`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 65Then 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 0.0218For 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 0.00763I 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 0.0086Additional note: kudos to Steve Taylor for the following graphical depiction of the interaction.

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