Photo: New England Patriots’ QB Tom Brady

Photo: New England Patriots’ QB Tom Brady

1 Introduction

Tom Brady and the Patriots famously keep the rest of the league on their toes by varying the Patriots’ offense, to great success. Sometimes Brady and the Pats charge down the field in series of short and lethal slants; and on other occasions, thanks to the skill of receivers like Julian Edelman and Rob Gronkowski, Brady and Co. give us some beautiful deep passes that will be played on highlight reels for years to come. For our research project, we seek to discover if one kind of offense has been more successful than the other; are Brady’s average yards related to the number of touchdowns he will throw in a given game?

In the 2014 season, where our data set begins, I felt as though the Patriots’ short passing game was on full display. As I wrote in an essay for my freshman seminar, “Brady’s starring role in another kind of American Sniper film was particularly prominent in the 2014 season. Brady would lead slow, deadly marches down the field with series of short passes. The most beautiful pass of the season, arguably, was a three-yard touchdown pass to Julian Edelman; clean, simple, and good enough to clinch Super Bowl victory.”

()

Every game will have a mix of long and short passes. Average yards may not be a perfect measure of the prevalence of long or short passes in a given game, but it was the best available statistic recorded uniformly for every game. Brady has been successful at both, but has had a definite tendency toward short passes. As Adam Kilgore wrote in The Washington Post, “Brady burnished his legend with short, precise passes to quicksilver receivers darting over the middle of the field. He became a master of manupulating defenses and spewing quick throws.” Kilgore also noted that for years, “Compared to his peers, Brady relied heavily on short passes.” In his article, Kilgore notes that Brady is “a quarterback amid personal evolution…this year [the 2017 season], Brady has turned gunslinger” and (successfully) thrown more deep passes.

In 2014, the season that culminated in that three-yard pass to Edelman, Brady’s mean average yards in the regular season were 7.06 yards, 6.56 in the playoffs. In the 2017 season, which culminated in the Patriots’ Super Bowl loss to the Eagles despite Tom’s record-setting 505 yard performance, his mean average yards were 7.88 in the regular season and 8.14 in the playoffs (ESPN). Despite the increase of .82 yards in the regular season and 1.58 in the playoffs, the seasons almost ended the exact same way. Since the 2014 and 2017 Patriots defenses are hardly comparable, it is fair to say that Brady has had similar levels of success with both styles of play. All in favor of getting Brady a bumper sticker that reads, “Defense loses championships,” please say aye.

We selected touchdowns as our outcome variable thinking that’s how the Patriots measure the success of a drive. Since there are three phases of the game, it would be difficult to assess Brady’s performance on winning and losing (even if that’s how he might see it). For our categorical predictor variable, we chose to split the games by the playoffs and the regular season. There are fewer data points for the playoff games than there are for the regular season. However, in the four seasons included in our data, Brady and the Patriots played in every single playoff game except one (Super Bowl 50–the Broncos, who came out victorious over the Carolina Panthers, defeated the Patriots in the AFC Championship in Denver).

This data and analysis does have some limitations. Of course, it is almost impossible to isolate the performance of one player alone–as Gisele infamously said, Tom “can’t throw the ball and catch the ball at the same time.” Football is a team game, and the receiving core around Brady has a lot to do with the style of the offense. Kilgore notes that Julian Edelman is “an option route artist in the NFL” and “Brady’s favorite target on quick passes.” Edelman played in every season of our data set save the most recent one (2017). Since our data set begins with the 2014 season, many seasons of Brady’s career are excluded–including the ones when Randy Moss played for the Patriots. If you’re wondering why I might think Randy Moss’s play might affect Brady’s average yards, watch this…

Overall, we hope to assess the importance of Brady’s average yards in the Patriots’ most recent run of success over the last four seasons.

2 Exploratory data analysis

First six rows of our data set:
Date Opponent home_away win_loss score playoff_reg total_yards avg_yards comp_per td sk
2018-02-05 PHI N/A L 33-41 Playoff 505 10.52 0.583 3 1
2018-01-21 JAX Home W 24-20 Playoff 290 7.63 0.684 2 3
2018-01-13 TEN Home W 35-14 Playoff 337 6.36 0.660 3 0
2017-12-31 NYJ Home W 26-6 Regular 190 5.14 0.486 2 2
2017-12-24 BUF Home W 37-16 Regular 224 8.00 0.750 2 2
2017-12-17 PIT Away W 27-24 Regular 298 8.51 0.629 1 2

Our dataset was made up of 71 rows, with each row representing a single game, either regular or playoff, where Brady had a certain number of average passing yards and touchdowns. Using this data we constructed the above visualizations that show the distribution of Brady’s average passing yards per game, the distribution of the number of touchdowns scored per game, and a scatter plot of touchdowns plotted against average yards, separated by playoff versus regular season game, where each point represents a game. Based on these visualizations we can make some preliminary observations. Most significantly, our scatter plot makes it appear that the slopes aren’t parallel, which suggests that there is some interaction between the categorical variable (playoff versus regular game) and the numerical variable (average yards). The visualization suggests that there is a stronger positive relationship between average yards and touchdowns scored in the regular season than there is in the playoffs. It is also important to note that both the touchdown and the average yards data appear to be approximately normally distributed (and hence symmetric).

3 Multiple regression

We seek to examine the relationship between the number of touchdowns scored by Tom Brady in a single game (the outcome variable \(td\)), Tom Brady’s average yards per game (the numerical explanatory variable \({avg\_yards}\)), and whether the game was a playoff or regular season game (the categorical explanatory variable \({playoff\_reg}\)).

We initially predicted that there was an associated effect of average yards on number of touchdowns depending on whether the game is a regular game or a playoff game, as shown by the scatter plot in our in our visual exploratory data analysis. However, when we fit the interaction model, the resulting regression table gave confidence intervals that included zero for our interaction term. This implies that the extra complexity of an interaction model is not beneficial for our analysis. Thus, we reverted to a parallel slopes model which is displayed in the table below.

term estimate std_error statistic p_value conf_low conf_high
intercept -0.614 0.656 -0.936 0.353 -1.924 0.696
avg_yards 0.356 0.083 4.309 0.000 0.191 0.521
playoff_regPlayoff 0.510 0.354 1.441 0.154 -0.196 1.217

3.1 Statistical interpretation

Based on the output of our table, we can use the following equation to describe our model:

\(\widehat{td} = -0.614 + 0.356*{avg\_yards} + 0.510*1[is\ Playoff]\)

To break this down, there are two possible relationships depending on whether the game is a playoff or regular season game:

For a regular season game: \(\widehat{td} = -0.614 + 0.356*{avg\_yards}\)

For a playoff game: \(\widehat{td} = -0.104 + 0.356*{avg\_yards}\)

The intercept here has no practical interpretations, but statistically speaking, it would mean that in a regular season game where Brady’s average yards equal 0 (none of our data points are even close to that), he would score -0.614 touchdowns (which is not possible in a football game).The intercept for the equation that models playoff games can be interpreted similarly: where Brady’s average yards equal zero, he would score -0.104 touchdowns (which is again an impossible result). Lastly, based on the slope of this equation, our model predicts that for every one-yard increase in average yards, there is an associated increase of on average .356 touchdowns (translating to roughly 2.136 points). This is consistent with the positive relationship that we observed in our visual exploratory data analysis between average yards and number of touchdowns. However, since the data led us to choose a parallel slopes model, the model predicts an identical relationship between the two variables regardless of whether the game is during the regular season or the playoffs, which doesn’t match our initial observations. If we had more data points there could prove to be an interaction between our numerical and categorical explanatory variables, but based on our dataset we are assuming that there is no significant interaction effect.

3.2 Non-statistical interpretation

As you saw in the early data visualization, the line representing the relationship between average yards and touchdowns for the regular season (blue) was steeper than the line representing that same relationship for playoff games (red). This would indicate that for each incremental increase in average yards, the associated increase in touchdowns scored is higher in the regular season. But when we ran a model with that in mind, the confidence intervals and p-values indicated that the differences between the playoffs and the regular season were not statistically significant. Thus, the extra detail of an interaction model was not worth it.

Our parallel slopes model indicates that Brady scores more touchdowns for the same number of average yards in the playoffs (since the intercept for the playoff line is less negative), but we will soon see that the difference is not statistically significant.

The big takeaway so far is that there is a slight positive association between Tom Brady’s average yards and the number of touchdowns he will score. For every one-yard increase in average yards, there is an associated increase of on average .356 touchdowns.

4 Inference for multiple regression

In order to proceed with our regression analysis, we first had to check if our data met the necessary conditions. Since the initial scatter plot of our data was fairly linear, the Linearity Condition is met. Our residual analysis shows that the residuals are randomly scattered for all values of x, and that they have a pretty consistent vertical spread. Thus, the Independence and Equal Variance Assumptions are met. The histogram of the residuals shows that they are pretty normally distributed, so the Normal Population Assumption is met. Thus, all necessary conditions are satisfied, which validates the inferences we can make from our multiple regression.

The intercept and the difference in intercept for the playoff games have no practical interpretations, so we will not interpret their confidence intervals and p-values. Realistically, a game where a QB has an average yards of zero means that he will probably score zero touchdowns; we should take zero as our intercept in both cases.

One interesting note on the intercepts: this is how we know that the difference in intercept between the regular season and the playoffs is not statistically significant. First, we found that an interaction model was not worth it because the difference in slope between the regular season and playoffs was not significant. Now, we know that the difference in intercept is not statistically significant either. Basically, the distinction between the playoffs and the regular season is not statistically significant; Tom Brady plays consistently in both.

Let’s construct a p-test for our explanatory variables, avg_yards. Our null hypothesis would be that the population slope of average yards for all of Brady’s games would be zero; our alternative hypothesis would be that the population slope is not zero. Our P-value for avg_yards was zero. No matter how small an alpha we chose, we would reject the null hypothesis and say that the true slope of avg_yards is not zero.

The confidence interval for our explanatory variable, avg_yards, did not include zero; it was [0.191, 0.521]. Thus, we are 95% confident that the real average associated increase in touchdowns for a one-yard increase average yards is between .191 and .521 touchdowns. Based on our hypothesis test and our confidence interval, we are pretty sure of some positive association between the two.

5 Conclusion

Our results indicate that there is a slight positive relationship between Tom Brady’s average yards and the number of touchdowns he will score. Furthermore, this relationship is not statistically significantly different between the regular season and the playoffs. While it did not lead to very exciting results of our regression and inference, it does tell us that Brady’s trademark consistency in the postseason is grounded in the numbers.

The take-home message here is that although there is a positive relationship between Brady’s average yards and the number of touchdowns he’ll score, it is not major. A difference of one yard does not even have an associated increase of an entire touchdown. Of course, correlation is not causation, and it would not be right to say that Brady increasing his average yards would lead to more scoring on his part.

Of course, this analysis has its limits. We did not examine opponents’ defensive play at all. In games where Tom Brady has high average yards, he is probably reading the defense pretty well. Thus, he might have more and better chances at the goal line.

Future work might look at the offense as a whole instead of focusing solely on the quarterback. In the past few years, the Patriots have had running backs like LeGarrette Blount and Dion Lewis who contribute rushing touchdowns on top of Brady’s contributions. An even more detailed analysis might go through all of Brady’s different receivers over the years (and there have been many) and looked more closely at their performance.

Overall, our analysis tells us that while Tom Brady’s average yards and the number of touchdowns he will score have a slight positive association. But, in a sport as multi-faceted as football, it’s certainly not the only important factor. Furthermore, our analysis indicated that the difference in relationship between average yards and touchdowns was not statistically significant between the regular season and the playoffs. This makes perfect sense given Brady’s stellar performance in the playoffs (the fact that he’s played in every playoff game in the last four seasons except Super Bowl 50 should tell you that).

Nothing in our analysis leads us to believe that he could not thrive with shorter offensive schemes. All in all, our results were not hugely surprising; we’ve seen Brady’s hallmark consistency and versatility on the field for years, and it came through in the statistical analysis.

6 Citations and References

Game Logs from: ESPN : http://www.espn.com/nfl/player/gamelog/_/id/2330, and Pro Football Reference : https://www.pro-football-reference.com/players/B/BradTo00/gamelog/

Kilgore, Adam. “Patriots’ Tom Brady, still evolving at age 40, is throwing deep like never before.” The Washington Post Web. 25 Oct 2017. https://www.washingtonpost.com/news/sports/wp/2017/10/25/patriots-tom-brady-still-evolving-at-40-is-going-deep-like-never-before/?utm_term=.54ab7a4f4dc2

YouTube Videos: “Patriots Julian Edelman TD Superbowl 49. https://www.youtube.com/watch?v=edwQnL-BYAI

“The Best of Patriots Randy Moss”: https://www.youtube.com/watch?v=QOITria4XOA

Photo:

https://cbsboston.files.wordpress.com/2015/12/gettyimages-500269554.jpg?w=628