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Here’s a timely guest entry from Jeffrey Witmer (Oberlin College).
As the “Deflate Gate” saga was unfolding, Warren Sharp analyzed “touches per fumble” for NFL teams before and after 2006, when a rule was changed so that teams playing on the road could provide their own footballs (http://www.sharpfootballanalysis.com/blog/). Sharp noted that, for whatever reason, the Patriots went from being a typical team, as regards fumbling, to a team with a very low fumble rate. Rather than rely on the data the Sharp provides at his website, I choose to collect and analyze some data on my own. I took a random sample of 30 games played by New England and 30 other games. For each game, I recorded all rushing and passing plays (except for QB kneels), but excluded kicking plays (the NFL, rather than the teams, provides special footballs for those plays). (Data source: http://www.pro-football-reference.com/play-index/play_finder.cgi.) I also recorded the weather for the game. (Data source: http://www.wunderground.com/history/.) Once I had the data (in a file that I called AllBig, which can be downloaded from http://www.amherst.edu/~nhorton/AllBig.csv), I noted whether or not there was a fumble on each play, aided by the grep() command:
grep("Fumb", AllBig$Detail, ignore.case=TRUE)
I labeled each play as Late or not according to whether it happened after the rule change:
AllBig$Late <- ifelse(AllBig$Year > 2006, 1, 0)
Now for the analysis. My data set has 7558 plays including 145 fumbles (1.9%). I used the mosaic package and the tally() command to see how often teams other than the Patriots fumble:
require(mosaic)
tally(~Fumble+Late, data=filter(AllBig,Pats==0))
Late
Fumble 0 1
0 2588 2919
1 54 65
Then I asked for the data in proportion terms:
tally(Fumble~Late, data=filter(AllBig,Pats==0))
and got
Late
Fumble 0 1
0 0.9796 0.9782
1 0.0204 0.0218
For non-Pats there is a tiny increase in fumbles. This can be displayed graphically using a mosaiplot (though it’s not a particularly compelling figure). mosaicplot(Fumble~Late, data=filter(AllBig,Pats==0)) Repeating this for the Patriots shows a different picture:
tally(~Fumble+Late, data=filter(AllBig,Pats==1))
Late
Fumble 0 1
0 996 910
1 19 7
tally(Fumble~Late, data=filter(AllBig,Pats==1))
Late
Fumble 0 1
0 0.98128 0.99237
1 0.01872 0.00763
I fit a logistic regression model with the glm() command: glm(Fumble~Late*Pats, family=binomial, data=AllBig)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.8697 0.1375 -28.14 <2e-16 ***
Late 0.0650 0.1861 0.35 0.727
Pats -0.0897 0.2693 -0.33 0.739
Late:Pats -0.9733 0.4819 -2.02 0.043 *
I wanted to control for any weather effect, so I coded the weather as Bad if it was raining or snowing and good if not. This led to a model that includes BadWeather and Temperature – which turn out not to make much of a difference:
AllBig$BadWeather <- ifelse(AllBig$Weather %in% c("drizzle","rain","snow"), 1, 0)
glm(formula = Fumble ~ BadWeather + Temp + Late * Pats,
family = binomial, data = AllBig)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.23344 0.43164 -9.81 <2e-16 ***
BadWeather 0.33259 0.29483 1.13 0.26
Temp 0.00512 0.00612 0.84 0.40
Late 0.08871 0.18750 0.47 0.64
Pats -0.14183 0.27536 -0.52 0.61
Late:Pats -0.91062 0.48481 -1.88 0.06 .
Because there was suspicion that something changed starting in 2007 I added a three-way interaction:
glm(formula = Fumble ~ BadWeather + Temp + IsAway * Late * Pats,
family = binomial, data = AllBig)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.51110 0.47707 -9.46 <2e-16 ***
BadWeather 0.34207 0.30013 1.14 0.254
Temp 0.00831 0.00653 1.27 0.203
IsAway 0.14791 0.27549 0.54 0.591
Late 0.13111 0.26411 0.50 0.620
Pats -0.80019 0.54360 -1.47 0.141
IsAway:Late -0.07348 0.37463 -0.20 0.845
IsAway:Pats 0.94335 0.63180 1.49 0.135
Late:Pats 0.51536 0.71379 0.72 0.470
IsAway:Late:Pats -3.14345 1.29480 -2.43 0.015 *
There is some evidence here that the Patriots fumble less than the rest of the NFL and that things changed in 2007. The p-values above are based on asymptotic normality, but there is a cleaner and easier way to think about the Patriots’ fumble rate. I wrote a short simulation that mimics something I do in my statistics classes, where I use a physical deck of cards to show what each step in the R simulation is doing.
#Simulation of deflategate data null hypothesis
Late = rep(1,72) #creates 72 late fumbles
Early = rep(0,73) #creates 73 early fumbles
alldata = append(Late,Early) #puts the two groups together
table(alldata) #check to see that we have what we want
cards =1:length(alldata) # creates 145 cards, one "ID number" per fumble
FumbleLate = NULL # initializes a vector to hold the results
for (i in 1:10000){# starts a loop that will be executed 10,000 times
cardsgroup1 = sample(cards,119, replace=FALSE) # takes a sample of 119 cards
cardsgroup2 = cards[-cardsgroup1] # puts the remaining cards in group 2
NEPats = (alldata[cardsgroup2]) #reads the values of the cards in group 2
FumbleLate[i] = sum(NEPats) # counts NEPats late fumbles (the only stat we need)
}
table(FumbleLate) #look at the results
hist(FumbleLate, breaks=seq(2.5,23.5)) #graph the results
sum(FumbleLate <= 7)/10000 # How rare is 7 (or fewer)? Answer: around 0.0086
Additional note: kudos to Steve Taylor for the following graphical depiction of the interaction.
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