A Points Per Game Rating for NFL Quarterbacks

A Points Per Game Rating For NFL Quarterbacks

Thesis

Presented in Partial Fulfillment of the Requirements for the Degree Master of Sciences in

the Graduate School of The Ohio State University

By

Jon Michael Gober, B.A.

Graduate Program in Agricultural, Environmental and Development Economics

The Ohio State University

2009

Thesis Committee:

Professor Tim Haab, Advisor

Professor Brian Roe

Copyright by

Jon Michael Gober

2009

ii

Abstract

Fans, teams, and commentators frequently use the NFL quarterback rating to

evaluate quarterback performance. Linear programming models, tiered logistic

regressions, and ordinary least squares regressions have also been used to measure

efficiency but the NFL rating is the most frequently used metric. One deficiency of the

NFL rating is that it overvalues completion percentage and interception percentage

relative to passing yards per attempt. This creates a bias in favor of modern quarterbacks

in the rating. I use NFL teams’ season statistics from 1970 through 2006 to derive a

rating estimating a quarterback’s contribution to points-per-game with an ordinary least

squares regression. I find that the points-per-game rating has less historical bias than the

NFL rating and predicts winning percentage equally well.

iii

Acknowledgments

I wish to thank my advisor, Dr. Tim Haab, for his advice and assistance in the

development of this thesis.

I also thank Dr. Brian Roe and Dr. Alan Randall for their comments on the

preliminary drafts of this thesis.

iv

Vita

March 4, 1984…………………………………………..Born-Charleston, West Virginia

2007…………………………………B.A. Mathematics-Economics, Denison University

2008 – Present…………………Graduate Teaching Assistant, The Ohio State University

Field of Study

Major Field: Agricultural, Environmental, and Development Economics

v

Table of Contents

Abstract…………………………………………………………………..……………….ii

Acknowledgments…………………………………………………………..…....………iii

Vita……………………………………………………………………………..…….…..iv

List of Tables………………………………………………………………..……………vi

Chapters:

1. Introduction………………………………………………………………….……..….1

2. Literature Review…………………………………………………………………...…7

3. Methodology and Model Design……………………………………………....….….12

4. Model Results……………………………………………………………….…….….17

5. Comparisons with Modern NFL Quarterback Rating……………………….…….....20

6. The ―Costs‖ of Each Rating System………………………………………….……....24

7. Other Approaches to Estimating Quarterback Efficiency…………………….......….28

8. Conclusions……………………………………………………………………...…....31

Bibliography……………………………………………...………………………....…41

vi

List of Tables

Table Page

1 Time Trends in Certain NFL Statistics…………………………………..…………...33

2 Variables that Impact Points per Game………………………………………..……..33

3 NFL Quarterback Rating versus a Points-per-Game Rating (for 2008)……..…….....34

4 Top 20 Quarterbacks All-Time, Minimum 1,500 Attempts…………….…………...36

5 Quarterbacks Who Benefit From A Points-Per-Game Ranking……………..……....37

6 Quarterbacks Who Benefit From the NFL Quarterback Rating………………..…....38

7 Correlation with Winning Percentage……………………………………..………....38

8 Cap Value and the Quarterback Rating........................................................................38

9 Cap Value and the Points-Per-Game Rating………………………………………....39

10 Offensive Statistics and the Natural Log of Points-Per-Game………...…………....39

11 A Points-Based Quarterback Rating Using Individual Game Data…………….......39

12 Individual Game Stats and the Natural Log of Points…………………...……........40

13 A Probit Model for Wins and Losses……………………………………….…...….40

1

CHAPTER 1

INTRODUCTION

Comparisons of professional athletes’ performances are a common topic of

discussion. Frequently these comparisons are based on statistical metrics that correlate

strongly with team success. In the academic world, most of the economic literature

concerning statistics in sports centers on baseball. A reason for this is that the

performance of an individual baseball player is relatively independent of his

teammates— the skill of the player in question primarily determines such statistics as

batting average, on-base percentage, and slugging percentage. American football

statistics, on the other hand, are highly interdependent. For example, a successful

running back often has a good offensive line and a good wide receiver usually has a good

quarterback throwing him the ball. The statistics of the quarterback position are highly

dependent on other position players: a quarterback needs a good offensive line to attempt

passes without getting tackled, a good running back to draw defensive players away from

the wide receivers, and good wide receivers to increase his passing yards. The current

NFL quarterback rating system1 attempts to accurately measure quarterback performance

in light of these considerations. However, the current NFL quarterback rating system is

1 The National Football League and NCAA football have separate quarterback rating systems. This paper is

concerned with the NFL rating. All references to the quarterback rating system refer to the NFL rating unless stated otherwise.

2

convoluted, has no meaningful interpretation as a statistic, double-counts certain metrics

and excludes some important metrics.

A committee of representatives from the Pro Football Hall of Fame and the Elias

Sports Bureau created the modern NFL quarterback rating in 1973. Fixed statistical

benchmarks are the basis for the NFL rating, while the NCAA rating evaluates

quarterbacks based on statistical averages. The fixed performance benchmarks of the

NFL rating are based on the statistics of all qualified pro passers since 1960. To be

included in the rating, a quarterback must average fourteen pass attempts per team game

played. To demonstrate the convoluted nature of the quarterback rating formula, it is

useful to look at its components: completion percentage, yards per pass attempt, average

touchdowns per pass (i.e. touchdown percentage), and average interceptions per pass (i.e.

interception percentage). The quarterback’s rating is the sum of the following four

components, multiplied by 100/6:

1. (Completion Percentage-0.3) / 0.2

2. (Yards Per Pass Attempt-3)/4

3. Touchdowns Per Pass Attempt/0.05

4. (0.095-Interceptions Per Pass Attempt)/0.04

Each component is bounded between 0 and 2.375. In other words, negative

values are set to zero and values above 2.375 are set to 2.375. This makes the zero the

lowest possible rating and 158.3 the highest possible rating. The rating is one of the most

common measures of quarterback performance used today. It is important economically

because it is frequently used as an incentive in player contracts. For example, Donovan

3

McNabb, Akili Smith, and Tony Banks have all signed contracts with bonuses related to

their quarterback ratings reaching a given threshold. Given NFL quarterbacks’ high

salaries, the difference between a high rating and a low rating can be worth millions of

dollars. In light of the large amounts of money at stake, you would think that the rating

system would have relatively few problems, but, as I will demonstrate, it has several

deficiencies.

One problem with the rating is its creation in 1973 based on statistics after 1960.

From the rating’s creation to the 2007 season, completion percentage has increased from

52.1 to 61.2 percent (a seventeen percent improvement). During the same time frame,

interception percentage has dropped from 5.3 to 3.1 percent (a forty percent

improvement). The time trends in these two categories have caused the average

quarterback rating to rise from 64.9 in 1973 to 80.9 in 2007 (a 24 percent improvement).

This raises the question of whether NFL quarterbacks have improved 24 percent in the

last three decades or whether the nature of pro football has changed in such a way as to

make quarterback statistics better. Given the frequency of West Coast offenses in today’s

game (which focus on easily completed, low risk passes), I believe the latter hypothesis

to be true. The current system is so biased towards modern offenses that fourteen of the

top seventeen career quarterback ratings come from active players.2

When looking at the year by year data, there is further reason to believe that there

is a time trend in these statistics: completion percentage, interception percentage, and

quarterback rating. Table 1 shows that the independent variable year is significant at the

2 http://www.profootballhof.com/history/story.jsp?story_id=2664 Accessed 11-12-2008. Ratings are

from the start of the 2008 season.

4

one percent level for the three statistics, despite there only being 37 observations (the

NFL yearly averages from 1970 through 2006). The table also shows that the variable

year is not statistically related to the yards per attempt statistic. An ideal rating system

for quarterbacks should have less of a time bias.

Another problem with the rating is that it overvalues completion percentage.

Because yards per attempt equals completion percentage times yards per completion,

completion percentage is implicitly double-counted in the formula. For example, an

incomplete pass counts as zero yards per attempt and also negatively impacts completion

percentage. Since completion percentage is implicitly included in yards per attempt, it

would make sense to reduce the weight given to completion percentage in the rating.

Also, the weights of the four components of the rating are arbitrary and unlikely to

correlate precisely with quarterback productivity.

A third problem with the rating is the bounded values of its four components.

Once certain benchmarks are attained, improved statistics do not improve the rating. For

example, once a passer attains a 77.5 completion percentage, the value of this component

cannot be improved. As far as the rating is concerned, a one-hundred percent completion

rate is equal to an eighty percent completion rate. Similarly, once a passer attains a 9.5

interception percentage, the value of this component cannot decline. The rating treats a

ten percent interception percentage the same as a fifty percent interception percentage.

Finally, there are no diminishing returns for yards per attempt below 3 or increasing

returns for yards per attempt beyond 12.5. While bounded values for the rating

components are not an issue over the course of a career or a season, they do become

important when evaluating individual game performances. There is no explicit constraint

5

on points scored in a game (although there is an implicit constraint with the game clock);

therefore the quarterback rating should not be constrained beyond the constraints inherent

in a sixty minute contest on a one hundred yard field.

Finally, the NFL rating excludes several important categories such as quarterback

rushes, fumbles, and sacks. The ability to run for positive yardage adds to a

quarterback’s value. Fumbles are clearly harmful to a team’s success yet they have no

impact on the current rating. Although many sacks are the fault of the offensive line,

sometimes the quarterback is to blame for a sack because he wasn’t mobile enough or

held on to the ball for too long. A sack is clearly worse than an incomplete pass, yet the

former category does not change the rating while the latter category diminishes it.

Including a wider variety of plays would improve the current rating.

Also, the NFL quarterback rating is difficult to interpret because it is not clear

what exactly is being measured. In my opinion, this is the greatest flaw in the rating.

Presumably the rating measures passing efficiency, but in what sense? The current rating

system combines inputs (completion percentage, yards per attempt, etc.), but the output

has no units associated with it. Because the output can’t be explained in terms of a

dependent variable (points, winning percentage, etc.), comparisons between quarterbacks

are difficult. How much better is a 95 quarterback rating than a 75? Later I will explain

why points per game should be the dependent variable in a quarterback rating system.

As a caveat, I do not think there is a perfect quarterback rating system. No rating

can address every possible question or complaint. However, I think there are enough

problems with the current system that it can be improved using statistical tools. Although

6

there is relatively little literature on this topic, some researchers have attempted to

improve the quarterback rating using linear programming, tiered logistic regression, and

least squares regression.

7

CHAPTER 2

LITERATURE REVIEW

DeOliveira and Callum used a linear programming approach called Data

Envelopment Analysis (DEA) to evaluate NFL quarterbacks. Their basic idea is to divide

statistical outputs (multiplied by the corresponding weights) by statistical inputs (again

multiplied by the corresponding weights) to obtain quarterback efficiency. The six

outputs were passing yards, rushing yards, total TDs, attempts per interception,

completions, and passing yards per game. The inputs were passing attempts and rushing

attempts. The weights were chosen in order to maximize each player’s efficiency, which

was bounded to be less than or equal to one (i.e. 100 percent). Their DEA analysis found

many players to be 100 percent efficient. The authors then conducted a cross-evaluation

of all players by assessing each player using every other player’s output and input

weights. Each player was rated based on their average score among these cross-

evaluations.

One advantage of DeOliverira and Callum’s approach is that it calculates the

category weights mathematically instead of choosing them arbitrarily. Another

advantage is that it allows one to compare and rank quarterbacks easily. Also, the rating

can be used to evaluate other football position players. A disadvantage of their approach

is the exclusion of sacks and fumbles. If sacks are occasionally the fault of the

quarterback, then a sack (which results in negative yards) should decrease the

8

quarterback rating more than an incompletion. Also, if lost fumbles are occasionally the

fault of the quarterback, then they should negatively impact the rating because they are

turnovers to the other team and therefore a worse result than an incomplete pass. Also,

the time constraint in football places a premium on scoring quickly when behind late in

the game. In this situation there is an important distinction between scoring on one long

play or several short plays, contrary to what the authors claim.

White and Berry used a tiered logistic regression to evaluate NFL quarterbacks.

Their research uses a per-play analysis rather than a game or seasonal analysis. This

makes their paper much more data intensive. They determined the value of each possible

play by estimating how a given play changed a team’s expected points. In their model

they used eventual points as a proxy for expected points. Eventual points (the dependent

variable) took the value of the next score and could equal 7, 3, 2, 0, -2, -3, or -7. The

independent variables in the model were Down (a dummy variable representing one of

the four possible downs), ToGo (a variable representing the distance from a first down),

and ToGoal (a variable representing the distance to the goal line). The authors used this

model to estimate the expected points from a given situation and by extension the

expected point values for all possible plays. They then ranked quarterbacks by their

average contributions to expected point values per play. The model is a tiered logistic

regression because the dependent variable can take on multiple categories and can be

modeled using tiers for each of the possible outcomes. The authors found that adding

sacks significantly impacts the relative quarterback rankings while adding quarterback

runs does not have a major impact.

9

One problem with the tiered logistic regression method is that it does not take into

account risk aversion on the part of coaches. For example, a completed ten yard pass

would have a larger gain in expected points than a run of just a few yards. However, a

team leading very late in the game would often rather call a running play because a pass

attempt would be considered an unnecessary risk. To the extent that maximizing the

probability of winning (the ultimate objective of all football coaches) differs from

maximizing expected points, the authors’ model is flawed. The authors partially address

this by excluding plays in the last two minutes of a half (although their reason for doing

so is to address the problem of ―prevent‖ defenses); however, excluding plays in the last

two minutes of a half seems inappropriate for a situation based ranking.

I think the authors made reasonable choices with regard to the independent

variables they chose. However, I believe they made a mistake by not including time as a

variable. Early in a half, the odds of there being zero eventual points are very low. For

example, the only way the first play of the half could have zero eventual points would be

if the half ended without any points being scored, which happens very infrequently.

Similarly, late in a half the odds of there being zero eventual points are relatively high.

The only way the last play of the half could have non-zero eventual points would be if a

scoring play occurred on the last play of the half, which happens infrequently. Adding a

variable which gave the time remaining in the half could adjust for this better than simply

excluding plays from the last two minutes of a half.

Another problem with the authors’ method is that the expected points for a given

game situation can vary across teams. For example, a team with good pass blocking

would have a higher expected points value on third and long plays than a team with a

10

mediocre pass blocking. A team with a good running back and offensive line would have

a higher expected points value on third and short plays than a team that could not run the

ball effectively. A team with a good kicker would have a higher expected points value

when possessing the ball in the opponent’s territory than a team with a bad kicker. To the

extent that factors outside of the quarterback’s control (like the strength of the running

back, offensive line, kicker, etc.) determine a team’s expected points value for a given

situation, the authors’ rating is flawed. In spite of the problems with the authors’ method,

I thought their paper was good because it gave a reasonable estimation of a quarterback’s

value.

Allan Ingraham used a three-stage least squares regression to estimate quarterback

production. His first equation used winning percentage as the dependent variable and

points scored and opponents’ points scored as the independent variables. The second

equation has points scored as the dependent variable and various football statistics (yards

per pass, yards per rush, turnovers, etc.) as the independent variables. The third equation

relates team statistics to opponents’ points scored. The idea behind this method is to

estimate how certain quarterback statistics affect points and opponents’ points scored,

and by extension estimate how these statistics affect winning percentage. The

quarterback ratings are thus derived from how each quarterback contributes to winning

percentage. In his paper Ingraham found that rushing yardage, rushing yards per attempt,

passing yardage, and passing yards per attempt all increase points scored per game.

However, rush yards per attempt was not statistically significant. Turnovers and

takeaways (opponent turnovers) were both statistically significant in the expected

directions, but the positive effect of a takeaway was superior to the negative effect of a

11

turnover. Finally, sacks were statistically significant while penalty yards, punt average,

and third down efficiency were not statistically significant.

An advantage of this model is that it allows one to measure the productive ability

of quarterbacks and other football position players. It also can be used to estimate the

value (in terms of contribution to winning percentage) of specific statistics like rush

yards, pass yards, and turnovers, just to name a few. Unfortunately, the dataset used only

includes 127 observations from the years 2001 through 2004. Adding more observations

would provide a more detailed analysis. Also, in this paper predicted net points per game

greatly exceed actual points per game. The author suggests this is due to the

quarterback’s contribution to field goals not being included in actual points per game.

Finally, Ingraham notes three of the top four rated quarterbacks for 2004 played in

domes. This suggests that domes and/or weather may be meaningful variables which are

not included in the model.

12

CHAPTER 3

METHODOLOGY AND MODEL DESIGN

In this paper I chose to take an econometric approach to evaluating quarterback

performance. My approach places great importance on aggregate statistics and does not

evaluate performances in specific situations (down, distance, etc.). My objective is to

derive a quarterback rating based on the statistics that best correlate with points per game.

I plan to rank quarterbacks based on their estimated contributions to points per game.

Like Ingraham, I used least squares to estimate how a given statistic affects points scored.

I used points per game as my dependent variable instead of points scored because I

believe points per game has a clearer interpretation and is more precise.3 I have

expanded on Ingraham’s idea by using 1,061 observations instead of just 127

observations. I used team offensive data from the years 1970 through 2006. My data

came from the 2007 ESPN Football Encyclopedia and were converted into an Excel

spreadsheet.

One might wonder why I used points per game as the dependent variable instead of

winning percentage, which is the variable that all football strategists are really trying to

maximize. There are many factors which affect winning percentage that are outside of

the quarterback’s control: strength of the defensive unit, strength of special teams units,

3 I realize that since points per game is always greater than or equal to zero, the distribution may be

skewed. This issue is addressed later on.

13

the coaches’ ability, and strength of schedule, just to name a few. Quarterbacks have

much more control over their teams’ points per game that they do over winning

percentage. Also, Ingraham and other researchers have demonstrated that points scored

(either for the season or per game) are very strongly correlated with winning percentage.

Also, the idea that points scored per game affects winning percentage is pretty intuitive—

although maximizing points scored per game may cause strategists to be risk-loving in

such a way as to diminish winning percentage. However, football coaches generally do

not recklessly risk losing purely for the sake of scoring more points, so this is probably a

relatively minor problem. Finally, using points per game as the dependent variable gives

the results a concise interpretation: each quarterback is rated based on his estimated

contribution to points per game.

There are many possible independent variables that could be chosen in this model.

The four that compose the quarterback rating are completion percentage, yards per pass

attempt, touchdown percentage, and interception percentage. I decided against including

touchdown percentage as an independent variable because it does not give useful results.

To say that the percentage of passes which result in touchdowns (i.e. points) affects

points scored per game is already obvious and thus not interesting. If this statement were

not true in the model there would a major problem with how the data were assembled

and/or how the model was designed. Perhaps a more questionable decision is to exclude

completion percentage. The reason for this is that completion percentage and yards per

attempt are very strongly correlated, as yards per attempt is equal to completion

percentage times yards per completion. Because yards per attempt incorporate both

passer accuracy and distance, it has more explanatory power than completion percentage

14

does. Therefore completion percentage is excluded to avoid multicollinearity. I am

assuming also that quarterbacks with high yards per attempt will be able to frequently

gain first downs (i.e. I am discounting the possibility that a quarterback could have a high

yards per attempt with a few ―long bombs‖ while gaining first downs infrequently.)

Yards per attempt are included for the reasons mentioned in the previous paragraph.

Interception percentage is included to penalize quarterbacks who frequently turn the ball

over to the other team. Rushing yards per game is included to reward quarterbacks with

running ability. The decision of whether or not to include sacks in the analysis was a

difficult one. Most, but probably not all, sacks are attributable to the offensive line

instead of the quarterback. Penalizing a quarterback for having a poor offensive line

seems unfair. On the other hand, yardage lost from sacks does affect points per game.

Ideally, we would want to know how many times the ―average‖ quarterback would get

sacked with a given offensive line and use that information to estimate the significance of

sacks in a quarterback rating. Unfortunately, such data is very difficult to find.

Therefore, sacks are excluded from this model to avoid any arbitrary assumptions over

what percentage of sacks is the fault of the quarterback and what percentage is the fault

of the offensive line. Takeaways per game are included in the model to incorporate the

importance of improved field position from interceptions and fumble recoveries. Finally,

fumbles lost per game is included in the model due to its statistical significance but not

included in the quarterback rating, as I assume that lost fumbles are generally the fault of

the offensive line for not blocking adequately. Also, there was little variation in

quarterbacks’ fumbles lost per game. These independent variables do not form an

exhaustive list of what statistics affect points per game, but they explain a lot of variation

15

in the dependent variable while avoiding the problem of multicollinearity. Next I will

provide a more thorough explanation of my independent variables.

Many critics of the quarterback rating believe that the yards per attempt statistic

should be given more weight. Yards per attempt is defined as gross passing yards

divided by pass attempts. The average yards per attempt from 1970 through 2006 was

about 6.84. There does not appear to be a time trend in the year-by-year data (see Table

1), although the NFL yearly averages for this statistic range from a low of 6.49 (in 1974

and 1977) to a high of 7.18 (in 1983). The team averages range from 9.49 (San

Francisco, 1989) to 4.88 (Seattle, 1992). It is expected that yards per attempt would have

a positive sign and be statistically significant.

Interception percentage is defined as interceptions thrown divided by passes

attempted, multiplied by 100. The average interception percentage from 1970 through

2006 was 4.03. Over the last few decades, interception percentage has shown a declining

trend from about five to three percent. This trend (along with improved completion

percentages) explains a great deal of the trend towards improved quarterback ratings in

recent years. The NFL yearly averages range from 5.8 percent (in 1971) to 3.1 percent

(six times, all since 1995). The team averages range from 9.45 (Green Bay 1971) to 1.11

(Kansas City 1990). It is expected that interception percentage would have a negative

sign and be statistically significant.

Despite the fact that it is not a quarterback statistic, takeaways per game are included

to incorporate the positive effect of improved field position from interceptions and

fumble recoveries. This variable is defined as interceptions plus fumble recoveries,

16

divided by games played. The average number of takeaways per game for each team

from 1970 through 2006 was 2.14, with a high of 3.94 and a low of 0.75. In the

regression takeaways per game should have a positive sign and be statistically significant.

Rushing yards per game is included to incorporate a quarterback’s running ability into

the rating. The average rushing yards per game per team over the dataset is about 121.7,

with a high of 220.6 (Buffalo 1973) and a low of 66.4 (San Diego 2000). I used rushing

yards per game instead of rushing yards per attempt in my analysis. The reason for this is

that most quarterbacks do not have a lot of rushing attempts. When quarterbacks do run,

the objective is usually to get a first down (―move the chains‖ in football speak) and then

avoid getting hurt. A higher priority is placed on quarterback safety than on getting a

huge running play, making the yards per attempt statistic less important. Including

rushing yards per game is designed to reward quarterbacks who consistently gain first

downs with their feet. It is expected that this statistic will have a positive coefficient and

be statistically significant.

Finally, fumbles lost per game is included in the model due to its statistical

significance. Because it is difficult to determine what proportion of lost fumbles are the

fault of the quarterback (as opposed to the offensive line), this statistic was not included

in my quarterback rating in order to avoid making arbitrary assumptions about this

question. Also, it is possible that whether or not a fumble is lost to the other team is

random, meaning that adding fumbles lost per game may unnecessarily add an element of

chance to the rating. It is expected that fumbles lost per game will have a negative

coefficient and be statistically significant.

17

CHAPTER 4

MODEL RESULTS

Table 2 gives my regression results. The dependent variable is points per game. All

independent variables have the expected signs and are significant at the one-percent level.

To compare the variables’ significance in terms of producing points, one needs to look at

the parameter estimates for each of the variables.

The parameter estimate for yards per attempt is 3.34. This means that an increase of

one yard per pass attempt increases points per game by 3.34. Over the course of a sixteen

game season this translates to about 53 more points. This might not seem like a

significant increase; however, when one takes into account that in 2007 twenty-two

percent of NFL games were decided by three points or less4, this is a meaningful statistic.

In 2007, out of the thirty-three quarterbacks who qualified to be rated, the highest yards-

per-attempt was 8.3 (Tom Brady) and the lowest was 5.5 (Brodie Croyle). This translates

to a difference of about nine and a half points per game. This difference is very

important when one considers that forty-seven percent of NFL games in 2007 were

decided by eight points or less.5

The parameter estimate for interception percentage is -0.82. This means that if

interception percentage increases by one percent, points per game decline by -0.82. This

4 “Parity Pairs with Drama in NFL Saga”, Jarrett Bell, USA Today 10-31-2008. Accessed 11-24-2008.

http://www.usatoday.com/printedition/sports/20081031/scover31.art.htm?loc=interstitialskip 5 Ibid.

18

statistic is useful for evaluating individual games, where interception percentages vary

widely. Over the course of a season, interception percentages among quarterbacks vary

less. In 2007 the best interception percentage among qualifying quarterbacks belonged to

David Garrard (0.92 percent), while the worst percentage belonged to Sage Rosenfels

(5.0 percent). This difference of 4.08 percent translates to a difference of about 3.3

points per game. This is a meaningful difference when one considers the proportion of

NFL games that are very close. However, the data suggest that interception percentage is

relatively less important than yards per attempt.

The parameter estimate for rushing yards per game is 0.04. This means that one

rushing yard per game is worth about 0.04 points per game, or that an increase of twenty-

five rushing yards per game translates to one point per game. In 2007, Vince Young led

qualifying NFL quarterbacks in rushing yards per game with 26.3, while Peyton Manning

was last in the same category with -0.32. This 26.62 yard difference translates to a

difference of about one point per game. The data thus imply that a quarterback’s ability

to rush for positive yards is relatively less important to points per game than a

quarterback’s yards per pass attempt and interception percentage.

Fumbles lost per game has a coefficient of -2.25. This means that an increase of one

lost fumble per game is worth about -2.25 points per game. The best performance in

2007 in this statistic was zero (by several players) and the worst performance was 0.19

fumbles per game (by two players). This difference translates to about four-tenths of a

point per game, which is relatively smaller than the corresponding differences for the

other quarterback statistics. This suggests that the variation in lost fumbles per game

19

among NFL quarterbacks is not wide enough to create meaningful disparities in scoring

averages.

Finally, the coefficient for takeaways per game is 1.75. This means that an increase

of one takeaway per game translates to an increase of about 1.75 points per game. This

statistic is more useful for evaluating individual game performances than season

performances. Still, the difference between the best team performance in this category in

2006 (2.75 takeaways per game) and the worst performance (0.75 takeaways per game)

translates to a difference of about three and a half points per game, which is very

meaningful in a close contest.

20

CHAPTER FIVE

COMPARISIONS WITH MODERN QUARTERBACK RATING

Table 3 compares the 2008 rankings of NFL quarterbacks under the modern

quarterback rating system and the points per game system used in the model. The left

column lists the quarterbacks. The middle column gives the official quarterback ratings

for each player, followed by the rank. The right column gives the points per game rating

for each quarterback, followed by the rank. The points per game rating is the

quarterback’s contribution to points per game and equals 3.34* (Yards per Pass Attempt)-

0.82*(Interception Percentage) +0.04*(Rush Yards per Game). Takeaways per game are

excluded because it is not a quarterback specific statistic. Fumbles lost per game are

excluded because some unknown proportion of lost fumbles is not the quarterback’s

fault. A plus sign indicates where the points-per-game ranking for a quarterback is five

or more slots above his quarterback rating rank. A minus sign indicates where the points

per game ranking for a quarterback is five or more slots below his quarterback rating

rank. Nine of the thirty three quarterbacks who qualified for the NFL rating had plus or

minus signs with their ranking, which suggests some correlation between the NFL rating

and the points per game rating. The advantage of a points based rating is that it has a

meaningful output (the quarterback’s estimated contribution to his team’s points per

game), whereas the NFL rating is not directly associated with any kind of team or

21

individual output. The points rating also provides a rough estimate of the value added

(i.e. increase in points per game) when one quarterback replaces another.

An interesting question to ask is whether the relative historical rankings of

quarterbacks differ under the two rating systems. As mentioned previously, the NFL’s

quarterback rating has a time bias, with fourteen of the top seventeen all time ratings

coming from active players.6 This time bias is caused by the inclusion of completion

percentage and interception percentage as elements of the rating (see Table 1). Because

the points-per-game rating excludes completion percentage, it is expected that it should

have less of a time bias. Table 1 demonstrates that there is a statistically significant time

bias with the points-per-game rating (due to interception percentage being included), but

it is less than the time bias with the official quarterback rating. The coefficient for the

independent variable Year is about eight times bigger for the official quarterback rating

than the points per game rating. If we assume that one point per game is roughly

equivalent to four quarterback rating units (see Table 3), then it is reasonable to estimate

that the points-per-game rating has about half of the time bias of the official quarterback

rating.

Table 4 appears to confirm the idea that the points-per-game rating has less of a time

bias. The table shows the top twenty quarterbacks of all time under both rating systems

and only includes quarterbacks with at least 1,500 pass attempts. Fifteen of the top

twenty quarterbacks under the NFL rating played in 2008, while only ten of the top

twenty in the points-per-game rating played in 2008. Of the six quarterbacks who are

6 http://www.profootballhof.com/history/story.jsp?story_id=2664 Accessed 11-12-2008. Ratings are

from the start of the 2008 season.

22

only in the top twenty of the NFL rating, five played in 2008 and the other (Rich Gannon)

last played in 2004. Of the six quarterbacks who are only in the top twenty of the points-

per-game rating, none played after 1990 and only two (Fouts and Lomax) played after

1980. This information suggests that the points-per-game rating has less of a time bias

than the NFL rating.

My historical list of quarterbacks includes 158 players. Generally, I included a

quarterback if he was in the top one hundred all time in a major statistical category or if

he had played within the last ten years. Table 5 provides a list of the quarterbacks whose

points-per-game ranking was at least twenty-five slots higher than their relative

quarterback ranking. Two notable characteristics of most of these twenty-one

quarterbacks are a high yards-per-attempt (average of 7.46) and a high interception

percentage (average of 5.29). The averages for all 158 quarterbacks I’ve included are

6.99 yards per attempt and an interception percentage of 4.19. All except Michael Vick

have been out of the NFL for at least ten years. Vick, who jumped from 73rd

overall in

the quarterback rating to 29th

overall in the points-per-game rating, is helped by having a

career average of 52.1 rushing yards per game. This average is approximately seventy-

percent higher than the second place finisher in this category (Randall Cunningham, 30.6

rushing yards per game).

Table 6 provides a list of the quarterbacks whose points-per-game ranking was at

least twenty-five slots lower than their relative quarterback rating. Two notable

characteristics of most of these sixteen quarterbacks are a low yards-per-attempt (average

of 6.56) and a low interception percentage (average of 3.19). Nine of the sixteen were on

an NFL roster in 2008. When Michael Vick’s rushing average is excluded, there is little

23

difference between the Table 5 and Table 6 quarterbacks in this statistic, with the former

group averaging 6.02 rushing yards per game and the latter group averaging 5.44 in the

same category. The information in Tables 5 and 6 suggest that yards-per-attempt is an

undervalued statistic in the NFL quarterback rating while interception percentage is an

overvalued statistic. Also, it appears that only truly exceptional running quarterbacks

gain enough rushing yards to significantly impact their relative points-per-game rating.

Another way to compare the NFL quarterback rating and the points-per-game rating

is to measure which of the two correlates better with winning percentage. To do this, I

used the same 1,061 observations used in Table 2. Unfortunately, this dataset only

contains team rushing yards per game (not quarterback rushing yards per game). Thus

the rushing yards component of my formula must be excluded from the comparison to

avoid inflating the points-per-game rating’s significance by including non-quarterback

statistics. In my comparison I use winning percentage as the dependent variable,

quarterback rating or points-per-game rating as a measure of offensive efficiency, and

points allowed per game as a proxy for defensive efficiency. Ties count as half of a win

and half of a loss. Table 7 shows the results of the two regressions. Despite only

including two independent variables, both equations explain about seventy percent of the

variation in winning percentage. Based on the results in Table 7, it is reasonable to

assume that the two ratings explain winning percentage equally well. The advantages of

the points-per-game rating are that it is easier to calculate and does not have bounded

values.

24

CHAPTER 6

THE ―COSTS‖ OF EACH RATING SYSTEM

Another way to compare the two ratings is to estimate which is ―cheaper‖ to acquire.

Many quarterback contracts contain financial incentives for achieving a given

quarterback rating, so one would expect there to be a statistically significant relationship

between salaries and the rating. Since there is some correlation between the points per

game rating and quarterback rating, one would also expect the points per game rating to

positively correlate with salary even without any corresponding incentives. Other

variables that affect quarterback salaries are experience (with older quarterbacks expected

to earn more on average), number of Pro Bowl appearances, and time (because the salary

cap increases most years, we would expect quarterback salaries to increase from year to

year regardless of performance). The idea behind this kind of analysis is to see if the

market (i.e. the 32 NFL teams) is undervaluing the points per game rating.

One problematic aspect of this kind of analysis is that there are multiple ways to

measure how much a quarterback is paid. NFL quarterbacks have base salaries, signing

bonuses, bonuses for performance, and bonuses for appearing at off season workouts.

Each quarterback also has a cap value, which usually differs from the total salary. Teams

frequently manipulate contracts to gain an advantage in the salary cap, which means that

the payout of contracts may be front-loaded or back-loaded without regard to player

performance. In my regression I used the quarterback’s cap value to measure how much

25

he was paid. The cap value may not be the most important measurement from an

economic standpoint, but it is the most important measurement from a competitive

standpoint because it determines how much ―cap room‖ is remaining for other players on

the roster. If a team overpays for a quarterback, less cap room remains for signing other

players and team competitiveness suffers. A quarterback who is overpaid from a salary

standpoint but is correctly paid in terms of cap value creates less damage for his team’s

salary cap situation and overall competitiveness.

My dataset for this analysis is the USA Today Salaries Database, which is available

online at the USA Today website.7 The database can be used to find quarterback salaries

back to 2000. My dataset includes 183 observations from 55 quarterbacks over the last

nine years (2000 through 2008). I only included observations in which the quarterback

had qualified for the NFL rating the previous year (minimum of 14 pass attempts per

game, or 224 pass attempts per season). The reason for restricting the data this way is to

avoid making any incorrect assumptions about what the quarterback’s passer rating

would have been if he had played a full season.

My dependent variable is the quarterback’s cap value, which is defined as the player’s

signing bonus plus salary and other bonuses for the season. The average quarterback’s

cap value is about 4.9 million dollars, with a range of 328 thousand to 18.7 million

dollars. My independent variables are experience, pro bowl, quarterback rating (or

points-per-game rating) and time. Experience is the number of years the player has been

in the league (including the given season), with a rookie’s experience being equal to one

(although the way the regression is set up, no rookies appear in the dataset). Pro Bowl is

7 http://content.usatoday.com/sports/football/nfl/salaries/default.aspx, last accessed on 4-7-2009.

26

a binary variable which equals one if the player made the Pro Bowl the previous season

and equals zero otherwise. Quarterback rating is defined as the NFL quarterback rating

from the previous season. Likewise, points-per-game rating is defined as the points-per-

game rating from the previous season. I am implicitly assuming that teams make their

expectations of future performance from the previous season’s performance. Finally,

time is included as a variable to account for yearly increases in the salary cap (with the

year 2000 being one, 2001 being two, etc.).

Tables 8 and 9 show the cap value regression results. All variables have the expected

positive signs. Every variable except the points-per-game rating is significant at the one

percent level, with that rating being significant at the five percent level. Table 8 shows

that one year of experience can be offset by improving the quarterback rating by just 2.7

units, making experience a relatively unimportant variable. Likewise, one year of time

(i.e. a typical salary cap increase) can be offset by improving the quarterback rating by

just 3.8 units, making time a relatively unimportant variable. However, the Pro Bowl

variable is very important as it takes about 20.9 quarterback rating units to offset a Pro

Bowl selection. For the points-per-game rating in Table 9, it takes about 1.1 points per

game to offset one year of experience and about 1.6 points per game to offset one year of

time. Also, it takes about 10.3 points per game to offset a Pro Bowl selection. This

information suggests that the experience and time variables can easily be negated by

improved performance, but a Pro Bowl quarterback will make a high salary even if he has

a low passer efficiency rating (which is unlikely). In addition, the low r-squared for

Tables 8 and 9 suggests that salary cap manipulations probably explain most of the

variation in a quarterback’s cap value.

27

Both the quarterback and points-per-game ratings appear to be accurate forecasters of

salary, but which is cheaper? Because the ratings are derived differently, simply

comparing the parameter estimates can’t tell us which measure of performance is cheaper

to acquire. For the 2008 season, the average ratio of quarterback rating to points-per-

game rating was about 3.94 (for qualifying quarterbacks). For the 2007 and 2006

seasons, this ratio was 3.98 and 3.90 respectively. Based on this information, I believe it

is reasonable to assume that one point-per-game is roughly equivalent to four quarterback

rating units. If that is the case, then an additional point-per-game ―costs‖ about $225,000

and the equivalent quarterback rating improvement costs about $346,000 (about fifty

percent more expensive). This suggests that the points-per-game rating is greatly

undervalued by NFL teams even though it correlates with winning percentage about as

well as the official quarterback rating does.

28

CHAPTER 7

OTHER APPROACHES TO ESTIMATING QUARTERBACK EFFICIENCY

A potential problem with the points-per-game rating is that if the independent

variables have particularly extreme values, estimated points per game could be zero or

negative. For example, a team with 3 yards per pass attempt, an interception percentage

of 15, zero takeaways per game, twenty rushing yards per game, and one lost fumble per

game would ―average‖ negative 3.7 points per game. This is obviously not realistic.

Also, given that the data set uses season data, averaging zero points per game is not

realistic. To see if the bias inherent in using points-per-game as the dependent variable

affects the results, one can compare the results with a new regression using the natural

log of points-per-game as the dependent variable. Using the natural log forces points per

game to be positive, thus removing this bias in the dependent variable.

Table 10 is a replication of Table 2 using the natural log of points-per-game as the

dependent variable. Although the parameter estimates are different, the significance

levels of the independent variables do not change significantly. Also, the explanatory

power of the overall regression does not change significantly. This suggests that the

directional bias in Table 2 is not very meaningful. The practical reason for this is that

over the course of a season, teams will not do so poorly statistically that they would be

projected to average zero or negative points per game.

29

Another way to derive a quarterback rating is to use individual game data instead of

aggregate season data. I did this by using game results from the 2008 NFL regular

season. The data set consists of 256 games, which results in 512 lines of data (one line

for each team). I omitted one game which ended in a tie. The most important

difference between the game and season data is that the game data shows more variation.

Yards per pass attempt ranges from a low of 1.4 to a high of 14.5. Interception

percentage ranges from a low of zero to a high of 18.2. Finally, rushing yards per game

varies from a low of 14 to a high of 332. Another difference is that the greater likelihood

of extremely poor offensive statistics increases the chance of the dependent variable

(points) having a significant directional bias.

Tables 11 and 12 give the regression results from using individual game data from the

2008 season. Table 11 has points as the dependent variable, while Table 12 has the

natural log of points as the dependent variable. The independent variables are the same

ones used in Table 2. From looking at Table 11, we see that interception percentage is

not quite significant at the five percent level while fumbles lost is significant at the five

percent level. The other three variables are significant at the one percent level. When

comparing the results with those in Table 2, it appears that yards per pass attempt and

interception percentage become less important in the rating while rushing yards become

more important. Table 12 shows that changing the dependent variable to the natural log

of points does not change the results much, although interception percentage becomes

significant at the five percent level. The regressions using individual game data have less

explanatory power than the seasonal data regression in Table 2. This suggests that a

30

situational data approach (as White and Berry used) may be more appropriate to use in

the case of individual games.

Another way to evaluate quarterbacks is to estimate how their performances affect the

probability of winning. A probit model is one way to estimate this. If we make the

dependent variable a binary categorical variable (1 for a win, 0 for a loss), we can

estimate how changes in a given statistic impact a team’s probability of winning. Table

13 shows how certain statistics impact the probability of winning. In addition to the

statistics used previously, a variable for points allowed is included as a proxy for

defensive performance. The results in Table 13 are not as robust as the previous

regressions were, with interception percentage and fumbles lost being statistically

insignificant. It is possible that the results could be made stronger with more

observations. However, whether a team wins a particular game depends on more than

aggregate statistics. When a particular play is made (or not made) has a meaningful

impact on the game’s outcome. Also, it is likely that the probability of winning does not

change linearly with statistical changes. For example, improving from three to four yards

per pass attempt likely would not impact the probability of winning much (as the latter

statistic is still very mediocre), but improving from seven to eight yards per pass attempt

could greatly increase the odds of winning. For these reasons, aggregate statistics should

probably not be used to calculate a probit model of the probability of winning. A

situational analysis involving variables such as the current score, the time, the down, the

distance to a first down, and the distance to a touchdown would probably be a better

approach.

31

CHAPTER 8

CONCLUSIONS

Teams, commentators, and fans have used the NFL quarterback rating to evaluate

quarterbacks for several years. Even though the quarterback rating is commonly used,

there are many problems with it. The rating overvalues completion percentage,

rewarding quarterbacks who attempt short, easy passes. The rating also overvalues

interception percentage relative to yards per pass attempt. This results in most of the

modern quarterbacks having ratings that would be considered outstanding three or four

decades ago. Many hall of fame quarterbacks from the sixties and seventies had

quarterback ratings that would be considered mediocre today. Also, the official

quarterback rating has no units associated with it, which makes it difficult to quantify

how valuable a given change in the rating is.

Linear programming, logistic regression, and ordinary least squares regression

models can be used to estimate a quarterback’s value. I used ordinary least squares

because it can be conveniently used to calculate a quarterback’s value using aggregate

statistics. DeOliveira and Callum’s linear programming method requires the researcher

to use every quarterback’s statistics to calculate one quarterback’s rating. White and

Berry’s tiered logistic model requires the researcher to know the given situation (down,

yards to a first down, and yards to the goal line) for every play involving a quarterback

running or passing before a rating can be calculated. I used season data to derive my

32

points-per-game rating. Individual game data could have been used instead, but the

increased randomness of individual games makes the results less robust. The

advantages of my points-per-game rating are that it has measurable output units, has less

of a time bias, is relatively simple, and is undervalued by the market of NFL teams. The

points-per-game rating also correlates with winning percentage just as well as the

official quarterback rating does. Because NFL quarterbacks have arguably the highest

profile position in professional sports, it is surprising that there hasn’t been more

scrutiny of how their performances are measured. I hope there will be more academic

papers addressing this issue in the future.

33

Table 1: Time Trends in Certain NFL Statistics

Number of Observations= 37

Dependent Variable Relationship P-value Adj. R-squared

Completion Percentage -403.51+.231Year <.0001 .853

Interception Percentage 154.32-.076Year <.0001 .867

Yards Per Attempt 3.47+.002Year .5451 -.018

Quarterback Rating -890.27+.485Year <.0001 .819

PointsPerGame Rating -114.96+.068Year <.0001 .486

Table 2: Variables that Impact Points per Game

Adjusted R squared = .672

Number of Observations: 1,061

Variable Parameter Estimate Standard Error P-value

Intercept -5.30 0.82 <.0001

Yards Per Pass

Attempt

3.34 0.11 <.0001

Interception

Percentage

-0.82 0.06 <.0001

Takeaways Per Game 1.75 0.16 <.0001

Rush Yards Per Game 0.04 0.003 <.0001

Fumbles Lost Per

Game

-2.25 0.27 <.0001

34

Table 3: NFL Quarterback Rating versus a Points-per-Game Rating (for 2008)

Average Ratio of QB Rating to Points-Per-Game Rating=3.94

Player NFL QB Rating Points-per-Game Rating

P. Rivers 105.5 (1) 26.3466 (1)

C. Pennington 97.4 (2) 24.543 (4)

K. Warner 96.9 (3) 23.6932 (8) (-)

D. Brees 96.2 (4) 24.4368 (5)

P. Manning 95 (5) 22.3298 (13) (-)

A. Rodgers 93.8 (6) 23.6998 (7)

M. Schaub 92.7 (7) 24.8686 (2) (+)

T. Romo 91.4 (8) 23.1684 (10)

J. Garcia 90.2 (9) 23.2626 (9)

M. Cassel 89.4 (10) 22.8676 (11)

M. Ryan 87.7 (11) 24.6962 (3) (+)

S. Hill 87.5 (12) 21.9292 (15)

S. Wallace 87 (13) 20.4702 (23) (-)

D. McNabb 86.4 (14) 21.722 (16)

E. Manning 86.4 (15) 20.8816 (21) (-)

J. Cutler 86 (16) 22.671 (12)

T. Edwards 85.4 (17) 22.2352 (14)

J. Delhomme 84.7 (18) 24.194 (6) (+)

J. Campbell 84.3 (19) 21.0706 (20)

D. Garrard 81.7 (20) 21.449 (17)

B. Favre 81 (21) 18.8746 (28) (-)

J. Flacco 80.3 (22) 21.3336 (18)

K. Collins 80.2 (23) 20.2714 (24)

Continued

35

Table 3 Continued

B. Roethlisberger 80.1 (24) 21.142 (19) (+)

K. Orton 79.6 (25) 19.3414 (27)

J. Russell 77.1 (26) 20.512 (22)

T. Thigpen 76 (27) 19.4662 (26)

G. Frerotte 73.7 (28) 19.8734 (25)

D. Orlovsky 72.6 (29) 18.7496 (29)

M. Bulger 71.4 (30) 18.2904 (30)

R. Fitzpatrick 70 (31) 15.1236 (32)

D. Anderson 66.5 (32) 16.9954 (31)

36

Table 4: Top 20 Quarterbacks All-Time, Minimum 1,500 Attempts (Bold=Active)

Source: http://www.pro-football-reference.com/leaders/pass_rating_career.htm

Name QB Rating Name PPG Rating

1. Steve Young 96.8 1. Steve Young 25.5416

2. Peyton Manning 94.7 2. Otto Graham 24.8894

3. Kurt Warner 93.8 3. Kurt Warner 24.3296

4. Tom Brady 92.9 4. Daunte Culpepper 23.8424

5. Joe Montana 92.3 5. Ben Roethlisberger 23.628

6. Chad Pennington 90.6 6. Peyton Manning 23.477

7. Drew Brees 89.4 7. Joe Montana 23.3492

8. Ben Roethlisberger 89.4 8. Roger Staubach 23.2922

9. Daunte Culpepper 89 9. Trent Green 23.2204

10. Carson Palmer 88.9 10. Bart Starr 22.8954

11. Jeff Garcia 87.5 11. Tom Brady 22.4426

12. Otto Graham 86.6 12. Chad Pennington 22.3406

13. Dan Marino 86.4 13. Johnny Unitas 22.2528

14. Trent Green 86 14. Norm Van Brocklin 22.2234

15. Donovan McNabb 85.9 15. Dan Fouts 22.2128

16. Marc Bulger 85.6 16. Neil Lomax 22.1378

17. Brett Favre 85.4 17. Jake Delhomme 22.0812

18. Jake Delhomme 85.1 18. Marc Bulger 22.0664

19. Rich Gannon 84.7 19. Dan Marino 22.0552

20. Matt Hasselbeck 84.5 20. Jeff Garcia 22.0518

37

Table 5: Quarterbacks Who Benefit From A Points-Per-Game Ranking

Name Rank (QB Rating) Rank (PPG Rating) Difference

Ed Brown 147 65 82

Norm Van Brocklin 79 14 65

Steve Grogan 119 63 56

Sid Luckman 81 28 53

Earl Morrall 92 40 52

Jay Schroeder 108 58 50

Michael Vick 73 29 44

Don Meredith 85 42 43

Joe Namath 140 97 43

Johnny Unitas 55 13 42

Bill Nelsen 115 74 41

Billy Wade 105 71 34

Doug Williams 121 90 31

Dan Fouts 46 15 31

Bobby Layne 143 113 30

Bart Starr 39 10 29

Lynn Dickey 112 83 29

Craig Morton 94 66 28

Charley Johnson 122 95 27

Terry Bradshaw 113 88 25

John Hadl 134 109 25

38

Table 6: Quarterbacks Who Benefit From the NFL Quarterback Rating

Name Rank (QB Rating) Rank (PPG Rating) Difference

Eli Manning 71 128 -57

Steve Bono 78 134 -56

Brad Johnson 29 82 -53

Tim Couch 80 129 -49

Brian Griese 26 72 -46

Jon Kitna 68 110 -42

Brett Favre 17 55 -38

Kyle Orton 111 149 -38

Erik Kramer 69 105 -36

Joey Harrington 120 153 -33

Byron Leftwich 45 76 -31

Elvis Grbac 49 80 -31

Neil O'Donnell 32 62 -30

David Carr 83 112 -29

Drew Bledsoe 65 94 -29

Bobby Herbert 58 86 -28

Table 7: Correlation with Winning Percentage

Number of Observations=1,061 (for both)

All p-values <.0001 for both

Dependent Variable Relationship Adjusted R-squared Mean squared error

Winning Percentage .642+.007QBRat-

.031OppPtsPerGame

.6955 .10634

Winning Percentage .554+.029PPGRat-

.031OppPtsPerGame

.6984 .10583

39

Table 8: Cap Value and the Quarterback Rating

Adjusted R-squared=.3212

Number of Observations=183

Variable Parameter Estimate Standard Error P-value

Intercept -6,239,498 1,909,319 .0013

Exper 230,533 63,797 .0004

ProBowl 1,806,134 602,913 .0031

QBRat 86,570 23,727 .0003

Time 331,901 89,955 .0003

Table 9: Cap Value and the Points-Per-Game Rating

Adjusted R-squared=.2953

Number of Observations=183

Variable Parameter Estimate Standard Error P-value

Intercept -4,218,561 1,921,096 .0294

Experience 257,313 64,161 <.0001

ProBowl 2,321,811 584,939 .0001

PPGRat 224,516 89,650 .0132

Time 350,643 91,815 .0002

40

Table 10: Offensive Statistics and the Natural Log of Points-Per-Game

Adjusted R-squared =.6631

Number of Observations=1,061

Variable Parameter

Estimate

Standard Error P-value

Intercept 1.706 .043 <.0001

Yards Per Pass Attempt .168 .006 <.0001

Interception Percentage -.044 .003 <.0001

Takeaways Per Game .089 .008 <.0001

Rush Yards Per Game .002 .0002 <.0001

Fumbles Lost Per Game -.115 .014 <.0001

Table 11: A Points-Based Quarterback Rating Using Individual Game Data

Dependent Variable: Points

Adjusted R-squared=.5260

Number of Observations=510

Variable Parameter Estimate Standard Error P-value

Intercept -5.15 1.57 .0011

Yards Per Pass

Attempt

2.61 0.18 <.0001

Interception

Percentage

-0.21 0.11 .0509

Takeaways 2.34 0.25 <.0001

Fumbles Lost -0.89 0.40 .0271

Rushing Yards 0.05 0.01 <.0001

41

Table 12: Individual Game Stats and the Natural Log of Points

Dependent Variable: Natural Log of Points

Adjusted R-squared=.4508

Number of Observations=504 (excludes shutouts)

Variable Parameter

Estimate

Standard

Error

P-value

Intercept 1.604 .093 <.0001

Yards Per Pass

Attempt

0.132 .010 <.0001

Interception

Percentage

-0.013 .006 .0386

Takeaways 0.110 0.015 <.0001

Fumbles Lost -0.051 0.023 .0290

Rushing Yards 0.0028 0.00035 <.0001

Table 13: A Probit Model for Wins and Losses

Dependent Variable: Win

Number of Observations=510

Variable Parameter Estimate Standard Error Pr>ChiSq

Intercept -1.9712 .4613 <.0001

Yards Per Pass

Attempt

.4261 .0542 <.0001

Interception

Percentage

-.0435 .0277 .1169

Takeaways .3301 .0647 <.0001

Fumbles Lost -.1417 .1029 .1685

Rushing Yards .0100 .0017 <.0001

Points Allowed -0.1117 .0119 <.0001

42

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Verlag, Berlin, Germany, 2004.

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kip

http://www.profootballhof.com/history/story.jsp?story_id=2664 Accessed 11-12-2008.

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