGA Tour average (subplot b), which speaks to the point about how there simply is less room for improvement for elite young golfers.

The other dimension of aging profiles we examine is the age at which a golfer entered our dataset (i.e. a proxy for when they started their professional careers). It seems plausible that golfers who start their professional careers at age 20, compared to those who started when they were aged 25, would have different profiles from age 28-32, for example. We do see that the early starters appear to have aging curves that start to decline at slightly younger ages, but overall their aging curves look pretty similar.

A final point is that the older end of the aging profile may be harder to interpret. It is likely that we are selecting for players who did not experience a huge decline in their performance as they moved into their mid-to-late 40s. Players who really drop off are unlikely to show up in our data, even though we do have data for the developmental tours. There are plenty of examples of professionals who end their playing careers in their 40s to pursue other things (e.g. commentating). To the extent that this happens in our data, we may underestimate the drop off in performance that occurs at the tail-end of the aging curve.
(Astute readers may wonder.. if aging effects are important, don't you need to incorporate this to properly adjust scores in the first place? This thinking is correct, and in theory we have incorporated aging into our score adjustment method [2].)
Part 2: Understanding our career projections
Next, let’s move on to the discussion of how best to predict career trajectories. As was eluded to earlier, for this prediction exercise we are going to be using the "relative” strokes-gained measure as the measure of performance. Recall that the interpretation of this measure is strokes-gained relative to the average PGA Tour professional in each season, while time-adjusted strokes-gained is a measure of performance relative to a specific year (e.g. the average PGA Tour player in 2000). The main reason we focus on relative strokes-gained is that an important part of this project is finding comparable golfers to the current professionals using our historical database. If we were to use the time-adjusted strokes-gained measure, it would be the case that most of the top comparisons for today’s players are recent players. For example, using the relative measure, we find that Rickie Fowler and Davis Love III were fairly similar golfers at the start of their careers. They both performed at a level between 1-2 strokes better than the average PGA Tour player (at the time) in most of their seasons between the ages of 22-27. However, using the time-adjusted measure, we would say that Fowler was performing at a significantly higher level than Love III (because we estimate that the PGA Tour was much stronger in 2010 than in 1990), and that they therefore are not that comparable. Using the relative measure also has the benefits of being more intuitive and not requiring that readers believe we have correctly adjusted scores from 1983-present. We also don't think that the projections would be significantly altered using the time-adjusted strokes-gained measure instead of the relative one [3].
(Here are rough guidelines for inferring performance level from relative strokes-gained averages: +2 is typically near the level of a top 5 player in the world, 0 is the PGA Tour average, and -1 is the European Tour average.)

The historical database includes data from 1983 on the PGA Tour, from 1990 on the Web.com Tour, and from 1996 on the European Tour. To find the top comparisons for each present-day golfer, we use a few (subjectively chosen) characteristics. The most important is age: for a 27-year-old Rickie Fowler, all comparisons are chosen from the 27-year-old versions of each golfer in our database. The characteristics used to calculate similarity amongst this set of 27-year-old golfers are: average strokes-gained and rounds played in each of the previous 3 seasons, career average strokes-gained, number of years as a professional golfer, and the fraction of their career spent on the PGA Tour (as opposed to other tours).
To actually form the projections, we rely on regression models. We project performance 5 years into the future, and for each year we project a golfer’s mean performance level (i.e. what we expect), their 90th percentile performance level (i.e. the performance level that is better than 90 percent of the possible career trajectories they “could” follow), and finally their 10th percentile performance level (i.e. the performance level that is worse than 90 percent of the possible trajectories they “could” follow). This therefore involves estimating 15 regression models: 1-5 years into the future and at 3 points of the distribution. To project the mean performance level we use the standard regression model as you know it, and to project performance at different percentiles we use quantile regression (see endnote [4] for a primer). The main variables used to predict future performance are a golfer’s current age, the number of years as a professional, their strokes-gained averages over the last few years as well as the number of rounds played, and their career average strokes-gained (as well as various interactions of these variables).
For intuition into the projections, there are really just two concepts to consider. The first is the aging profile - recall that, because we are using the relative strokes-gained measure, the relevant age profiles to look at are the red lines in the figures above. All else equal, we should expect younger golfers to improve and older golfers’ performance to decline. Using the relative strokes-gained measure, this decline seems to start around age 32-33. The second concept is regression to the mean: we always expect present-day performance gaps between golfers to narrow over time. Why should we expect this? Let's think of golf performance as mainly the result of golfer skill, but also as something influenced by “luck” (where “luck” should not be necessarily thought of as fortunate bounces off of trees, but as unusually good or poor stretches of performance by a golfer, which could occur for any number of reasons). Under this model, we will observe regression to the mean: if two golfers are separated by, for example, 3 strokes on average per round in one season, we expect them to be separated by less than that the following season. The reasoning is simple: some of that performance gap is likely due to luck and not true skill differences, and we do not expect luck to persist to the following season.
Another important, and related, point is that the further the projections are into the future, the closer together they become. A present-day 3 stroke difference between two golfers might be projected to narrow to 2 strokes by next season and to just 1 stroke 5 seasons into the future. (The best projection for next season belongs to Dustin Johnson at +2.3, while the best projection for 5 seasons from now belongs to Jon Rahm at just +1.76.) Again you could think of several models of lifetime performance that could produce this pattern. Intuitively, the further into the future we look, the more opportunity there is for a golfer’s ability level to change. These changes will exhibit regression to the mean: golfers with a current ability level that is in the right tail of the ability distribution will be statistically more likely to experience changes for the worse. (If you are befuddled by the concept of regression to the mean, start here, and if after that you want to take the full dive into the rabbit hole, see [5]).

Finally, we should point out that while the writing above would seem to imply we are making choices about how much regression to the mean to apply to our projections, this is not the case. The projections are data-driven: they are the values that best fit the data, conditional on using the class of models we use (i.e. regression models). The concepts of regression to the mean and aging profiles are useful for understanding the career trajectories of golfers, but they are not imposed on to the data in any way when forming the projections.

Let’s break down a specific example. Here is Justin Thomas’ projection:

Notes: Shown is a screenshot of Justin Thomas' projection from our interactive projection page. The solid black lines are Thomas' actual strokes-gained numbers up to the most recent season (2018). The dotted black lines are our projected strokes-gained for Thomas. The faded lines (which become solid when the name on the left-hand side is hovered over) are the actual strokes-gained numbers for Thomas' top comparisons.
Thomas will spend the majority of 2019 as a 26-year-old. This puts him on the increasing side of the aging curves shown above. However, Thomas has also performed at a very high level to begin his career, especially in the last 2 seasons. We have Thomas’ performance projected to be lower next season than it has been the last two years. Therefore, in his case, the (negative) effects of regression to the mean outweigh the (positive) aging effects. Thomas’ projection is also lower due to the fact that in seasons before 2017, he was not performing at the level he did in 2017 and 2018. If he had performed at the level of the previous two seasons for each of his first five seasons on tour, then we would not project as much regression to the mean (see Jon Rahm’s projection for an example). As the projection moves farther out, we see more regression to the mean for Thomas, which is typical of all the top players. In terms of projecting quantiles, Thomas’ 90th percentile projection basically hovers around 2.5. This is also characteristic of a lot of the top players: it is simply very rare for a golfer (other than the younger versions of Tiger) to on average gain more than 2.5 strokes per round in a season. The final point, which is common to most golfers’ projections, is that the gap between the 10th and 90th percentile projections widen as we move further into the future; this is simply because the longer the timeframe under consideration, the more uncertainty there is around a player’s future performance. Even though we do project Thomas' performance to decline slightly in the next 5 years, he still has 3rd highest projection 5 seasons from now (behind just Rahm and Spieth).

One interesting question is the following: for good young golfers, at which point do the negative effects of regression to the mean outweigh the positive effects of aging? The answer will depend on several characteristics of the golfer at hand, but a rough answer appears to be around +1 strokes-gained. Take Daniel Berger as an example: he will be 26 years old next season and has averaged around +1 strokes-gained in each of his last four seasons. Our projections for him for the next 5 years are basically flat at +1. You can think of this as the result of the offsetting effects of regression to the mean and aging.
In general, the regression models we use seem to provide a lot of nuance in the projections. This is mainly due to the inclusion of several interaction terms [6]. For example, the interaction between age and various measures of past strokes-gained (e.g. last season, two seasons ago, career to that point) allows for separate age profiles for golfers who have performed differently in the past. We saw this with Thomas above: young elite golfers are expected to improve less (or even regress) than young average or below-average golfers. Another important interaction term in these models is the number of years spent on tour interacted with the golfer’s career strokes-gained average. Averaging +2 strokes-gained over a career that has spanned 20 years is worth more than averaging that over a 5-year career. This is largely what is driving Tiger Woods’ excellent projection for the next few seasons.

If you've made it through the document, awesome, hopefully it was useful and insightful. You can explore all of the projections and the historical comparisons for today's players here.

1. Plotted on these graphs is the set of coefficients from age dummy variables, obtained from a regression of yearly average strokes-gained on a set of age dummies with player fixed effects (i.e. a dummy variable for each player in the sample). The interpretation of the coefficient on a given age dummy variable is the expected difference in strokes-gained relative to some baseline age (in the graphs above, that baseline is always the youngest age in the sample). The player fixed effects control for any differences in strokes-gained at different ages due to level effects (i.e. the fact that some players are better at all ages than others). Including fixed effects is similar to taking differences across years to identify age effects, in that you are comparing the same player at different ages and not making any comparisons across different players. [Back to text]
2. When we adjust scores we allow each golfer's ability to vary over time. In practice, this means including a polynomial in each player's "golf time" - i.e. their sequence of rounds. This likely is confusing unless you've followed the link to our field strength article and you have experience with fixed effect regression. The point is simply that we allow for each player's ability to vary flexibly over time, and this will capture any aging effects. [Back to text]
3. By using the strokes-gained measure that is corrected within years but not across years we are relying on some assumption like "we expect the average player quality on the PGA Tour to continue to improve at the same rate as it has in the past". We are relying on the aging curves of historical players to predict the aging curves for current players; for this to be accurate using "relative" strokes-gained, the strength of the PGA Tour needs to keep increasing at the same rate it has. If you look at the first figure in this article, it does appear that the rate of improvement in the average PGA Tour player is approximately linear since 1983, which bodes well for the assumption we require. [Back to text]
4. Regression as you likely know it can be thought of as a statistical model for the conditional mean. The predicted values from a regression of Y on X are, under a couple conditions, an estimate of the conditional expectation of Y given X. In our context, X includes all of the variables mentioned above: a golfer’s strokes-gained over the last few years, their career strokes-gained, etc. Therefore, the standard linear regression is giving us, roughly speaking, the expected value of next year’s strokes-gained (Y) given the characteristics of that player (X). This is what we use as each player’s projection. Similarly, quantile regression can be thought of as a statistical model for the conditional quantile. The mechanics of quantile regression are very different from standard mean regression, but their interpretations have some similarities. The predicted values from a quantile regression at the 90th percentile is (with some assumptions) the 90th percentile of Y given the covariates, X. In our context, think about the hypothetical distribution of next season’s strokes-gained conditional on a set of historical variables (past strokes-gained, years on tour, etc.). Quantile regression at the 90th percentile estimates the value for next season’s strokes-gained that is better than 90 percent of all other possible values given the set of characteristics (e.g. given that the player averaged +2 SG this past season, that they are in their 5th season on tour, etc.). [Back to text]
5. A common reaction to these projections will likely be that they are too pessimistic on the top young golfers. Why do we project that a young golfer like Justin Thomas will regress rather than improve in the next 5 years? The answer to this requires coming to grips with the idea of regression to the mean, which is no easy task in my experience. Consider two variables: strokes-gained in 2018 for a set of players, and strokes-gained in 2019 for that same set of players. If these two variables are not perfectly correlated, then they will exhibit regression to the mean. That is, we will predict the highest 2018 values to decline a bit in 2019, and the lowest to increase a bit. This can be seen right away if you were to plot the regression line from the 2019 values regressed on the 2018 values. The predicted 2019 values will exhibit less variance than the observed 2019 values. This is the definition of regression to the mean. The final piece to coming to terms with this concept is then thinking about when these two variables would be perfectly correlated. The answer is that they would only be perfectly correlated if golf was a game of 100% skill. If there is any luck involved, we will not have perfect correlation between 2018 and 2019 scoring averages, and will therefore have regression to the mean. (One technical detail that is important here is that the two variables are scaled the same, e.g. if they are normally distributed, they have the same variance. This should be roughly true in our example: the overall distribution of 2018 and 2019 strokes-gained averages for the same group of golfers is similar as they age). [Back to text]
6. Including interaction terms in a regression model allows for more complex relationships between your covariates and the outcome variable. If you have two regressors X and Z, and you decide to include the term X*Z, this allows the effect of Z on Y to vary depending on what the value of X is. One example in our context is the interaction of age and past strokes-gained average. We expect a 25-year-old to experience less of an increase in next season's strokes-gained (Y) if they have averaged +1 SG over the past season than if they have averaged -1 SG. [Back to text]