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November 12, 2007

Assist Rates

Posted by Eli in Stat Theory

To follow-up on my discussion of rate stats, I’m going to look at how this theoretical foundation can help evaluate passing stats created from the starting point of assists.

The basic assist-related player stats are assists per game and assist-to-turnover ratio. Assists per game is a time-period rate, while assist-to-turnover ratio is an opportunity rate (technically it’s an opportunity ratio of successes/failures, but it can easily be transformed into an opportunity rate of Ast/(Ast + TO)).

A lot of the advanced stats in basketball are simply refinements to traditional stats to remove potential biases. So from Assists/Game we can instead shift to Assists/Minute, which controls for playing time, or to Assists/Team Possession, which controls for pace (and playing time). We can even go one step further and shift to Assists/Team Play, which also controls for offensive rebounding (possessions don’t keep track of the extra plays that result from offensive rebounds).

If we turn to the opportunity rate of Ast/(Ast + TO), a flaw is noticeable. Some turnovers have nothing to do with passing - they may be the result of a player trying to score and traveling or committing an offensive foul. In other words, turnovers are not the corresponding failures to the successes of assists. So to make a better opportunity rate for assists, we first need to determine what constitutes an assist opportunity.

Assist Opportunities

One initial thought might be that turnovers are too broad - we need to limit assist opportunities to assists plus turnovers committed while attempting to get assists. 82games has a stat that seems to fit here - they divide turnovers into Offensive Fouls, Bad Passes, Ball Handling Turnovers, and Other Turnovers. So one could create an opportunity rate of Ast/(Ast + BadPassTO). 82games actually does list this on their player pages, though they give it as a ratio of assists to bad passes (in an upcoming post I will discuss why I find it highly preferable to present stats as rates rather than ratios). This isn’t perfect - the opportunities of assists plus bad pass turnovers leave out a lot of passes, but one can look at assists as a proxy for good passes, bad pass turnovers as a proxy for bad passes, and Ast/(Ast + BadPassTO) as a player’s percentage of eventful passes (i.e. those that result in either a made shot or a turnover) that are good passes, or his Good Pass Percentage. I haven’t examined Ast/(Ast + BadPassTO) in detail (mainly because calculating it requires compiling data from all of 82games’ individual player pages), but in theory I think it can tell us something useful.

Alternately, one could say that assist opportunities are better viewed in terms of made shots, as those are the only thing that can be assisted. Each assisted FGM is a success, each unassisted FGM is a failure, and total FGM are the opportunities. On a team level, this seems to make sense, and one occasionally hears or reads about a team’s percent of made shots that are assisted. Ken Pomeroy and Ed Küpfer have taken this approach to assist opportunities and applied it to the player level. Ken’s formula for what he calls Assist Rate is Ast/(tmFGM*Min% - FGM) where Min% = (5*Min/tmMin) (to approximate how many shots the player’s teammates made while the player was in the game).

This formula can be seen as an opportunity rate (successes/(successes + failures)) of this form:

Pomeroy's Assist Rate = Ast/(teammates' FGM assisted by player + teammates' FGM not assisted by player)

This seems a little strange in that the “failures” are shots that the player’s team made (a good thing) and had assists on (presumably a good thing). At the player level, these assist opportunities are in some sense a limited quantity that teammates compete for. One player’s success (assisting on a made shot) is another player’s failure (having a teammate assist on a made shot). Even if a player can increase the total opportunities by making better passes (leading to more successes of assisted teammates’ FGM), the failures seem to be beyond the player’s control. One can look at this stat as an opportunity rate of how large of a distributing role a player plays for his team (e.g. Player X does 30% of the assisting for his team). From this perspective I think it’s a valuable tool, though I wonder just how it translates when we start using it to compare players from different teams.

One objection to the previous approach would be to say that it’s not just made shots that could have been assisted - missed shots also could have been assisted FGM if a better pass was thrown that led to the shot being made. This would be an argument for including all teammate shot attempts (or perhaps just misses? or only those shots coming after a pass from the player in question?) among the assist opportunities. I’m not aware of anyone taking this approach, however.

One could also say that the failure that corresponds to the success of an assist is ball-hogging - a player taking too many shots of his own. Here one could measure Ast/(Ast + FGA) (I’ve occasionally seen this stat used, though normally in the form of a ratio of assists per shot attempt). The logical next step to this is to add turnovers to the opportunities (since we’re now looking at scoring tries as well as passing, these don’t have to be limited to just bad pass turnovers), and arrive at the point where assist opportunities are equal to all player touches/possessions. The idea is that every time a player handles the ball he has the opportunity to get an assist. This is the theory behind John Hollinger’s Assist Ratio.

Hollinger's Assist Ratio = Ast/(Ast + (FGA + .44*FTA + TO))

Assists are the successes, and times the player shoots (FGA + .44*FTA) or turns it over are the failures. It’s an opportunity rate measuring the percent of touches on which a player gets an assist (note that touches here mean player possessions - times the player was statistically involved in a play, rather than literal “touches” which would include passes that didn’t result in assists). It should be noted that here again we have the oddity of some positive events (made shots) being included among the “failures.”

What’s Next?

Theory only gets you so far, and at some point you have to get your hands dirty and look at how these different stats apply to real players and games. Are the league leaders players who are subjectively viewed as good passers? Do the same players top the leaderboard every year? Do players at different positions rate differently? What’s the average year-to-year correlation of the stat? How consistent is the stat when players change teams? Do any types of players appear to be systemmatically over- or under-rated? And so on. I may look at these questions and what they tell us about the different types of assist rates in a future post.

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