Monday, March 25, 2013

The USGA Bogeys Its Explanation of the Slope System

Previous posts (e.g., see The Mathematical Underpinnings of the "New" Slope Rating, February 4, 2013) have documented the confusion at the USGA about the Slope Rating.  This post takes another whack in the spirit of flogging a dead horse.  Let’s start with documenting the USGA’s arguments as put forth by Dean Knuth, former USGA Director of Handicapping:

The USGA coined the term which numerically describes the difference in course rating difficulty between the bogey and scratch players as the “Slope Rating.”…Slope Rating is the slope of the regression line of total score versus USGA Handicap for a particular golf course.  The Y-intercept is the USGA Course Rating which is the better half of score average of scratch golfers… The slope of the scores line of an average course has been observed to be 1.13 and the USGA Slope Rating is referenced as 113 to deal with whole numbers.  A scratch golfer will score proportionally fewer strokes than a 10, 20, and 30 handicapper.  The slope of the line through their score indicates the difficulty of the course, and the slope is steeper for more difficult courses and less steep for a less difficult course.[1] (Note: USGA Handicap refers to a player’s handicap before the introduction of the Slope System).

A minor error is that Knuth uses the regression line of “total” score versus USGA Handicap.  He probably meant to use “average” score as shown in the work of Stroud and Riccio.[2]  (The term “total” score has no obvious meaning, so “average” score is substituted for the purposes of this post.)  The larger error is the assertion that the slope of the regression line of average score versus USGA Handicap will be the Slope Rating.  This is not the case.  The slope of this regression line at any and every course should be 1.13 if Stroud and Riccio are correct.

Why isn’t the slope of the regression line steeper at more difficult courses as Knuth maintains?  The answer is straightforward.  If a player only plays his home course, his handicap is independent of the Slope Rating.  The relationship between average score and USGA Handicap is only a function of the procedures for reducing an average score to produce a handicap (i.e., taking only the ten best scores out of twenty, equitable stroke control, and the 96 percent factor).

If Knuth was correct, a Slope Rating would have an absolute value.  Raters could simply use the regression line relating average score and handicap in order to estimate the Slope Rating.  But the Slope Rating only has relative value.  The Slope Rating is dependent on the Slope Rating of the reference course.  What Knuth should have done is taken players from the reference course and had them play the course in question.  He should have then plotted the player’s average score versus his handicap on the reference course.  The Slope Rating would then be the slope of the regression line relating average score versus the handicap of the player from the reference course.

Knuth compounds his error by arguing the Y-intercept of the regression line is the Course Rating.  The Y-intercept, however, is the “average score” of a player with a scratch handicap.  The Course Rating however is the average of a player’s ten best scores out of the last twenty.  The Y-intercept is clearly not an unbiased estimate of the Course Rating.

Several conclusions can be drawn from this critique of USGA research:

1. The research papers appear to have been dashed off in order to qualify for a conference at St. Andrews in July of 1990 (e.g., greek symbols were entered by hand).  I suspect the authors were more concerned with locking down their tee times on the Old Course than quality control.  Why else would members of the USGA Research Team (Knuth, Riccio, and Stroud) arrive at such disparate reasons for the ubiquitous number 113 that pervades the USGA Handicap System?

2. The Conference editors (Cochran and Farrally) should have been responsible for the quality of the papers presented.  They overlooked this duty, probably for monetary reasons.  I suspect the USGA, a major financial sponsor of the conference, could have submitted the menu from Rusack’s Hotel and it would have been accepted.

Well that’s enough flogging for now. The next post will examine the difficulty in empirically verifying Course and Slope Ratings.

[1] Knuth, D, “A two parameter golf course rating system,” Science and Golf, edited by A. J. Cochran and M. R. Farrally, E & FN Spon, 1990, pp. 142-143.
[2] Stroud, R.C. and Riccio, L.J., “Mathematical underpinning of the slope handicap system,” Science and Golf,  edited by A. J. Cochran and M. R. Farrally, E & FN Spon, 1990,  pp. 135-140.