## YAC Receiver Correlation

In the last post, I compared the year-to-year correlations of yards after catch (YAC) of quarterbacks and receivers. This follows a series of posts regarding an improved QB rating system.

I found that the correlation for receivers was signficantly stronger than for quarterbacks. This result bolstered my theory that QBs do not contribute to the YAC of their receivers by virtue of their abilities.

Another author had found that between '05 and '06, the year-to-year correlation of QB YAC was 0.33 for all QBs, and was 0.41 for QBs who remained on the same team. I previously discussed why this data could not be used to conclude that QBs are consistent in producing YAC, and in fact, suggests the opposite conclusion.

My analysis of year-to-year receiver YAC was for all receivers, regardless of QB or team changes, in the '02-'06 seasons. I found that the correlation between year-pairs ranged from 0.41 to 0.57, and averaged 0.47.

Since I posted that, however, I realized it was an unfair comparison because I included receivers who may have only had a handful of catches in any given year. This would cause the random component of variance to dominate the calculations. So I repeated the analysis using only receivers with at least 20 catches in every year from '03 through '06. The resulting sample contained 65 receivers. The critical level of significance for n=65 is 0.25.

 Year Pair Receiver r '03 - '04 0.71 '04 - '05 0.78 '05 - 06 0.69

Those are strong correlations by most conventional standards. And they are much stronger than the 0.33 correlation for QBs, especially considering the exponential nature of correlation coefficients (r=0.8 is four times stronger than r=0.4).

But the correlations of QBs and receivers still cannot yet be directly compared. Here's why: Most QBs have several hunded completions in a season. But my receiver list contains players with as few as 20 receptions, which leaves a lot more room for random variance in the receiver variable than the QB variable. If the influences are independent, the variance of YAC can be written as:

var(YAC) = var(QB) + var(rec) + var(def) + var(random)

The smaller the number of repitions, the larger the share of var(random) would be, and therefore the smaller the share of var(rec) would be. But when we look at r for quarterbacks, var(QB), the high number of repititions means the var(random) will be much smaller as a share of the total variance. We'd therefore expect to see a stronger coefficient for the QBs. (This is why I said statisticians grate their teeth when a couple correlations are used to make a conclusion).

In other words, the bar should be set higher for QB correlations because they have so many more repititions in a season. If QBs and receivers had equal influences on YAC in each play, we'd expect to see a much higher correlation for QBs than for receivers. But we don't, we see the opposite.

Accordingly, a correlation as high as 0.70 for the receivers is stunningly strong, particularly when compared to the QB correlation of 0.33. Had the receivers tended to have the same number of receptions as QBs have completions, the total variance would not have so much random luck within it, and the correlation coefficient would be significantly higher than 0.70.

[The sabrmetricians in baseball have gone around and around on this issue for years. The year-to-year correlation methodology was used for decades to disprove the existence of "clutch" hitting. The theory was that if clutch performance was a skill possessed by batters, it should endure from year to year. The resulting correlations were very small, and the conclusion was there is no such thing as clutch. Not until recently did a researcher point out how much luck influences batting in the first place, and the low number of potential clutch plate appearances exacerbated the effect. So if clutch hitting existed, we'd see a small correlation anyway. (As it turns out clutch performance has indeed been disproved, but through other methods.)]

We can now take a step further. Previously we had evidence that YAC does not belong to the QB. Now we have evidence that the lion's share of it indeed belongs to the receiver.

### 2 Responses to “YAC Receiver Correlation”

1. Anonymous says:

Are your Rs in this post Rs or R^2? In other posts you've been pretty clear, but this one left me a bit muddled. Great analysis!

2. Brian Burke says:

Miles-These are all r's.