Thursday, October 23, 2014

Can the USGA Slope Rating Decrease as Yardage Increases?

There are cases where the USGA Slope Rating decreases as the yardage increases.   It is never clear, however, whether this anomaly is due to an error by the Rating Committee or an oddity in the course design.  This post examines one such situation to determine the most likely explanation.
Our example is drawn from the files of the Oregon Golf Association (OGA) in 2006.  The yardages and Slope Ratings of the course in question are shown in Table 1 below.

Table 1
Yardages and Slope Ratings for Women

Slope Rating

As Table 1 shows, the Slope Rating decreases as yardage increases.  A player with a 14.0 Index would receive 19 strokes if she played the Green/Silver tees, but only 18 Strokes if she moved back to the Green tees.   An examination of the Slope Rating formula reveals why the decrease in Slope Rating is unlikely.

                Slope Rating = 4.24·((Y/120 + BOV +51.3) – ( Y/180 + SOV +40.1))
                                    = .0118·Y + 4.24·(BOV –SOV) +47.5
                             Y = Effective Course Yardage
                       BOV = Bogey Obstacle Value
                       SOV = Scratch Obstacle Value

The Slope Rating is an increasing function of yardage.  For each 100 yard increase in yardage, the Slope Rating—all things being equal—will increase by approximately 1.2 rating points.  For the 310 yard increase in length in the example, the Slope Rating would increase by 3.7 rating points.  The only way for the Slope Rating to increase with distance is for the SOV to increase more than the BOV.  It is difficult to conceive of a course design where this could happen.

Where an error may have occurred can be found by examining the SOVs and BOVs implicit in the new ratings.  Table 2 shows the BOV for the Green tees is actually lower than from the Green/Silver tees. This is highly unlikely.  The bogey player should have shorter approach shots from the Green/Silver tees which should reduce the BOV.  If the shorter tees bring more hazards in to play, the bogey player can just use less club so that her landing area is the same as from the longer tees (i.e., there is no change in obstacle values). 

Table 2
Obstacle Values


The case for an error in ratings is strong.  The OGA, however, did not see it that way. 
Jim Gibbons, Executive Director, of the OGA wrote the following:

We have received your review of the course ratings. Nancy Holmes has
started the process to double check our ratings, but initial review shows we
are correct.  The bogey rating for women from the Green/Silver tees of 110.6
slopes to a 150 based upon the course rating of 75.3 at 6249 yards.  This
relates to a 19 handicap to shoot(sic) a net 75.3 for women.
The Green tees at 6559 have a course rating of 77.6 with a bogey rating of
112.5 (because some obstacles are located with less impact).  This provides a
handicap of 18 to shoot (sic) the net score of 77.6.
The rating process has changed since the previous ratings were done. Green
speeds and rough heights may be different. A check of the BOV between the
Green/Silver and the Green (tees) will be made and if there is a change that will
be posted.  Realize that the BOV sometimes is lower from the longer tees, but
many organizations will adjust the numbers to avoid having to explain the
reason.  We choose to do the rating correctly. 

Mr. Gibbons does not really give a defense of the Slope Ratings. He is correct that the various course and bogey ratings yield the peculiar Slope Ratings. This math was never in question. The issue was whether the Obstacle Values were estimated correctly. He states this will be addressed in the future, but never did. Gibbons ends with the audacious claim that the BOV is frequently lower from the longer tees than reported because other associations fudge the numbers. I suspect he never filed a complaint to the USGA to that effect.

The course was re-rated in 2014. The new BOVs and Slope Ratings are shown in Table 3. Both the BOV and the Slope Rating increase with distance as expected. The difference in the Slope Ratings between the two sets of tees is 7 rating points (+2 to -5). Since the course was essentially unchanged between ratings, a difference of 7 rating points is too large to be attributed to random error. Given the more reasonable 2014 ratings, it is likely that the 2006 ratings were due to Committee error rather than course design.

Table 3
2014 BOVs and Slope Ratings

Slope Rating
While the 2014 ratings are more sensible, it is difficult to prove they are more accurate than the ratings of 2006. Rating Committees take measurements, assign number to subjective judgments, and plug those results into a model that has never been empirically verified. Ratings come out of the computer and are sent to the golf course. You can’t argue with a rating, but only the logic behind the rating. Since the OGA never presents that logic, there can be no debate.

Thursday, October 2, 2014

Why Does the USGA Treat Women Differently?

The USGA does not have an enviable record when it comes to its view of women:
  1. The USGA has a “separate but equal” handicap system for women that codifies them as the weaker sex. 
  2. While the United States Tennis Association provides equal prize money for men and women at its Opens, the USGA awards more than twice as much prize money to men than women at its Opens.
  3. No woman has ever served on the USGA’s Handicap Research Team.
  4. In recommended handicap allocations, the USGA seems to presume women and men are different psychologically (i.e., men are bigger risk takers and women are more conservative on the golf course). 
The first two actions of the USGA can at least be defended on physiological and economic grounds.  The third result could stem from no women wanting to work in area where the possibility of publishable research is nil.  It is the fourth action—the USGA’s perceived difference is the psychological make-up of men and women –that is the subject of this post.
In many competitions, the USGA recommends different handicap allowances for men and women.  For example, in four-ball stroke play men are allowed 90 percent of their handicap while women are allowed 95 percent of their handicap.  Why are women treated differently?  Much of the USGA’s research on multi-team events was done over 35 years ago and there appears to be no mention of any differences due to the gender of the player.[1]   It is likely the USGA had no empirical evidence for the women’s allocation, and the percentage was just a consensus guess by members of the Handicap Procedure Committee.   If women were studied, it is probable any difference in the estimated optimal allowance for men and women was not statistically significant.  Remember, all of the studies used to justify four-ball allowances were completed long before the introduction of the Slope System.  With this error and others, it is likely any difference as small as five percent was not significant.  Since the USGA does not release its research for peer review, the accuracy and validity of the USGA’s allowance may never be known.
The typical reason given for reducing handicaps in multi-ball events is that the higher handicap player has a larger standard deviation in his/her scores and hence an advantage.  Given that women get a smaller reduction in handicap, the USGA must believe women have as smaller standard deviation in their scoring.  Women must be steadier and/or less prone to risk taking as noted above.  In the appendix below, it is shown that the difference in standard deviations between teams is the same regardless of gender.  Therefore, it is difficult to defend the different allocations based on differences in standard deviations of scoring.[2]

While the allocations should be reviewed and revised, it is doubtful the USGA will take such action.  The allocations were never based on sound science, but rather on the internal politics at the USGA.  The allowances are considered “settled law” by the myriad of attorneys that guide the USGA.  To make a small step toward the equal treatment of women, however, the USGA could keep the hallowed men’s allowances and simply eliminate any allowance specific to women. 


The Slope Handicap System assumes that the standard deviation of a player’s scores increases linearly with handicap.  The standard deviation for each gender would be:
1)            σ (m,h) = σ(m,0)·(1 + a·h)
                                σ(m,h) = standard deviation of a male player with handicap h
                                σ(m,0) = standard deviation of a scratch male player
                                          h = handicap of the player
2)            σ (f,h) = σ(m,0)·(1 + b·h)
                                σ(f,h) = standard deviation of a female player with handicap h
                                σ(f,0) = standard deviation of a scratch female player
                                        h = handicap of the player
The USGA assumes that the line plotting average scores versus handicap would have a slope of 1.13.  The equation for males reflecting this assumption is:
3)            1.13 = (Average Score(h) –Average Score(0))/h
If a normal distribution of scores is assumed, then:
4)            Average Score(h) = ATBD(h) + .8 ·σ(m,h)
                                ATBD(h) = Average of Ten Best Differentials of a player with a h-handicap
Substituting eq. 4 into eq. 3:
5)            1.13 =(ATBD(h) + .8·σ(m,0)·(1 +a·h)  - (ATBD(0) + .8·σ(m,0))/h
6)            h = ATBD(h)·.96
Eq.  5 can be rewritten as:
7)            1.13 = (h/.96 +.8·σ(m,0)·(1 + a·h) – .8·σ(m,0))/h
                1.13 = 1.04 + .8·σ(m,0)·a                               
Since the same equality must hold for women, it follows that:
8)            σ(m,0)·a = σ(f,0)·b
Using eq. 8, the equations for the standard deviations can be rewritten as:
9)            σ(m,h) = σ(m,0) ·(1 +a·h)
10           σ(f,h) = σ(m,0)·(a/b) (1 +b·h) = σ(m,0)·(a/b +a·h)
For simplicity, assume we have a team where both players have a handicap of h1, and another team where both players have a handicap of h2.  The difference in the average standard deviation of the two teams is:
11)          Average Male Difference = σ(m,0)·(h1 - h2)
12)          Average Female Difference = σ(m,0)·(h1 - h2)
Therefore, the difference in average standard deviation between teams is the same regardless of gender (i.e., any advantage a team has is the same for both genders).  This finding makes it difficult to justify different handicap allocations for men and women in four-ball stroke play.


[1] Ewen, Gordan, What the Multi-ball Allowances Mean to You,, Far Hills NJ, 1978.  The USGA has not released the original research for peer review. 
[2] The USGA has the data to examine if there are differences in scoring patterns between men and women.  It has chosen not to do so.