"Establish the run" might be the three most over-used words in football analysis. Bad football analysis, that is. You would give yourself one heck of hangover playing one of those fraternity-type drinking games going bottoms-up every time you heard that phrase on a Sunday.

Establishing the run could mean a lot of different things. To most people I think it means that offenses will demonstrate the willingness to run frequently, hoping that defenses will bias towards stopping run later in the game. I think this interpretation has been debunked fairly thoroughly, so I'm going to look at it from a different angle.

Establishing the run could mean running often early in a game in hopes that a defense will weaken, and runs will be longer later in the game. I think this is more plausible theory. It makes sense on the surface. Running plays tend to batter the defense while passing plays allow the defense to go on the attack. By the fourth quarter, it's believable that defenders would be battered and fatigued. Defenders would get off of blocks slower and tackles would be sloppier. Watching Baltimore's two consecutive long TD runs late in the fourth quarter at Dallas in week 16 made me wonder: Does running frequently lead to longer gains?

To answer the question, I compared the average gains from running plays based on when they took place in a game. By "when," I mean which run play it was, not what time it took place in the game. In other words, if running frequently fatigues a defense, then the gain of a team's 30th run should tend to be longer than its first run.

We'd expect that the more runs an offense calls, the longer the subsequent runs should tend to be. The graph below depicts the average gain of each rushing play based on its order in which it was run. It plots the average gain of each team's first run of a game, 2nd run, 3rd run, ...etc.

(Data is from all regular season games from 2000-2007. All runs except kneel-downs were included.)

There's no increase in average gain as the number of runs increase. A team's very first run of a game is just as long as, if not longer than, the 20th, 30th or even 40th.

This result is despite the expectation that teams that are good at running would naturally run more often. Teams that are ahead toward the end of the game Those teams would therefore be the ones that we'd expect would accumulate more attempts. If so, we'd see the average gains increase as the run attempts mount. But we don't.

So this is evidence that runs are just runs, no matter how many have come before them. An offense can expect the same average gain on the first snap of the game as on the 80th, after 30 previous run plays.

The run may not set up the run, but does it set up the pass? Does running frequently allow longer gains on passes later in the game? I'll look at that question next.

Edit: Follow-up here that addresses many of the comments below.

In his Tuesday Morning Quarterback column from a week ago, Gregg Easterbrook tells us about his favorite high school state champions, the Pulaski Academy of Arkansas. They never punt on 4th downs, and they almost always win. To his credit, Easterbrook has been out in front of the go-for-it movement. (I'm not a regular reader of Easterbrook, and now I remember why. His columns are the length of short novels, frequently meandering into bad politics and pop-culture commentary.) Thanks to all the readers who alerted me to the Pulaski article at Rivals.

FO has a good article on the hidden value of pass interference penalties. Devin Hester is used to illustrate how the value of speedy deep-threat WRs is obscured by the fact that they aren't credited with the penalty yards they draw.

Dave Berri from Wages of Wins posted his weekly QB and RB rankings based on econometric models like my own. Note who the top QB is--Pennington. Dave may have an updated ranking by the time you read this. He also adds his thoughts on an ongoing discussion between amateur/internet sabermetricians and academic researchers.

Last week I wrote that the Redskins collapse this year may have been due to too few interceptions. In other words, their very low interception rate may have become an end in itself, rather than a by-product of a good passing attack.

I wrote, "You can guarantee zero interceptions by playing in an extremely conservative way, tossing short passes, taking sacks, or throwing the ball away anytime a defender is in the same zip code as the receiver. You can minimize interceptions, but you'll lose every game doing it. At some point in risk-reward continuum, there is an optimum level of risk in passing strategy."

The PFR blog appears to have picked up on the theory a few days later and added some evidence that supports it. We appear to be of like mind on the topic, as the PFR post says, "If you never throw an interception, you’re taking too many sacks, throwing too many balls out of bounds, and getting too many four yard gains on 3rd-and-9. So if zero is not the optimal turnover rate, then what is?"

I have a couple small constructive suggestions. The post finds that teams that have high rates of turnovers per non-scoring drive score more points than teams with low turnover rates. First, I'd suggest looking at interceptions per non-scoring drive instead of all turnovers. Interceptions more than fumbles are functions of an offense's risk-reward balance. I'm not sure if it really matters, though. The effect is the same no matter what kind of turnover it is. But it's worth looking at.

Second, and more importantly, I'd look at point differential rather than points scored. In other words, do teams with more turnovers per non-scoring drive outscore opponents? It may be that they score more points themselves, but they may be allowing even more points due to handing favorable field position over to their opponents.

Smart Football, the best Xs and Os site on the web, dissects Paul Johnson's 'Flexbone' offense at Georgia Tech. I'm a big Paul Johnson fan because he brought Navy's program a lot of success over the past several years. I'm not a big Xs and Os guy, meaning I'm not an expert. But I'd like to learn more. Can anyone suggest a good book that digs deep into NFL-style offensive or defensive systems?

The updated Super Bowl probabilities are out. Despite the #1 vs. #2 match-ups in week 16, there aren't any big changes this week. The big drop-off between the top 5 and everyone else is mostly due to two factors: 1) just clinching a playoff spot, and 2) getting a first round bye.

Courtesy of Chris at NFL-Forecast.com, here are the latest playoff probabilities for each team.

These are calculated using the NFL-Forecast software mini-app that runs thousands of simulated seasons. The outcomes are based on game-by-game probabilities with every crazy tie-breaking scenario factored in. Chris has used the probabilities from Advanced NFL Stats as his default game probabilities for the past two seasons.

There are two tables below. The first lists the probability that each team will finish in each place in their division. The second table lists the overall playoff probabilities, broken down by seed.

Quite a few people thought I was nuts when I said Miami and San Diego had the inside tracks on winning their divisions halfway through the season. If there was ever a week to have 'Sunday Ticket,' this would be it. Eight of the 16 games have do-or-die playoff consequences--not just seeding implications. Eleven of the 32 teams are still on the playoff bubble in the final week of the season.