#### stats junkie

Earlier this season the Canadian Football League unveiled QUAR, the league's new quarterback rating system. So what is QUAR and how effective is it?

QUAR is based on CFL data for the period 2009-2016. We have been told that QUAR incorporates the existing pass efficiency rating and then adds 8 new categories which results in a rating scored out of 100. As the season progressed, we have been given sample data and other details about QUAR. The one thing that is sorely lacking is the formula.

Fortunately, enough data has been provided to backwards engineer the QUAR formula - it looks like this:

How effective is it?
One of the first considerations is how does a formula like QUAR deal with statistical outliers, the data points that are so far from the average that they skew the end results. The method that the CFL chose is to establish ranges of relevant data and apply minimum and maximums to each aspect of the formula. This very effective - in fact it is the same method that is used in the Pass Efficiency Rating.

Pass Efficiency Rating (40%)
Speaking of Pass Efficiency Rating (PER), perhaps we should start our analysis with that component of QUAR - PER comprises 40% of the end result for QUAR.

PER was developed in the early 1970's and takes into account four categories; completion percentage, average gain per pass attempt, TD frequency and interception frequency. For each of these categories, a score of one represented an average score (average in the 1970's). Each component is also subject to a minimum score of zero and a maximum score of 2.375.

The minimum possible score for PER is zero; this is achieved by scoring 0 in each of the 4 components. A perfect score for PER is 158.33; this is achieved by scoring a maximum 2.375 is each of the 4 categories.

When it comes to pass efficiency rating and the CFL, it should be noted that there are 3 variations of the formula in play. There are 2 versions used by the CFL and the third version is the correct one (detailed above).

The more common version of the formula that the CFL uses was described in the weekly stats package earlier this season as an "uncapped" version. That is to say, the min/max parameters that prevent statistical outliers from skewing the results are ignored by the CFL.

Various articles and stats packages issued by the CFL also indicate that the maximum score for PER is a "mysterious" or "theoretical" value. It is also suggested that the 158.33 score has been exceeded several times in recent years - of course this in NOT possible if the correct formula is used.

The second variation of the PER used by the CFL shows up in the post game stats packages. This version of the formula is simply the "uncapped" version with an arbitrary maximum set at 158.3 and an arbitrary minimum set at 0.

To demonstrate the difference between the 3 formulas, let's look at a simple example. A quarterback throws one pass which is completed for 19 yards; no TDs and no interceptions.

For completion percentage a score of 3.5 results which should be reduced to the maximum score of 2.375. For average gain a score of 4.0 is the result which again should be reduced to 2.375. For the TD calculation, the minimum score of 0 is attained (no adjustment required) and for the interception calculation, the maximum score of 2.375 results (again, no adjustment needed).

The "uncapped" version of the formula yields a PER of 164.58. The post-game version of the formula is 158.33. The correct formula results in a PER of 118.75.

Here we have 3 variations of the same formula with 3 very different results. The only difference between the formulas is how the min/max parameters are applied.

I would argue that since the CFL uses a broken version of the PER that QUAR is also a broken formula.

Let's assume that the CFL was to use the correct version of PER in its QUAR formula. The PER portion of QUAR would yield a score of 46.23 which is reduced to the maximum score of 40.

Instead of using PER as a single score, let's look at the 4 categories of PER as separate calculations with a maximum score of 10 for each. In the example above, each score of 2.375 would become a score 10 while the score of zero would remain a zero. That means that the 3 scores of 2.375 become 3 scores of 10 for a net score of 30.

Why is there a difference? Well 118.75 is 75% of 158.33 just as 30 is 75% of 40. The PER portion of QUAR applies min/max parameters to a calculation that has already had min/max parameters applied. By applying a second set of parameters, the PER component of QUAR becomes unbalanced.

We now know that the pass efficiency rating portion of QUAR is poorly constructed, perhaps the remainder of QUAR has some redeeming qualities. Let's take a look at the other components.

Winning Percentage (10%)
Many stats are used as a predictive measure for winning percentage. By incorporating winning percentage into the QUAR calculation it removes the predictive measure from the formula.

-   QUAR is NOT valid as a single game calculation. In statistics, a large sample size is required to make a result significant. When it comes to winnings percentage, there is a sample of one for a single game which makes this portion of the formula statistically insignificant.

-   For most of the components of QUAR, the average score is 70-80% of the maximum possible score for the category. For winning percentage, 50% is the best that can be achieved. A score of less than 50% is actually possible.

Example: Earlier this season, Calgary defeated Hamilton 60-1. In that game, 5 quarterbacks each led at least one drive to qualify for a QUAR rating. Bo Levi Mitchell recorded the win (10 points) and Zach Collaros was tagged with the loss (0 points). The three other QBs (Jeremiah Masoli, Andrew Buckley & Ricky Stanzi) each received a no decision which results in a score of zero - the same as a loss. As a result, the average score for winning percentage from that game was 2.0 or 20% of maximum (hardly consistent with the other QUAR categories).

Drive based calculations
Points per drive (15%)
1st downs per drive (5%)
Yards per drive (10%)
Sacks taken per drive (5%)
Fumbles per drive (5%)
QB rushing yards per drive (5%)

I commend the CFL for delving into drive based stats for the QUAR formula.

1.   It is possible for two quarterbacks to have the same stats for a game. If one QB achieves his stats on a lesser number of drives, then he is actually more efficient when it comes to stats based on yards, first downs, points scored, etc. The QB with a greater number of drives is better for stats based on sacks taken, turnovers, etc.
2.   In recent years, we have seen CFL teams use a specialist QB for short yardage situations and goal line packages. It makes sense to have all statistics accumulated on a drive attributed to the drive QB.

Three components of QUAR are well designed - 1st downs per drive, yards per drive and points per drive.

There are three other QUAR components which are also described as drive based - QB rushing yards per drive, fumbles per drive and sacks taken per drive. Although they are described as a per drive calculation, they are actually individual stats disguised as drive based numbers.

Example: Earlier this season, Andrew Buckley entered the game on a 3rd and 1 situation and he galloped 60 yards for a TD. Bo Levi Mitchell was the drive QB for Calgary and he was credited with the 1st down attained on the play, 60 yards of net offence and 6 points for the unconverted TD. It was Andrew Buckley and not Mitchell who received credit for the QB rushing yards (60) although Buckley was not credited with the drive.

This means that the specialist quarterbacks are accumulating stats to be applied against other drives that these men lead. When these specialist QBs are tagged with a sack or fumble, these stats are applied against other drives as well.

To put this into perspective, Andrew Buckley would have maximum score (5) in the QB rushing yards component of QUAR for his next 40 drives.

2nd Down Conversion Rate (5%)
The final component of QUAR is 2nd conversion rate. It makes sense to have this stat apply to the drive QB because in many instances it the drive QB who accumulates the first 8-9 yards. In practice, it is the QB who plays on the 2nd down play who accumulates the stats. As a result, these specialist QBs often achieve a maximum score in the 2nd down conversion category because they are always attempting to convert 2nd and short situations.

In conclusion, my initial concerns about QUAR were that a formula was not provided and the lack of historical data for comparison - it was based on data from 2009-2016. After breaking QUAR down into its components, I am just as concerned about the poor execution of the formula in general.

@stats_junkie

#### sully

Interesting analysis! I assume you wouldn't mind if I forwarded this the way of the league's stats people, stats junkie?
I'm the Senior Director of Technology for the Canadian Football League. Got technology concerns/ideas for the league? E-mail me at ssyed@cfl.ca and we'll talk.

#### ExPat

Stats junkie:

First of all, I'm impressed that you've actually reverse-engineered the formula.  I remember reading about it and wishing for more detail at the time.

But you've lost me with the commentary on winning percentage.

1) Why would the maximum be 50%?  For any single game, yes, there is a winner and a loser.  But there is no reason to average out the scores of the winner and the loser.  Over the course of the season, any QB's QUAR will reflect the cumulative winning percentage he achieves.  e.g. BLM is higher than Matt Nichols who is higher than Ray who is higher than Durant.  The first two of those will be well above 50%, while the latter is well below.

2) Why would you say there is a sample of one game?  As I noted above - I assume the QUAR takes into account all the games started, not just a single one.  Anyway, I don't think it was ever positioned as a single-game evaluation tool - like any other stat, it is more valid the more games it covers.

As for your drive-based concerns, many of these sound like they are only concerns for the second and third string QB who rarely get to play.  No disagreement there if the rating truly works as you've described.  But I noticed that the league doesn't even list these bit players on its QUAR page, which may be an admission that the rating is only really applicable to the starters.  And that's what most people will be interested in anyway, no disrespect to the Buckleys of the world.

Lastly ... please don't take any of this as a critique of your critique.  It just struck me that you're probably the type of person who likes to discuss this sort of thing, so I'm playing along.

#### stats junkie

Thanks for the feedback

I am sorry for not being as clear as I could have been. The maximum that an individual can score for winning percentage is 10 out of 10 (Bo Levi Mitchell has a winning percentage of 89.2% so he scores 8.92).

When considering the average for all QBs, 5.0 is the best average that can be scored. For each win there is an equal number of losses (i.e. one QB scores a 10 while another QB scores a 0). The average will trend lower than 5.0 for each non-starter who leads a drive and qualifies for a QUAR score because a no decision also results in a score of 0.

I agree 100% with your comment that statistics gain validity as the sample size increases. That is why a minimum of 300 pass attempts is required to qualify for pass efficiency rating and completion percentage records. A minimum of 90 drives has been declared to qualify for QUAR records.

Although these minimums exist for record purposes, pass efficiency ratings, completion percentages and QUAR ratings do exist for non-qualifiers. Here is an example from the Calgary team stats package that the CFL has published:

Earlier in the season, the weekly QUAR packages included a week by week ranking of QUAR ratings for each QB. I mentioned the sample size of 1 to indicate that a weekly QUAR rating is not statistically significant.

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