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# Validating Strokes-Gained Method for Measuring PGA Tour Player Success

Contributed by Ben Townson. He enrolled in the NYC Data Science Academy 12-week full time Data Science Bootcamp program taking place between July 5th to September 23rd, 2016. This post is based on their third project - Web Scraping, due on 6th week of the program.

Background In 2016, the PGA Tour, the leading professional golf tour in the world, began using an upgraded version of the Strokes-Gained methodology pioneered by Mark broadie as the primary statistic used to describe player success on tour.  The strokes-gained statistic is calculated by comparing the improvement in a player’s position for each shot, compared to a baseline average improvement of the average PGA tour player from the same position.  While a simplified version of the metric was used for several years, statistics are now being collected for 4 specific areas of the course: off the tee, approaching the green, around the green, and putting.  By keeping a cumulative score in each area for each golfer, one can measure success in various parts of the game, or by summing across statistics, one can expect to measure the success of a golfer in a specific round vs the average golfer.  The full details of the calculation can be found here: http://everyshotcounts.com/.  More color about the metrics and the adoption of them by the PGA tour can be found here: http://www.pgatour.com/news/2016/05/31/strokes-gained-defined.html. The resulting metrics provide a uniform way of explaining the success of a tour player vs the field despite the differences in the types of shots taken or the player’s position on the course.  Yet, for decades, the golf community has relied upon statistics separate from scoring to explain success in the sport.  For instance, rather than talking about how many strokes have been gained off the tee, the golf community relied upon descriptors such as driving distance and driving accuracy to measure performance in this area of the game..  These statistics can offer more description of the actual shots taken, while the strokes-gained metric captures how much each shot improved the golfers relative position to the field without describing “why” or “how” that shot was played. Because of this, as strokes-gained has been adopted and used increasingly as the metric of choice by the Tour, there has been backlash from the fanbase about the transition.  Many feel that the traditional statistics are just as sufficient as strokes-gained in describing a golfer’s success, and provide qualitatively a better description of a Tour players game.  So, do they have a point? Using data from www.pgatour.com/stats, we can test whether strokes-gained can better explain success than traditional statistics.  Using data from 2009 through 2015, we can try to build predictive models using traditional stats and these new stats and compare how well they perform at predicting a golfer’s success in 2016.  “Success” on Tour can be measured by the number of Fedex Cup Points (FCPs) accumulated during the year.  As we will be transforming the data to allow us to predict a normally distributed variable (the raw Fedex Cup Point distribution is not normally distributed), we will not be able to directly predict the number of FCPs accumulated, but will be able to predict the rank compared to the field and compare how well the models can predict a players standing. Code used to analyze the data in R can be found below.

The Models We built and optimized two multiple linear regression models in R, predicting Box-Cox transformed Fedex Cup Points per tournament.  The best-fit model using the old statistics called for inclusion of many traditional statistics: driving distance, driving accuracy, greens in regulation, sand saves, proximity to hole around the green, putts made distance, and putts per round.  However, when evaluating the model, there was evidence of multiple collinearity amongst several of the variables.  By dropping the driving accuracy and putts made distance, we were able to produce a well-fit model with an adjusted R-Squared of 0.4977.  Qualitatively, this model makes sense, as it uses data from the four major areas of the game, which correspond with the four areas of the game which are captured by the strokes-gained methodology: driving (off the tee), greens in regulation (approaching the green), proximity to hole around the green (around the green), and putts per round (putting). In the strokes gained model, we attempted to predict the same Box-Cox transformed Fedex Cup Points data, using the four strokes-gained metrics.  The best-fit model relies upon each metric with similar weighting and confidence for each, and upon inspection we see that there is no multiple collinearity.  The model produces an R-Squared of 0.7548. The evidence suggests that the strokes-gained model is a better predictor of success, as defined by relative Fedex Cup Points gained per round played, than the traditional model.  However, the ultimate test is in predictability of results in the current year, 2016.  Again using data from www.pgatour.com/stats, we are able to capture statistics for Tour players in 2016, and can predict their ranking in the average Fedex Cup Points gained per tournament.

The Conclusion We tested each of the predictions by comparing the model fit to the actual standings in 2016.  Applying the same Box-Cox transformation on the average Fedex Cup Points earned in 2016 provided a target for the model.  When comparing the errors to the actual results vs the fit from the models, we found that the mean error squared for the strokes gained model is 0.38 with a standard deviation of 0.94, while the standard model returns a mean error squared of 0.75 and standard deviation of 0.98.  It appears that the Strokes-Gained model is a better predictor of results in 2016 and we can conclude that the better fit model is in fact strokes gained.  The PGA Tour has wisely decided to use this metric to describe a golfer’s success.

The Code

```library(stats)library(VIM)library(mice)library(car)
library(Hmisc)
library(dplyr)
setwd('~/R')

#import and prepare the master data.
row.names(full_golf)=paste0(full_golf\$name,full_golf\$year)
full_golf=full_golf[,-which(names(full_golf) %in% c('X','name'))]
golf_fedex = full_golf[is.na(full_golf\$rk_fedex)==F,]
golf_fedex\$rk_fedex=golf_fedex\$rk_fedex/golf_fedex\$rk_fedex_variable2
summary(golf_fedex)
aggr(golf_fedex) #Lots of missing data, but is it relevant data?  Get a good subset of data first

########################################################################################################
################################Shots Gained v fedex Earned#############################################
########################################################################################################

#Prepare the dataset
sg_data=golf_fedex[golf_fedex\$year!=2016,c('rk_fedex','sg_ott','sg_aptg','sg_artg','sg_putt')]
sg_test=golf_fedex[golf_fedex\$year==2016,c('sg_ott','sg_aptg','sg_artg','sg_putt')]
sg_test_actual=golf_fedex[golf_fedex\$year==2016,c('rk_fedex','sg_ott','sg_aptg','sg_artg','sg_putt')]
sg_test_actual=sg_test_actual[which(complete.cases(sg_test_actual)),]
aggr(sg_data)
md.pattern(sg_data) #Missing Predictor Variables... not really imputable, so we won't test these, fair to drop
#because this data only recorded for top 200 players, the others will be OUTSIDE of the other data
sg_data_complete=sg_data[which(complete.cases(sg_data)),]
sg_test_complete=sg_test[which(complete.cases(sg_test)),]
aggr(sg_data_complete)
summary(sg_data_complete) #Data is prepped for testing

#Fit a multiple linear regression
sg.saturated=lm(rk_fedex~.,sg_data_complete)
summary(sg.saturated)
vif(sg.saturated)
avPlots(sg.saturated)
plot(sg.saturated)

#Better Box-Cox transform this
sg_sat.bc=boxCox(sg.saturated)
lambda = sg_sat.bc\$x[which(sg_sat.bc\$y == max(sg_sat.bc\$y))]
sg_data_complete\$rk_fedex.bc = (sg_data_complete\$rk_fedex^lambda - 1)/lambda
sg_sat_mod.bc=lm(rk_fedex.bc~sg_ott +sg_aptg +sg_artg +sg_putt,sg_data_complete)
summary(sg_sat_mod.bc)
plot(sg_sat_mod.bc)
vif(sg_sat_mod.bc)
avPlots(sg_sat_mod.bc)
#Assumptions look good
#Looks like multiple linear regression model may hold.
#But, let's see if we need all of these variables...

#prepare less fit models
model.empty=lm(rk_fedex.bc~1,sg_data_complete)
model.full=lm(rk_fedex.bc~.-rk_fedex,sg_data_complete)
scope=list(lower = formula(model.empty), upper = formula(model.full))

forwardAIC = step(model.empty, scope, direction = "forward", k = 2)
backwardAIC = step(model.full, scope, direction = "backward", k = 2)
bothAIC.empty = step(model.empty, scope, direction = "both", k = 2)
bothAIC.full = step(model.full, scope, direction = "both", k = 2)

#Stepwise regression using BIC as the criteria (the penalty k = log(n)).
forwardBIC = step(model.empty, scope, direction = "forward", k = log(196))
backwardBIC = step(model.full, scope, direction = "backward", k = log(196))
bothBIC.empty = step(model.empty, scope, direction = "both", k = log(196))
bothBIC.full = step(model.full, scope, direction = "both", k = log(196))

#Looks like combination of the four stats will work fine, let's predict 2016 output
fedex_predict_2016=predict(sg_sat_mod.bc,sg_test_complete,interval='confidence')
sg_test_actual\$rk_fedex.bc = (sg_test_actual\$rk_fedex^lambda - 1)/lambda
sg_test_actual=cbind(sg_test_actual,fedex_predict_2016)
sg_test_actual\$fit_error=sg_test_actual\$rk_fedex.bc-sg_test_actual\$fit
sg_test_actual\$error_sq=sg_test_actual\$fit_error**2
sg_total_error=sum(sg_test_actual\$error_sq)

#LET'S MAKE OUR OWN MODEL

########################################################################################################
##############################Other Variables v fedex Earned############################################
########################################################################################################

#prepare a data set
raw_data=golf_fedex[golf_fedex\$year!=2016,c('rk_fedex','drd','dra','gir','ssv','scr','pth','pthatg','pmd','ppr')]
raw_test=golf_fedex[golf_fedex\$year==2016,c('drd','dra','gir','ssv','scr','pth','pthatg','pmd','ppr')]
raw_test_actual=golf_fedex[golf_fedex\$year==2016,c('rk_fedex','drd','dra','gir','ssv','scr','pth','pthatg','pmd','ppr')]
aggr(raw_data) #Again, the lower-ranked golfers, so hard to impute, since they all fall outside of the data ranges.
raw_data=raw_data[which(complete.cases(raw_data)),]
raw_test_complete=raw_test[which(complete.cases(raw_test)),]
aggr(raw_data)

#fit multiple linear model on all data
td_mod = lm(rk_fedex~.,raw_data)
summary(td_mod)
td_mod_summary = summary(td_mod)
avPlots(td_mod)
plot(td_mod)  #Definitely some relationship, but it's not really "linear", maybe we can make it so with box-cox transform
#First, let's build a model with reduced variable set and see how that looks.

##########################MAKE MY OWN REDUCED MODEL################################
td_mod_red = lm(rk_fedex~drd+dra+gir+ssv+pth+pthatg+pmd+ppr,raw_data)
summary(td_mod_red)
plot(td_mod_red)
vif(td_mod_red)

#Box Cox transform
td_mod.bc=boxCox(td_mod)
lambda = td_mod.bc\$x[which(td_mod.bc\$y == max(td_mod.bc\$y))]
raw_data\$rk_fedex.bc = (raw_data\$rk_fedex^lambda - 1)/lambda
td_mod.bc=lm(rk_fedex.bc~.-rk_fedex,raw_data)
summary(td_mod.bc)
plot(td_mod.bc) #Looks like a reasonable model, let's try a reduced model based on significant variables

#Test Box Cox transformed dependent variable on the reduced variable set
td_mod_red.bc=lm(rk_fedex.bc~drd+dra+gir+ssv+pth+pthatg+pmd+ppr,raw_data)
summary(td_mod_red.bc)
plot(td_mod_red.bc)
vif(td_mod_red.bc)
avPlots(td_mod_red.bc)
#Looks like a decent model, but let's see if there's a "best model"

#Check full vs reduced
AIC(td_mod_red.bc,td_mod.bc)
BIC(td_mod_red.bc,td_mod.bc)
#AIC and BIC don't show a clear advantage

#Alternatively, let's do a stepwise regression and see what set of variables is identified
model.empty=lm(rk_fedex.bc~1,raw_data)
model.full=lm(rk_fedex.bc~.-rk_fedex,raw_data)
scope=list(lower = formula(model.empty), upper = formula(model.full))

forwardAIC = step(model.empty, scope, direction = "forward", k = 2)
backwardAIC = step(model.full, scope, direction = "backward", k = 2)
bothAIC.empty = step(model.empty, scope, direction = "both", k = 2)
bothAIC.full = step(model.full, scope, direction = "both", k = 2)

#Stepwise regression using BIC as the criteria (the penalty k = log(n)).
forwardBIC = step(model.empty, scope, direction = "forward", k = log(196))
backwardBIC = step(model.full, scope, direction = "backward", k = log(196))
bothBIC.empty = step(model.empty, scope, direction = "both", k = log(196))
bothBIC.full = step(model.full, scope, direction = "both", k = log(196))

#BIC identifies a subset, similar to the SG model statistics but includes some overlapping areas, dra, pmd
#after reviewing VIF, we can safely remove dra and pmd on concerns about collinearity.  Resulting variable set follows:
td_mod_red2.bc=lm(rk_fedex.bc~drd+gir+pthatg+ppr,raw_data)
summary(td_mod_red2.bc)

#Looks like resonably good model, but check assumptions
plot(td_mod_red2.bc)
vif(td_mod_red2.bc)
avPlots(td_mod_red2.bc)

###Which Model is best?
AIC(td_mod_red2.bc,sg_sat_mod.bc)
BIC(td_mod_red2.bc,sg_sat_mod.bc)

#Test to see how well it predicts
raw_mod.predicrat=data.frame(predict(td_mod_red2.bc,raw_test_complete,interval='confidence'))
summary(raw_mod.predict)
sg_test_actual\$raw_mod_predict=raw_mod.predict\$fit
sg_test_actual\$raw_fit_error=sg_test_actual\$rk_fedex.bc-sg_test_actual\$raw_mod_predict
sg_test_actual\$raw_error_sq = sg_test_actual\$raw_fit_error**2

#Compare the two models
mean(sg_test_actual\$raw_error_sq)
sd(sg_test_actual\$raw_error_sq)
mean(sg_test_actual\$error_sq)
sd(sg_test_actual\$error_sq)The original article can be found here.```

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