Wednesday, May 10, 2017

Southern California Golf Association Errs Badly in Explanation of the Slope System

From time to time the USGA and regional golf associations publish articles explaining the USGA Slope System.  More often than not, the articles get it right.  On too many occasions, however, the articles demonstrate a lack of understanding of the Slope System.  Such articles tend to confuse rather than inform their readership.  The latest example comes from Fore Magazine, the publication of the Southern California Golf Association (SCGA).[1]  The article is entitled “Slippery Slope: Comparing Course Difficulty” and was written by Doug Sullivan, the Director of Course Rating for the SCGA.  The article contains three major errors.  First, the suggested criterion for measuring course difficulty is not adequate for many situations.  Second, the article incorrectly describes how the Slope Rating is determined.  Third, the article’s explanation of why a player gets more strokes is not consistent with the theory behind the Slope System. Each error is reviewed in turn
Error No. 1 – Misleading advice on measuring course difficulty
Sullivan follows in the footsteps of several other articles that claim the Course Rating is the dominant criterion for ranking courses by Course Difficulty.[2]  Sullivan is more adamant, however, and claims the Course Rating should be the sole criterion is measuring course difficulty.  He writes:

… Slope Ratings should not be used to compare the difficulty of different golf courses.  Instead a Course Rating is generally a better indicator in determining the difficulty of one course compared to another. 

To make the case for the Course Rating as the measure of difficulty, Sullivan creates a straw man.  He describes two courses.  Course A has a Course Rating of 68.5 and a Slope Rating of 130.  Course B has a Course Rating of 71.8 and a Slope Rating of 120.  Sullivan asks which course is more difficult.  Sullivan argues your answer should be “Course B because Course B has a higher Course Rating.”   Sullivan never defines a “difficulty.”  For the sake of argument, difficulty is defined here as the score a player needs to play to his handicap (i.e., Diff = Course Rating + Course Handicap).   Because of Sullivan’s fortuitous selection of ratings, a player’s Diff will always be higher on Course B.  Sullivan used this faux example as definitive proof the Slope Rating should not be used to compare the difficulty of courses.[3]
Sullivan concludes:

So the next time someone asks you which course you think is more difficult, compare the Course Ratings, not the Slope Ratings.  Not only will your answer be more accurate, but your friends will be more than a little impressed with your knowledge of SLOPE.    

Sullivan is wrong.  While the Course Rating may be the dominant factor in measuring difficulty, it is not the only factor.[4]  If the definition of difficulty shown above is accepted, then:

                            Diff = Course Rating + Index∙Slope Rating/113

If there is a large difference in Course Ratings (e.g., 3 strokes), the course with the highest Course Rating will also have the highest Diff.   If the difference in Course Ratings is small, then Diff will be influenced by the Slope Rating and the player’s Index.   To demonstrate how Sullivan is wrong, two fantasy courses are constructed.  Course A has a Course Rating of 70.0 and a Slope Rating of 69.0.  Course B has a Course Rating of 70.5 and a Slope Rating of 155.  The vast majority of players would score higher (i.e., have a higher Diff) on Course B since it has both a higher Course Rating and Slope Rating.  But what if a player’s Handicap Index was +4.0?  Course A would have a Diff of 67.6 while Course B would have a Diff of 65.0. This player would find Course A more difficult.  In essence, Course Difficulty for an individual player cannot be determined by simply comparing Course Ratings as Sullivan suggests.

Since Sullivan did not acknowledge the importance of Index and Slope Rating in determining difficulty, his knowledge of SLOPE should not impress his friends.  Moreover, though left unstated in Sullivan’s article, course difficulty will vary by a player’s characteristics (hooker, slicer, short hitter, etc.).  If you really want to determine which course is more difficult, you should play them and draw your own conclusion.  To try to determine the most difficult course solely by USGA Ratings that are prone to error will only yield answers of little reliability and even less consequence.

Error 2 – Incorrectly describing how the Slope Rating is determined
Sullivan demonstrates a lack of understanding behind the derivation of the Slope Rating when he writes:

This is the type of graph (shown below) that is created to compare scores of golfers of different abilities.  As you can see, scores increase as a player’s Course Handicap increases.  A line is created to connect as many scores as possible.  The steeper the line, the higher the Slope Rating.  The flatter the line, the lower the Slope Rating.  The concept of SLOPE simply refers to the slope of this line – that bit of trivia should help you make a few bucks on your next friendly bet.

Sullivan is simply wrong.  Since the advent of the Slope System, the slope of the line relating average scores to handicap is the same for all courses.  After all, that was the purpose of the Slope System. To be correct the horizontal axis in Sullivan’s graph should have been labeled Handicap Index and not Handicap.[5]   If you followed Sullivan’s advice and placed a bet—you lost!

Sullivan makes a technical error when he writes the “line (shown in the graph) is created to connect as many scores as possible.”  Connecting as many scores as possible is not the appropriate criterion for determining the slope.  The line is typically determined  by linear regression techniques where connecting the scores is definitely not a requirement.

Error No. 3 – Confusing the reader on how Course Handicaps are determined
In trying to explain how and why the Slope System assigns additional strokes, Sullivan writes:

In simple terms, SLOPE is designed to make sure a golfer receives more strokes when playing a more difficult course and fewer strokes when playing an easier golf course compared with the USGA Course Rating.

This sentence is so poorly constructed it is hard to discern what Sullivan means.  Since Sullivan has just proclaimed the Course Rating is the measure of difficulty, he seems to be saying a player should receive more strokes at the course with the higher Course Rating.  This would be incorrect.  The handicap player does not receive more strokes because the Course Rating is higher and the course more “difficult.”  Handicap strokes are given as a function of the difference between the Bogey Rating and the Course Rating at a course.  Even in Sullivan’s own example, most players will get more strokes at the course Sullivan has declared to be the easiest (Course A).

So why do articles such as Sullivan’s continue to be published?  It’s probably because neither the SCGA in this case nor the reader are sticklers about accuracy.  The SCGA is looking to fill its magazine and assume the Director of Course Rating must know what he is talking about.   And if he doesn’t, it doesn’t really matter.  SCGA members can be divided into three groups.  The first will pass on any article about the Slope System so the image of the SCGA will not be harmed.  The second group will assume the article must be accurate since it came from the experts at the SCGA.  The third group that recognizes the article as nonsense is a very small subgroup of the SCGA membership (i.e., approximately 1) that can easily be dismissed.  

[1]Sullivan, Doug, “Slippery Slope: Comparing Course Difficulty,” Southern California Golf Association’s Fore Magazine, Spring 2017, p. 94.
[2]Cowan, Jim. “An Explanation of Slope,” Northern California Golf Association website,  Metropolitan Golf Association, “How do the Course Rating and Slope numbers affect my Handicap Index?”
[3] It is not clear why the authors of these articles have chosen peculiar Course and Slope Ratings to make their point.  Jim Cowan of the Northern California Golf Association, for example, used a course with a Course Rating of 72.8 and a Slope Rating of 114.  Since Course Ratings are highly correlated with Slope Ratings, it is very likely such a course does not exist.  Real courses could have been selected to demonstrate how difficulty is measured, but no author has taken that path.    
[4] Cowan, loc. cit.
[5] Stroud, R.C. and L.J. Riccio,  “Mathematical Underpinnings of the slope handicap,” Science and Golf, E & FN Spon, London, 1990, pp. 129-140.  Stroud shows the Perfect Valley Handicap as the label for the horizontal axis.  The Perfect Valley Handicap is a player’s Handicap Index. 

Wednesday, April 12, 2017

Another Inequitable Tournament and So It Goes...

This blog has documented many cases where a misunderstanding of the USGA Handicap Manual has led to inequitable results. The latest example comes from a tournament at a club in Southern California.  The tournament consisted of four different stroke play competitions of nine holes each: 1) Four-ball, 2) Total score of partners, 3) Scramble, and 4) Pinehurst. 
The Tournament Committee imposed a limit of 8 strokes between the handicaps of the partners.  The handicap of the higher handicapped player was reduced until the 8-stroke limit was reached.  This, however, is not what the USGA recommends:[1]
It is recommended that the Committee considers it a condition of four-ball stroke play competitions that the Course Handicap (after allowance) of the members of a side may not differ by more than eight strokes. A side with a large difference has an advantage over a side with a small Course Handicap difference. If a difference of more than eight strokes cannot be avoided, it is suggested that an additional 10 percent reduction be applied to the Course Handicap of each member of the advantaged side.[2]
Moreover, this recommendation is only for four-ball stroke play.  Its application to the other forms of competition, as was done in this tournament, is an egregious error.  To see the size of the error, assume Player A has an Index of 3.5 and Player B has an Index of 14.8.  The Table below presents the handicaps that were used in the Tournaments and those that would have been used if USGA guidelines had been followed. The USGA handicap calculations are shown in the Appendix.
Tournament and USGA Handicaps

Player A (3.5 Index)
Player B (14.8 Index)
Four Ball
Total Score

In Four-ball, Player B would play as a 7-handicap under USGA guidelines instead of a 6-handicap in the 9-hole competition. The competition used a modified Stableford scoring so that a net par was worth 2 points and a net birdie would be worth 4 points.[3]  It is likely the loss due to the Tournament handicap is in the 1 to 2 point range. (Note: The difference in Tournament handicaps after the allowance is 7-strokes and not the 8-strokes as recommended by the USGA.)
In the Total Score event, Player B would play as an 8-handicap rather than as a 6-handicap that was assigned by the Committee.  The loss due to the Tournament handicap is probably in the 2 to 4 point range.  If Player B bogeyed each hole where he had an additional stroke, he would gain 2 points.  If he made par on the two holes he would add 8 points instead of 4 for a net gain of 4 points.
The Scramble competition would not be affected since both the Tournament and USGA handicaps are the same.  Similarly, the handicaps are equal in a nine-hole Pinehurst competition.  The Tournament Pinehurst handicap of 3.5 is rounded to 4.0, the same as the USGA handicap.
The number of points lost by this team due to the Tournament Committee’s handicaps is in the range of 3 to 6 points.   Would this have affected the outcome?  Probably, since no team that had its handicap reduced by the 8-stroke rule came in the money.  The bigger problem, however, is the Tournament Committee that failed to follow USGA guidelines.   Players have an expectation a tournament will be run fairly.  In this case, that expectation was not met.

USGA Handicaps
Four-BallPlayer A’s course handicap is 4. His handicap after the 90 percent allowance is still 4 (4x.9= 3.6 rounded up to 4.0).  Reducing his handicap by an additional 10 percent still leaves the player at a 4-handicap (4  - .1 x 4 = 3.6 rounded to 4.0).  Player B’s course handicap is 16.  His handicap after the allowance is 14.  After an additional 10 percent reduction, his handicap is 13 (14 - .1 x 14 = 12.6 which is rounded to 13).
Total Score of Partners – The USGA recommends players be assigned their full handicap.  Player A would be a 4-handicap and Player B a 16-handicap.
Scramble – The USGA recommends the team handicap should be 35 percent of Player A’s handicap and 15 percent of Player B’s handicap.  Player A would have a 1-handicap (.35 x 4 = 1.4 rounded to 1).  Player B would have a 2-handicap (.15 x 16 =2.4 rounded to 2.0).
Pinehurst –The USGA recommends the team handicap should be 60 percent of Player A’s handicap and 40 percent of Player B’s handicap. Player A’s handicap would have a 2-handicap (.6 x 4 = 2.4 rounded to 2).  Player B would have a 6-handicap (16 x .4 = 6.4 rounded to 6).

[1] The eight stroke limit stems from research done by Francis Scheid published in Golf Digest in June 1971.  Scheid never studied actual tournaments, but used scorecards from his home club to simulate matches.   In 1971 there was no Slope System and the bonus for excellence was .85 rather than .96 as it is today.  Nevertheless, the  8-stroke limit has been imposed in many four-ball events even though it has never been proven to lead to increased equity in studies of actual competitions. 
[2] USGA Handicap System, Sec. 9.4bii.  The USGA does not recommend the 8-stroke limit for Four-ball match play.  The USGA’s reasoning does not seem consistent.  If a large difference in handicap leads to low scores in Four-ball stroke play, it would seem that a large difference in handicap would also lead to low scores in Four-ball match play—i.e., the team with a large difference would always have an advantage.  The USGA has never explained why the 8-stroke limit should only apply to Four-ball stroke play.
[3] Modified Stableford scoring adds an element of serendipity in deciding the winner.  Under modified Stableford scoring, two players with the same handicap and gross score can have different point totals in the Four-ball competition.  It is not clear if the Tournament Committee purposefully wanted to add an element of chance to the scoring or simply made a mistake in selecting the modified Stableford scoring system.

Monday, August 22, 2016

Revising Rule 18-2: Ball at Rest Moved

No rule has caused so much consternation to the USGA and R&A as Rule 18-2.  Rule 18-2 states if a player causes a ball to move, the player incurs a penalty of one stroke.  In the end a rules official must determine what caused the ball to move, and there’s the rub.

The interpretation of Rule 18-2 caused what many describe as a fiasco at the 2016 U.S. Open.  Dustin Johnson’s ball moved on the 5th green, and it had to be determined what caused the ball to move.   Johnson declared he had not caused the ball to move.  On the 12th tee, Johnson was informed he may be assessed a one stroke penalty.  The question of the penalty became moot when Johnson won by four strokes. The USGA ruled that absent any other suspects (wind, gravity), Johnson was guilty.  The one stroke penalty only reduced Johnson's winning margin to three strokes.  If Johnson had tied with another player, would the USGA have assessed a penalty?   Probably not. The USGA looked bad enough without deciding the outcome of a major championship on the subjective judgment of a panel of rules officials.  The eighteen hole playoff would have been held on Monday.
The 2015 Open Championship encountered another problem with Rule 18-2.  Louie Oosthuizen addressed a tap-in putt, but a gust of wind started the ball moving and it did not stop until it was five feet away from the hole.  At that point, play was suspended, but too late to help Oosthuizen.  He had to play from the further distance.
The problem inherent in Rule 18-2 is that it does not make a distinction between a ball on or off the green.  With today’s fast greens, it is much more likely a ball can be moved by wind or gravity.  But whether wind, gravity, or the player is the culprit is still difficult to discern.  To bring more equity to the problem, a rule change (also suggested by others) could make a ball on the green not in play if has been addressed or marked.  It is only in play after the ball has been struck by a stroke. Under this rule change, if the ball moves for any reason other than a stroke, it must be replaced with no penalty.   This rule change meets the following requirements of a good rule.
Clarity – No need to call in an official if the ball moves on the green after it has been marked or addressed.  Simply replace it.  The adoption of this rule would eliminate many Decisions on when a penalty should be assessed. It does bifurcate the rule depending on whether the ball is on the green or elsewhere.   This is reasonable, however.  Through the green, a player could move his ball in an attempt to remove loose impediments around his ball. He should be penalized since the removal of the impediment would improve his lie.  On the green, however, the player is allowed to remove loose impediments without penalty.  In other words, the Rules already differentiate between a ball on the green and a ball through the green.
In determining whether a ball has moved a player is given some leeway in the rules.   If the ball moves by an amount not reasonably discernible to the naked eye, a player’s determination that the ball has not moved will be deemed conclusive, even if that determination is later shown to be incorrect through the use of sophisticated technology (Decision 18/4).[1]  The revised rule would eliminate this Decision, make the outcome independent of the leniency of the rules official  and minimize the number of call-ins from Rules Mavens who believe they detected ball movement.
 Fairness – The revised rule tends to minimize luck in determining tournament outcomes.  If the wind blows a ball that has been addressed or marked off the green or in the hole, the player would not be punished or rewarded for such random acts of nature.
Proportionality –The one stroke penalty for a player inadvertently moving his ball on the green appears to be disproportionate.  Currently, a player is assigned the same penalty for 1) dropping his marker on his ball and causing it to move or 2) hitting a ball into a water hazard.  The latter action is the result of a bad swing and/or judgment and should be penalized.   The first action is due to carelessness.   A player gains no advantage if he replaces his ball after inadvertently causing it to move on the green.  True, the current penalty of one stroke acts as a deterrent to such carelessness.  But any benefit from reducing the frequency of such behavior is more than offset by the elimination of disputes over what caused the ball to move.
Any change in the Rules needs to be seriously vetted.  There may be unintended consequences of having a ball on the green considered out-of-play.  Testimony should be taken from those most affected by Rule 18-2 (i.e., Tour Players).  USGA and PGA Tour officials responsible for making the call of when a ball has moved also need a voice.  Rules changes follow Newton’s First Law: A rule at rest tends to stay at rest.  Without a demand for change from players and officials, Rule 18-2 will be cut and pasted into the next edition of the Rules of Golf for the foreseeable future.  

[1] That same leeway test is not given when a player touches the ground in a hazard.  In the 2016 Women’s U.S Open, Anna Nordqvuist touched the sand with her club and was given a two-stroke penalty.  A strong argument could be made that the violation was not apparent to the naked eye.  No one noticed the small grain of sand take a tumble until Fox, using sophisticated technology, zoomed in on her address of the ball.  Should the USGA be consistent in its Rules and apply the same standards concerning sophisticated technology to ”ball moved” and “touching the ground?”  It is a debatable question, but one that has never been publically addressed by the USGA.

Thursday, July 7, 2016

Beating Your USGA Index or Beating Your USGA Handicap: Which is Easier?

There has always been some confusion about what the probabilities in Appendix E of the USGA Handicap System actually represent.  John Paul Newport of the Wall Street Journal, for example, wrote a player with a handicap between 13 and 21 will only play 3 strokes better than his handicap once in 43 rounds.[9]  Newport cites the website of Dean Knuth, former Senior Director of Handicapping at the USGA for his probability estimate.  Knuth cited the work of F.P.  Engel who defined the Net Differential as:[1]

         Net Differential = (Adj. Score – Handicap) - Course Rating

It appears Knuth used Engel's probabilities for a player with a 13-21 handicap.  These probabilities were for beating your Handicap by N strokes or better.

Some time later, the USGA changed the definition of Net Differential  to mean beating your Index, but the probabilities remained the same.  This change was made because either the USGA thought it expedient or did not understand that “beating your Index” and “beating your Handicap” were not the same thing. The USGA was informed of the problem, but chose to ignore it.

To demonstrate the difference, a player’s adjusted score under the two definitions of Net Differential is computed below:

Beat Your Handicap
1)      Net  Differential  = (Adj. Score – Course Rating)  - Handicap = -N  
2)      Adj. Score = -N + Course Rating + Handicap

Beat Your Index
3)      Net Differential = (Adj. Score – Course Rating)·(113/Slope Rating) – Index = -N
            4)  (Adj. Score-Course Rating)·(113/Slope Rating) - Handicap·(113/Slope Rating) = -N
            5)  Adj. Score = -N·(Slope Rating/113) + Course Rating + Handicap

Equations 2) and 5) show it takes the same score to beat your Handicap or Index (i.e., N = 0).  For other values of N, however, a player needs a lower score to beat his Index than he does to beat his Handicap if the Slope Rating is greater than 113.  For example, if the Slope Rating is 150 and N= 6, a player would have to score approximately 2 strokes lower to beat his Index by -6 than he would to beat his Handicap by -6.

[1] Knuth, D.L., F.J. Scheid, and F.P Engel, “Outlier identification procedure for reduction in handicap,”  Science and Golf II, E & F Spon, London, 1994, pp. 228-233. 

Friday, June 10, 2016

The USGA and R&A Bogey Distance Study

The USGA and the R&A recently released a study of driving distances on various professional tours.  With the resources of both organizations, one would expect original data and insightful analyses.   The reader will be disappointed on both counts.  The data was supplied by the professional tours.  It appears the USGA merely put the data in a spread sheet and produced charts like that shown in the Figure 1 below.

The major conclusion of the study is:

Between 2003 and the end of the 2015 season, average driving distance on four of the seven tours increased about 1%, or 0.2 yards per year.  For the same period, average driving distance on the other three tours decreased about 1%.

This sounds like the USGA has curtailed the rapid increase of driving distance.  Left unsaid—but shown in the charts—is the large increase in driving distance from 1980 to today.  Average driving distance during this period increased from about 255 yards to approximately 290 yards (13.7%).  While the charts do give a timeline of when club and ball innovations were introduced and when rules on equipment were implemented, there is no attempt to examine which were responsible for changes in distance.  That is left to the interested reader.

Future increases in distance will probably come from the change in the composition of players.   Short hitters do not do well on the PGA Tour and are the most likely group to lose their playing status.  When they do, they will be replaced by Web.Com players where the average driving distance is already greater than on the PGA Tour.  The study could have tested for this “composition” effect, but it chose not to.

Because of the increase in driving distances, courses have been stretched out.  Does this give the long hitter an even greater advantage?   Of the top ten money winners in 2015, six averaged over 300 yards.  Three others (Fowler, Spieth, and Stenson), were above average in distance and capable of reaching many par 5’s in two.  Only Zach Johnson was below average in driving distance.  Is driving distance more important in determining relative performance than it has been in the past?   Unfortunately, the USGA and R&A did not even attempt to assess the impact of driving distance on any aspect of the game.

As regards to the Handicap System, the USGA has not changed its assumptions about the driving distances of the scratch and bogey golfer from 1994 to today.  The scratch golfer has been frozen in time with an average tee shot of 250 yards.  The bogey golfer is assumed to have an average tee shot of 200 yards regardless of what equipment he buys.  If the standards by which Indexes are determined (Course Rating and Slope Rating) have remained the same, but an amateur’s driving distance has increased there should have been a resultant decrease in Indexes.  There is no strong evidence supporting such a decrease.  Nor has the USGA explained why its Course Rating System is impervious to changes in equipment.

The USGA and R&A stated they will produce this distance report every year.  If this means just adding another data point to the charts, then why bother?       

Friday, May 27, 2016

The R&A Battles for Gender Equality: Or Does It?

The Royal and Ancient Golf Club of St. Andrews(R&A) recently announced Muirfield Golf Club would no longer be considered as a site for The Open Championship.  The announcement was pure political theater since Muirfield was not up in the rotation for a considerable number of years.  The R&A’s intent was to show its enlightenment on issues of gender equality to placate sponsors of its golden goose—The Open.  The R&A had a lot of work to do to clean-up its bona fides as a civil rights champion.  After all, for 260 years the R&A had the same policy of barring women from membership that Muirfield was now being pilloried for.  Here are some alternatives the R&A might have considered for image control and why it chose to do what it did (i.e., alternative 4):

1. Cancel this year’s Open Championship at Royal Troon Golf Club which also does not allow women members - This step would dramatically demonstrate how seriously the R&A takes the issue.  While it might punish Royal Troon, it would wreak more economic harm on the R&A.  Better to argue Troon was awarded the Open by a previous administration of the R&A, have the television announcers “tsk, tsk” the men only policy, and reap the millions The Open will bring. 

2. Atone for years of discrimination by putting itself on probation and granting sole control over the Rules of Golf for 5 years to the United States Golf Association (USGA) - The problem with this alternative is some of the nabobs of the USGA were members of the R&A during the heyday of sexual discrimination.  The USGA should not benefit from a crime in which it abetted.  David Fay, former Executive Director of the USGA comes to mindWhile he famously resigned form the all male Pine Valley Club, he kept his membership in the R&A. It is more likely the notoriously parsimonious Fay left Pine Valley because he didn’t want to pay the dues rather than from any moral convictions.

Mike Davis, the current Executive Director of the USGA does not appear to have cleaner hands.  He too is probably a member of Pine Valley since he posts scores from there.  He was a member at Trump National, New Jersey until Donald Trump found disfavor with the liberal elites.  When The Donald was a fringe candidate and given little hope of winning, Davis was quick to announce the USGA was reconsidering holding the US Women’s Open at Trump National.[1]  Since Trump won the Republican nomination, Davis has been mute on relocating the tournament. 
If the R&A were to cede control, it must find an organization with a much better record on gender equality and that also has the courage of its convictions.  That would not be the USGA.

3. Assume control the Women’s British Open and give equal prize money to men and women - The R&A is in the process of merging with the Ladies Golf Union which owns the rights to the Women’s British Open.  Currently, the difference in total purses between the two Opens is about $7 million.  To get rid of the title sponsor of the Women’s Open (Ricoh) would probably cost an additional $1 million or more.  What price is the R&A willing to play to demonstrate its commitment to full gender equality?  It is probably somewhere south of $8 million a year.

4. Issue a press release chastising Muirfield and announce its removal from the rota until Club members vote for a more inclusive membership policy – This alternative would give the R&A moral high ground and be cost free.   In the meantime, Muirfield would be encouraged to follow the R&A’s lead by inviting a few women to join. The three main arguments would go like this:
  • Women want the right to be a member more than they want to be a member.   If the membership becomes open to women, the Club may have to recruit women to prove it has an inclusive membership policy. It is likely women memberships will top off at 1 percent of the total membership as they do at the R&A and Augusta.  Does Muirfield really want to take to the barricades over this minor incursion?
  • It’s not like Muirfield is all male club where women are not permitted on the grounds.  Female tourists and member’s wives are mulling around all the time.  A few more women will not be noticed and they have the upside of paying dues.  True, Sir Edgar will have to more careful about where he takes a leak.  Lose the Open or fit Sir Edgar with a catheter?  Take your pick.
  • If Muirfield is not in the rota, tourists will not pay as much to play it.  This will be an income hit for the club.  The only way to make up for the loss is for the members to pay more for dues and/or liquor.  Will single malt scotch taste as good at twice the price?  

There is little doubt the Muirfield membership will eventually find these arguments compelling.  With that settled, Muirfield will resume its place in the rota.  The R&A will be praised for its tough stand that brought about change, and Muirfield will be extolled for its enlightened, if belated, policy.   In essence, everything appears to change, but nothing really changes--the perfect bureaucratic solution.

[1] Mell, Randell, “Davis: USGA ‘evaluating things’ regarding Trump, Golf Central Blog, July 9, 2015.

Tuesday, May 17, 2016

The Probability of Beating Your USGA Index

Appendix E of the 2016-2017 USGA Handicap System presents estimates of the probability of a player of achieving various negative Net Differentials or zero or better.[1]  These estimates differ substantially from the estimates the USGA has presented in the past.  Specifically, players no longer have an equal chance of beating their Index, and the chances for having a large negative Net Differential are substantially increased.  This post examines possible causes for the discrepancies between the two sets of estimates.          

The Probability of Beating Your Index -The 2012-2015 Edition of the USGA Handicap System estimated the probability of beating one’s Index to be .20 for all Index ranges as shown in Table 1.  In the 2016-2017 Edition, that probability varies with a player’s Index (see Table 1).  The low-Index player now has a 64 percent better chance of beating his Index than his high-Index competitor.  What causes the high-Index player to perform so poorly?   Is he confused by too many tips from Golf Digest?  Can he not hit the pressure shots essential for low scores due to the low self-esteem of habitually being placed in the last flight? 
Table 1
Probability of Beating Your Index[2]

Index (Average Differential)
+2.0 (0)
3.0 (5)
7.0 (10)
17.0 (20)
27.0 (30)
2012-2015 Ed.
2016-2018 Ed.
Normal Dist.

No, psychological or physical traits do not explain the phenomenon.  The poor performance of the high-Index players is most likely the direct result of USGA handicap policy, specifically the Bonus for Excellence (BFE). 
 To see how the probability of beating  one’s Index is affected by the BFE, assume a player’s differential follows a normal distribution with a mean of X and a standard deviation of σ.[3]  It can be shown (see Appendix A) that the average of a player’s better half of differentials is X - .8·σ.  A player’s Index is then:
1)           Index = BFE·(X-.8·σ)
              BFE = Bonus for Excellence
A player’s scoring differential is
2)           Player’s Scoring Differential = X - S·σ 
S = Number of standard deviation from the mean.
For a player to beat or equal his Index: 
3)           X-S·σ ≤ BFE·(X - .8·σ)
Solving for S:
4)           S ≥ (1- BFE) ·X/σ + BFE .8
If the BFE was 1.0, all players would have the same chance of beating their Index.  They would have to have a scoring differential equal or better than .8·σ below their mean scores.  From the Normal Distribution Table, the probability of this occurrence is .21.  Substantially the same probability the USGA has maintained in previous years.
The BFE is not 1.0, however, but .96.  For the purposes of illustration, it is assumed that a player’s σ is a linear function of his average differential, X:[4]
5)           σ = 3 + (1.5)·X 
A player with an average differential of 0.0 would have σ of 3.0.  A player with an average differential of 30.0 would have σ = 4.5.
Substituting equation 5) into equation 4), the new equation for S becomes:
6)           S ≥ (1 - .96)·X/(3+(1.5/30)·X)  +.96·.8 
For a player with an average differential of 30, S equals 1.04.  From the Standard Normal Table, the probability of such a differential or better is .15.  The probabilities for a range of average differentials are presented in Table 1.  The estimates from the Normal Distribution analysis mirror the empirical findings in the 2016-2017 Appendix E.  Both estimates reveal the high-Index player has less of a chance of beating his Index than the low-Index player.   The Normal Distribution analysis identifies the BFE as the likely culprit.  There is no theoretical or empirical evidence suggesting the BFE promotes equity.[5]  It was born from political compromise and not statistical analysis.   

Why Do the Estimates Differ by So Much?  There is no apparent correlation between the two sets of probabilities.  There are major differences, however, between the estimates.   For example, in the 2012-2015 Edition the frequency of a player with a 20.0 Index having a Net Differential of -10 or better was 1 in 37,000.  In the 2016-2017 Edition, the frequency is now estimated at 1 in 1,950.   If one of the estimates is correct, the other is off by a factor of 19.
The best explanation is the old estimates were not accurate.[6]   They appear to be based on a study by Bogevold in 1974 (unpublished).   Most of the research back in the ‘70s and ‘80s depended upon collecting scorecards from various sources.   The cost of collecting and processing scorecards necessarily limited the sample size.  It is doubtful Bogevold had a sufficient sample to estimate the probability of large negative Net Differentials with any accuracy.  Moreover, these scores did not reflect the Slope System which was not in existence at the time and were reported using a different stroke adjustment procedure than used today.  All of these factors make the probability estimates suspect.   There is no evidence the USGA ever tried to replicate the results with another study so the probability estimates were always of questionable validity.
While the old estimates may be in error, the 2016-2017 Edition estimates also have problems:
Measurement Errors -The USGA Handicap System only estimates a player’s true Index.  If the USGA used an unbiased estimate of a player’s Index there would not be a significant problem.  If the USGA used a player’s Current Index, however, the probability estimates for negative Net Differentials would be biased downward.
Model Error - Appendix E assumes the probabilities are a function of a player’s Index.  For the most part, a player’s probability of achieving negative Net Differentials is determined by his standard deviation and not his Index.  It is quite possible a wild 15.0 Index has a better chance of having a -12 Net Differential than a steady 30.0 Index player.  The probabilities in Appendix E are for an average player within each Index range.  There can be a large error if they are applied to an individual’s performance.  
No Estimate of the Error - Appendix E does not give the reader any estimate of the accuracy of its results.  Most likely, the USGA just took 7.3 million differentials, placed them into cells, and did the necessary long division to produce a frequency.   Appendix E implies 100 percent confidence in the estimates, which is clearly not true.  The fluctuating nature of the probabilities as a player’s Index increases cannot be explained by any theory, and should be attributed to random error. 
Sample Size - The handicap Index Classifications do not include the same number of players. The number of players with a plus Index is approximately 1 percent of the population.  Appendix E shows such players have a 1 in 9,216 chance of having a Net Differential of -7 or better.  The frequency of a Net Differential of -8 or better is “off the charts.”  The frequency should not be “off the charts” since the chart goes up to 1 in 46,328 for the 10.0 to 14.9 Index player.  It is likely the sample of players with plus Indexes was too small to capture any Net Differential of -8 or better.
Women - It is unlikely women were included in the sample, but if they were it would corrupt the analysis.  If women were not included, the USGA should make clear Appendix E only applies to men.

Conclusion - There are two conclusions.  First, the BFE should go the way of the stymie since it does not serve its intended purpose of promoting excellence[7] and is a source of inequity within the USGA Handicap System.
Second, trying to estimate the probability of a Net Differential by a player’s Index is a fool’s errand.  There is too much variance within Index ranges to produce meaningful results.  Appendix E should be eliminated since it serves no real purpose in the Handicap System.[8]   A better approach would be to choose a player with an average standard deviation and estimate the probabilities for various Net Differentials using the Standard Normal Table.  This procedure produces estimates very similar to those found in the 2016-2017 Appendix E.  These results should be presented as a rule-of-thumb to give players a rough idea of the likelihood of various Net Differentials.   To strive for more precision is both misleading and a waste of USGA resources.   

Appendix A
The Probability of Beating Your Index Assuming a Normal Distribution

Assume the distribution of a player’s differentials has a standard deviation of σ.  The average of a player’s better half of differentials is found by multiplying possible differentials measured in standard deviations from the mean by the probability of making that score.   For example, the probability a player has a net differential between 0 and .1 standard deviations below his mean differential is 0.0396.  The probability that one of the player’s best scores is between 0 and .1 standard deviations below the average differential is 0.0796 (i.e., the probability is multiplied by two since only scores below the average are included).  The player’s average differential is found by summing all expected value for all intervals.  If the value within an interval is approximated by the mean value within the interval, a player’s average  of his best differentials can be estimated as shown in Table A-1. The estimate is approximately -.8·σ.
Table A-1
Calculation of Expected Differential (Abbreviated)

Mean Value
Expected Value
0 to -.1
-.1 to -.2
-.2 to -.3
-2.9 to- 3.0


[1] Net Differential = (Adjusted Score – Course Rating)·113/Slope Rating – Index
[2] The groupings of players are different from that shown in Appendix E.  The analysis presented here is concerned with average differentials.  The average differential consistent with the Index is shown in the heading of Table 1.  For example, an Index of 27.0 corresponds to an average differential of approximately 30.   Therefore USGA results for a player with a 27.0 Index are compared to a player with a 30.0 average differential.
[3] The USGA has maintained golf scores follow the normal distribution.  See Scheid, F.J., “On the normality and independence of golf scores with various applications,” Science and Golf, E &F Spon, London, 1990, pp. 147-152.
[4] From the probabilities presented in Appendix E, it is possible to estimate the standard deviation by Index range.  Equation 5) is a reasonable representation of the relationship between a player’s average differential and his standard deviation.
[5] In a rudimentary study, Dr. Francis Scheid estimated a BFE of 1.07 was the most equitable (“You’re not getting enough strokes!, “Golf Digest, June 1971).  Given the small sample size, a BFE of 1.0 probably could not be eliminated as the most equitable.
[6] A possible error was the old estimates were really the odds of negative Handicap Differentials (i.e., (Adj. Score – Course Rating) – Handicap = Handicap Differential) and not negative Net Differentials.  On a course with a Slope Rating greater than 113, however, a player has a better chance of scoring a Handicap Differential of –N than a Net Differential of –N (N>0).  If the old estimates were for Handicap Differentials, the old probabilities should be systematically higher than the new probabilities.  They are not, so the explanation for the differences must lie elsewhere.
[7] See Dougharty, Laurence, “The Bonus for Excellence Ruse,”, January 15, 2013.
[8] Appendix E has had many problems over the years as documented in “The Unmaking of the USGA’s Appendix E,”, March 2, 2016. 
[9] Newport, John Paul, “Fighting Back Against Sandbaggers,” Wall Street Journal, July 2, 2011.