1 Description of the Data Set

This is a data set of the results of all NFL regular season field goal attempts for the 2008 season. There are 1039 observations with 23 variables The variables are:

  1. GameDate

  2. AwayTeam

  3. HomeTeam

  4. qtr (quarter, 5=overtime)

  5. min (minutes remaining)

  6. sec (seconds remaining, added to minutes)

  7. kickteam (team kicking field goal)

  8. def (defending team)

  9. down

  10. togo (yards to go for 1st down)

  11. kicker (ID #)

  12. ydline (yardline of kicking team)

  13. name (kicker’s name)

  14. distance (yards)

  15. homekick (1 if kicker at Home, 0 if Away)

  16. kickdiff (kicking team lead +, or deficit -, prior to kick)

  17. timerem (Time remaining in seconds, negative = overtime)

  18. offscore (kicking team’s score prior to kick)

  19. defscore (defense team’s score prior to kick)

  20. season (2008)

  21. GOOD (1 is Success, 0 is Miss)

  22. Missed (Missed, not blocked = -1, 0 ow)

  23. Blocked (1 if Blocked, 0 ow)

The variables Missed, Blocked, season, and GameDate are removed from the data set because they do not affect the success of a field goal. The Good variable which measures if a field goal is good is the response varaible.

1.1 Question

Using the data, can we make a linear model that correctly predicts if a field goal is a success/good?

1.2 Exploratory Analysis

First, the following pairwise scatter plots are made to inspect the potential issues with predictor variables and colinearity.

## Warning: package 'psych' was built under R version 4.2.2

The variables that were removed due to colinearity or violating the normality assumption were: yardline, defscore, offscore, kickteam, quater, timerem, homekick and down.

1.3 Building the Logistic Regression Models

Based on the above exploratory analysis, we first build the full model and the smallest model and then the final model using backward selection. The goodness of fit of these models will be compared to chose the best final model.

## Warning: glm.fit: algorithm did not converge
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
Summary of inferential statistics of the full model
Estimate Std. Error z value Pr(>|z|)
(Intercept) 5.535457e+00 2.275154e+00 2.433004e+00 0.0149741
AwayTeamATL -1.556454e+00 2.387268e+00 -6.519810e-01 0.5144134
AwayTeamBAL -3.971946e+00 5.036447e+00 -7.886405e-01 0.4303222
AwayTeamBUF -7.125008e+00 7.561394e+00 -9.422876e-01 0.3460454
AwayTeamCAR -9.890033e+00 9.549332e+00 -1.035678e+00 0.3003524
AwayTeamCHI -8.926903e+00 1.159639e+01 -7.698005e-01 0.4414182
AwayTeamCIN -1.363555e+01 1.334355e+01 -1.021883e+00 0.3068361
AwayTeamCLE -1.485387e+01 1.520253e+01 -9.770658e-01 0.3285366
AwayTeamDAL -1.731829e+01 1.710124e+01 -1.012692e+00 0.3112073
AwayTeamDEN -1.986115e+01 1.898034e+01 -1.046406e+00 0.2953735
AwayTeamDET -7.840539e+00 1.359663e+02 -5.766530e-02 0.9540152
AwayTeamGB -2.242242e+01 2.275244e+01 -9.854950e-01 0.3243810
AwayTeamHOU -2.341128e+01 2.467544e+01 -9.487687e-01 0.3427383
AwayTeamIND -2.702596e+01 2.656083e+01 -1.017512e+00 0.3089099
AwayTeamJAC -2.866608e+01 2.843761e+01 -1.008034e+00 0.3134381
AwayTeamKC -3.231267e+01 3.031489e+01 -1.065901e+00 0.2864684
AwayTeamMIA -3.318976e+01 3.412020e+01 -9.727303e-01 0.3306873
AwayTeamMIN -3.459917e+01 3.597854e+01 -9.616612e-01 0.3362198
AwayTeamNE -3.910455e+01 3.787649e+01 -1.032423e+00 0.3018742
AwayTeamNO -4.409573e+01 4.170996e+01 -1.057199e+00 0.2904208
AwayTeamNYG 1.700967e+13 3.445357e+14 4.936980e-02 0.9606246
AwayTeamNYJ -4.446868e+14 4.341656e+14 -1.024233e+00 0.3057252
AwayTeamOAK -5.163533e+01 5.109699e+01 -1.010536e+00 0.3122388
AwayTeamPHI -5.236801e+01 5.298997e+01 -9.882627e-01 0.3230240
AwayTeamPIT -5.600648e+01 5.677777e+01 -9.864158e-01 0.3239291
AwayTeamSD -5.858989e+01 5.867164e+01 -9.986067e-01 0.3179853
AwayTeamSEA -6.021790e+01 6.056500e+01 -9.942690e-01 0.3200919
AwayTeamSF -6.241997e+01 6.244749e+01 -9.995592e-01 0.3175239
AwayTeamSTL -6.260563e+01 6.430006e+01 -9.736481e-01 0.3302313
AwayTeamTB -6.611665e+01 6.623164e+01 -9.982639e-01 0.3181514
AwayTeamTEN -6.772260e+01 6.814420e+01 -9.938132e-01 0.3203138
AwayTeamWAS -5.607026e+01 5.485902e+01 -1.022079e+00 0.3067434
HomeTeamATL -8.457193e-01 2.320184e+00 -3.645052e-01 0.7154808
HomeTeamBAL -5.237059e+00 5.185105e+00 -1.010020e+00 0.3124857
HomeTeamBUF -8.549868e+00 7.711028e+00 -1.108784e+00 0.2675232
HomeTeamCAR -8.321023e+00 9.554322e+00 -8.709172e-01 0.3837994
HomeTeamCHI -9.240068e+00 1.147406e+01 -8.053008e-01 0.4206461
HomeTeamCIN -1.269760e+01 1.330980e+01 -9.540041e-01 0.3400816
HomeTeamCLE -1.548193e+01 1.521054e+01 -1.017842e+00 0.3087528
HomeTeamDAL -1.480517e+01 1.705416e+01 -8.681268e-01 0.3853249
HomeTeamDEN -1.963814e+01 1.902365e+01 -1.032301e+00 0.3019310
HomeTeamDET -3.856699e+00 1.359755e+02 -2.836320e-02 0.9773725
HomeTeamGB -2.087110e+01 2.272722e+01 -9.183307e-01 0.3584457
HomeTeamHOU -2.340382e+01 2.463167e+01 -9.501519e-01 0.3420351
HomeTeamIND -2.374972e+01 2.656899e+01 -8.938885e-01 0.3713815
HomeTeamJAC -2.801876e+01 2.846929e+01 -9.841750e-01 0.3250295
HomeTeamKC -3.087148e+01 3.034373e+01 -1.017393e+00 0.3089667
HomeTeamMIA -3.394577e+01 3.417018e+01 -9.934324e-01 0.3204993
HomeTeamMIN -3.327315e+01 3.598718e+01 -9.245835e-01 0.3551826
HomeTeamNE -3.760994e+01 3.791022e+01 -9.920792e-01 0.3211589
HomeTeamNO -4.219991e+01 4.171806e+01 -1.011550e+00 0.3117532
HomeTeamNYG 1.700967e+13 3.445357e+14 4.936980e-02 0.9606246
HomeTeamNYJ -4.446868e+14 4.341656e+14 -1.024233e+00 0.3057252
HomeTeamOAK -5.023005e+01 5.114142e+01 -9.821792e-01 0.3260115
HomeTeamPHI -5.038272e+01 5.293466e+01 -9.517907e-01 0.3412031
HomeTeamPIT -5.586343e+01 5.679272e+01 -9.836372e-01 0.3252939
HomeTeamSD -5.625791e+01 5.869768e+01 -9.584351e-01 0.3378434
HomeTeamSEA -5.876997e+01 6.054782e+01 -9.706372e-01 0.3317290
HomeTeamSF -6.102912e+01 6.241582e+01 -9.777829e-01 0.3281817
HomeTeamSTL -6.291857e+01 6.411716e+01 -9.813062e-01 0.3264418
HomeTeamTB -6.481341e+01 6.624534e+01 -9.783844e-01 0.3278842
HomeTeamTEN -6.644168e+01 6.815422e+01 -9.748726e-01 0.3296235
HomeTeamWAS -5.492697e+01 5.485732e+01 -1.001270e+00 0.3166964
min 1.169800e-03 7.131400e-03 1.640312e-01 0.8697066
sec 8.748500e-03 6.537400e-03 1.338232e+00 0.1808210
defATL 2.845128e-01 2.403261e+00 1.183862e-01 0.9057617
defBAL 6.111634e+00 5.209426e+00 1.173188e+00 0.2407205
defBUF 8.766517e+00 7.794142e+00 1.124757e+00 0.2606920
defCAR 9.058731e+00 9.574430e+00 9.461379e-01 0.3440783
defCHI 8.444654e+00 1.151458e+01 7.333881e-01 0.4633217
defCIN 1.482772e+01 1.338027e+01 1.108179e+00 0.2677847
defCLE 1.481924e+01 1.524541e+01 9.720463e-01 0.3310275
defDAL 1.653002e+01 1.704775e+01 9.696308e-01 0.3322306
defDEN 2.101824e+01 1.903944e+01 1.103932e+00 0.2696227
defDET 6.827769e+00 1.359700e+02 5.021530e-02 0.9599509
defGB 2.128853e+01 2.278216e+01 9.344387e-01 0.3500776
defHOU 2.381394e+01 2.475499e+01 9.619858e-01 0.3360567
defIND 2.587322e+01 2.659784e+01 9.727562e-01 0.3306745
defJAC 2.916836e+01 2.849628e+01 1.023585e+00 0.3060314
defKC 3.288330e+01 3.036123e+01 1.083069e+00 0.2787780
defMIA 3.500721e+01 3.415395e+01 1.024983e+00 0.3053713
defMIN 3.379215e+01 3.600769e+01 9.384705e-01 0.3480027
defNE 3.728241e+01 3.791662e+01 9.832734e-01 0.3254729
defNO 4.361795e+01 4.172730e+01 1.045309e+00 0.2958800
defNYG -1.700967e+13 3.445357e+14 -4.936980e-02 0.9606246
defNYJ 4.446868e+14 4.341656e+14 1.024233e+00 0.3057252
defOAK 5.142767e+01 5.116452e+01 1.005143e+00 0.3148279
defPHI 5.255102e+01 5.291160e+01 9.931853e-01 0.3206197
defPIT 5.624705e+01 5.681264e+01 9.900447e-01 0.3221523
defSD 5.856069e+01 5.873291e+01 9.970677e-01 0.3187317
defSEA 6.084801e+01 6.055589e+01 1.004824e+00 0.3149817
defSF 6.211009e+01 6.242307e+01 9.949861e-01 0.3197430
defSTL 6.542996e+01 6.423115e+01 1.018664e+00 0.3083625
defTB 6.673347e+01 6.628931e+01 1.006700e+00 0.3140788
defTEN 6.549158e+01 6.817038e+01 9.607044e-01 0.3367008
defWAS 5.572992e+01 5.485211e+01 1.016003e+00 0.3096279
togo -1.156760e-02 2.870510e-02 -4.029808e-01 0.6869623
kicker 1.861579e+00 1.891689e+00 9.840831e-01 0.3250747
nameC.Barth -1.816808e+00 2.380101e+00 -7.633322e-01 0.4452653
nameD.Rayner 2.060280e+01 4.359682e+05 4.730000e-05 0.9999623
nameG.Hartley 4.503600e+15 1.862078e+07 2.418588e+08 0.0000000
nameJ.Carney -1.700967e+13 3.445357e+14 -4.936980e-02 0.9606246
nameJ.Feely 4.446868e+14 4.341656e+14 1.024233e+00 0.3057252
nameL.Tynes -1.700967e+13 3.445357e+14 -4.936980e-02 0.9606246
nameM.Gramatica 3.637611e+00 2.628972e+00 1.383663e+00 0.1664617
nameM.Nugent 4.446868e+14 4.341656e+14 1.024233e+00 0.3057252
nameM.Stover 8.924217e-01 1.922320e+00 4.642421e-01 0.6424743
distance -1.445343e-01 1.593330e-02 -9.071209e+00 0.0000000
kickdiff -6.291900e-03 1.409080e-02 -4.465250e-01 0.6552181
Summary of inferential statistics of the reduced model
Estimate Std. Error z value Pr(>|z|)
(Intercept) 6.4956640 0.5703859 11.388191 0.0000000
distance -0.1192415 0.0123408 -9.662395 0.0000000
sec 0.0080947 0.0054667 1.480729 0.1386788
Summary of inferential statistics of the final model
Estimate Std. Error z value Pr(>|z|)
(Intercept) 6.4956640 0.5703859 11.388191 0.0000000
distance -0.1192415 0.0123408 -9.662395 0.0000000
sec 0.0080947 0.0054667 1.480729 0.1386788
Comparison of global goodness-of-fit statistics
Deviance.residual Null.Deviance.Residual AIC
full.model 557.9044 809.6515 775.9044
reduced.model 684.7092 817.7227 690.7092
final.model 684.7092 817.7227 690.7092

Both the reduced model and the final model have the same variables in them so they both have the same AIC, deviance residual, and null deviance residual. Also, both of these models have the best AIC value so I will chose the final model as my model.

1.4 Final Model

In the exploratory analysis, only the variables distance and second are linearly correlated. After automatic variable selection, all the other variables were dropped out of the model besides distance and second so we chose the final model as our model.

Using the information below the final model is \(\hat{y} = 6.496 - 0.119*distance + 0.00809*second\). We interpret this as the distance increases by 1 yard, having a good field goal decreases by 0.119. And as the second increases by 1, having a good field goal increases by 0.00809.

Summary Stats with Odds Ratios
Estimate Std. Error z value Pr(>|z|) odds.ratio
(Intercept) 6.4956640 0.5703859 11.388191 0.0000000 662.2638362
distance -0.1192415 0.0123408 -9.662395 0.0000000 0.8875935
sec 0.0080947 0.0054667 1.480729 0.1386788 1.0081276

Because the odds ratio for distance in less than 1, so as the distance increases, the odds of having a successful field goal decrease. Because the odds ratio for second is almost 1, it means that it doesn’t affect the success of a field goal.

1.5 Summary and Conclusion

The case study focused on the association analysis between a set of potential factors that might influence a field goal’s success. The initial data set has 18 numerical and 5 categorical variables.

After exploratory analysis, some highly correlated values were removed from the model.

After automatic variable selection, the final model was obtained with 2 factors: distance in yards and seconds. As this distance increases, the success of a field goal decreases and as seconds increase the success of a field goal increases.