Tuesday, February 27, 2007


On the right side of the page you'll find a series of links to various statistics. Some of them are intuitive, others not so much. The one I'm most proud of is PAPER, which stands for Player Adjusted Probabilistic Effectiveness Rating. It is my attempt to condense all of an individual's box score numbers into one unified number.

PAPER represents the number of points a player would contribute to a league-average team over the course of a 16-game conference season, relative to the expected contribution of a hypothetical league-average player. Only statistics from conference games are used. PAPER does not simply assign a set value to the various statistics that individuals accumulate. Instead, it uses a model of a typical league possession and introduces the player's net contribution to determine what the expected scoring output would be.

When a team gains possession of the ball, one of three things is going to happen: they will turn the ball over, a player will be fouled and sent to the free throw line, or they will take a field goal attempt. (Actually, there is a fourth possible outcome--the end of a period or game--but we're not going to concern ourselves with that right now.) Within all but the first, there are additional possible outcomes. Free throws and field goals can be made, in which case points are scored, and the possession ends. They can also be missed, in which case either the defense grabs the rebound, and the possession ends, or the rebound goes to the offense, in which case the possession is renewed.

The first step in calculating PAPER is to determine the frequency with which each of these events occur. This is simply a matter of dividing the number of times an event took place in all conference games by the total number of possessions in all conference games.

The next step is to similarly find the frequency, on a per-possession basis, with which a player causes an event to occur. To determine offensive PAPER we'll need to know each of the following:
  • Turnover rate: how often the player turns the ball over
  • Foul rate: how often the player gets sent to the free throw line
  • Free throw percentage: self explanatory
  • Shot rate: the percentage of his team's shots a player takes
  • Field goal percentage: again, self explanatory
  • Make value: the average value of a player's make (if a player makes 20 shots from the field and 8 of them are 3-pointers, his make value is 2.4 points)
  • Setup rate: how often the player passes the ball to a teammate in position to make the shot (much more on this will follow in a later post)
  • Offensive rebound rate: the percentage of his team's offensive misses that a player rebounds.
Defensive PAPER makes use of the following individual statistics:
  • Steal rate: the percentage of defensive possessions on which a player records a steal
  • Team turnover share: all five players on the floor at the time of a non-steal turnover are assumed to deserve equal credit for its creation; this represents the frequency with which a turnover was forced while the player was in the game.
  • Foul rate: how often a player commits a defensive foul that results in free throw attempts
  • Player defensive unblocked FG%: the percentage of unblocked field goal attempts that opponents make while a player is on the court; much more on the philosophy behind this will follow in a later post. Unblocked FG% is used so as not to double-credit blocks.
  • Block rate: the percentage of opponent field goal attempts a player blocks
  • Defensive rebound rate: the percentage of his opponents' misses that a player rebounds
There are two key adjustments to be made so that PAPER will accurately compare players to the league average.

Pace is built into the system, since all statistics are rate stats based on per-possession (or per-miss, for rebounding) rates. Possessions are estimated based on the formula Ken Pomeroy lays out at the bottom of this page. Hopefully in the near future play-by-play data will be posted by more schools, and I'll be able to use counted possessions rather than estimates. (I'm not counting on this, though; some schools still aren't posting the standard box score for their home games, and others will only post them in PDF format. This is annoying.)

Size is the second key adjustment. Size functions as a basic proxy for position in this analysis. For the purpose of PAPER, which attempts to place each player in the context of an average team, this is an important adjustment to make. A team on the floor is not made up simply of five 6'7" 205# players. A 6'2" point guard has different responsibilities than a 6'10" post player, and both should be judged by how they fill their roles rather than how they perform against a generic standard. More importantly, the other four players on the floor will have different profiles in each case.

Once all the adjustments have been made, we're ready to construct the new model and find out how many points per possession the player would contribute to the average team. This can be found in the columns labeled RATE. Multiply RATE by the number of possessions a player would participate in, e voila! you've got PAPER.

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