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,” www.ongolfhandicaps.com, January 15, 2013.
[8] Appendix E has had many problems over the years as documented in “The Unmaking of the USGA’s Appendix E,” www.ongolfhandicaps.com, March 2, 2016. 
[9] Newport, John Paul, “Fighting Back Against Sandbaggers,” Wall Street Journal, July 2, 2011.

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