# The Power of the (Home) Crowd

There is a lot of ink that has been used (or wasted if you prefer) on the home field advantage in sports. Many reasons have been put forward to explain this observation.  For example, Scorecasting argues that the primary benefit for the home team is referee favoritism.  Regardless of various reports that this advantage might be declining, these have been disputed for similar reasons that scientists have judged medical research studies for not being reproducable.  In fact even by simply computing the fraction of games that home team have won during the past decade in a variety of sports (without considering factors such as team strength etc.) we can see that the home court bias exists!

NylonCalculus.com presented an interesting analysis that concludes that NBA players seem to get a slight edge in their shooting from playing at home.  In particular, they get an approximately 4% improvement (after controlling for various fixed effects and other covariates). This discussion inspired me to examine the performance of quarterbacks in the same context, i.e., home and road games. The quarterback might be the single most important position in a team sport (maybe together with the golie in hockey); ask the Steelers’ fans how they feel about having to see Big Ben sitting out for a few weeks.

How does one goes about evaluating the QB performance though? Even more ink might have been used for this topic, but for my analysis I will just use the passer rating, not because it is better than other alternatives, but because I had easy access to every QBs game passer rating for the past 7 NFL seasons.  Passer rating is a single number that is calculated through various different performance dimensions/statistics of the QB (all passing related, which makes for the main criticism with respect to the usefuliness of this metric).

Using the QB ratings from the past 7 NFL seasons, we statistically compare (through the non-parametric U-test)  the median home and road passer rating of the QBs.  Our results provide a difference between the medians of home and road passer ratings of 3.4% (p-value < 0.05).  These inital results are not on paired samples, but rather capture the overall performance of home and road QBs. There seem to be some  difference in the performance of the home and road QBs.  However, the significant result might be due to the large sample size that is able to pick up even small differences – which in reality do not have any practical meaning.  We have performed the same test with random samples of smaller size (in particular 50 data points for the home QB performances and 50 data points for the road QB performances) and following are the results.  We see that the p-values for these smaller samples are much higher and it appears that the difference picked by the original test was mainly due to the large sample size.

We then focus on specific QBs (this analysis also should not suffer from the problem of a big dataset).  We only use the QBs with at least 15 games.  The figure below presents the difference between the median home rating of the QB and his median road rating.  Hence, a positive value corresponds to a QB that performs better at home.  Furthermore, each bar is colored based on the p-value of the corresponding test.

As we can see, the vast majority of the QBs exhibit similar performance both at home and on the road (p-values > 0.05) However, there are only 5 QBs that exhibit a statistically significant (level 0.05) difference in their passer rating between their home and road appearances. These QBs are:

• Joe Flacco
• Drew Brees
• Matt Ryan
• Aaron Rodgers
• Ryan Fitzpatrick

What is going on here? These are all top-notch QBs! Do they bend under hostile crowd? There are many things that might be responsible for these results that are not necessarily related with the QBs abilities. Furthermore some intricacies of NFL schedule might need to be consider. In particular, contrary to other sports leagues where every team plays against every other team both home and on the road, in NFL teams have only one matchup with 10 teams either home or away (that is only one matchup). They only face the rest of the teams of their division twice a year – one home and one on the road. This means that there is a probability that a QB is facing easier opponents either at home or away and hence there might be a  difference in his ratings. The above analysis does not account for that.

Focusing on the above results for the individual QBs we need to first understand and accept that they  are based on the “p-value rule”.  Rejecting the null hypothesis does not necessarily mean that the alternative is true! There can be very well be false positives! In fact, the larger the dataset the higher the probability that you will identify flukes and label them as significant results (just as mentioned above).  In our case the dataset consists of 39 QBs, so finding 5 false positives might be too much but still a possibility.  Furthermore, for some of the QBs we might have a false negative results due to underpowered statistical test (many times a result of small sample size).  In any case, there is some evidence that some top-QBs might be putting up better performance in front of their home crowd.

Another reason for this observation might be that these specific QBs do not have to put huge passer ratings during their road games for their team to win. As we mentioned the QB is the most important position in American Football and in fact, passer rating is very highly correlated with winning a game.  In particular, the correlation between the final point differential of the game and the differential of the two opponents QB ratings is 0.75!  Furthermore, after fitting a simple linear regression between these two variables, the passer rating differential can explaine about 60% of the variance at the point differential between the two teams.

Based on that we can explore for each QB his rating differential with the opponent QB for both his home and road games. We could have used the winning percentage but American football is a team sport and even though QBs are important, so are wide receivers, running backs, tight ends and the offensive line (not to mention defense). The results are presented in the following figure and we can see that right now Drew Brees and Ryan Fitzpatrick  do not exhibit a difference in their home and road performances.  Flacco, Ryan and Rodgers still appear to perform better at home than on the road but again false positives are our “enemy” – our friend if you are trying to get them to sign a smaller contract

Overall there appears to be little evidence (if at all) that QBs perform better at home as compared to road games. Even if they do perform differently the difference in their median passer rating is overall less than 3.5% and can barely be deemed important from a practical point of view.