# Defensive Runs: the Method

This article describes the method I have used to convert Sherri Nichol's Defensive Average (DA) numbers, using Palmer and Thorn's Linear Weights, into something I call "Defensive Runs". This is the number of runs that a player would give up at his position over league average.

My method consists of determining how many hits (of each type) and double plays the player would have given up if he had had a league average number of opportunities for the season. I define "league average opps" (LAopps) as total opps per position for the whole league divided by the number of teams in the league, i.e. the average number of opps at each position for each team for the whole year.

Here are all the intermediate and final numbers I calculate. Most of the primary information comes from the DA data, and are described here. Note that these quantities all represent events allowed (or made, in the case of double plays) above the league average rate; therefore, a negative NorH means you gave up fewer hits than a league average fielder, and a negative NorDP means you made fewer double plays than the average fielder.

The number of non-outs the player would give up above average in LAopps.

Nor2B = LAopps*(2B/APO) - LA2B.
Number of doubles that would be given up above average in LAopps.

Nor3B = ...
Same as Nor2B, only for triples.

Number of double plays that would be made above average in LADPopps.

MDP = DPI - Popps * LADP%
"Marginal double plays." This is the number of double plays the player made over what a league average player would have made in the actual number of DP opps the player had.

DR(hits) = (0.27+0.46)*NorH + (.34)*Nor2B +(.56)*Nor3B
Defensive Runs from hits (using T&P's Linear Weights). Double plays aren't factored in yet.

DR/G = [DR(hits) + .5*NorDP]/AGP
"Defensive Runs per Game". The number of runs the player would give up per game (above average) if he had an average number of opps (both regular and DP) and played the full game.

TDR = DR(hits)*Popps/LAopps + .5*MDP
"Total Defensive Runs", for the player's season (i.e. number of actual opps). Essentially DR/G pro-rated by player opps.

The Nor prefix means "Normalized to league average opps". LA... means "League Average", and Popps is "Player opportunties".

The AGP in DR/G is "average number of games for each team", which is generally close to 162 except for strike affected seasons (1994 and 1995).

DR/G should give us a ranking of how "good" the fielder is, regardless of actual playing time or balls hit to him. TDR puts the Popps back in, giving us a measure of how valuable the player's entire defensive season was. This gives us defensive equivalents of Dave Tate's MR/G and TMR from the MLV method, which was what I was trying to accomplish.

The difference between NorDP and MDP is in the number of opps used for the comparison. NorDP uses the league's average DP opps over a full season at the player's position; MDP uses the actual number of DP opps the player saw in his playing time. I had to split it out like this because DR/G assumes an average number of opps, while TDR uses the player's real number of opps. In practice, double plays don't affect the final DR values very much, but I didn't know that when I started, and doing things this way is more rigorous.

The weights are from The Hidden Game of Baseball, and are slightly different than the ones in TBIII. Too bad - I started before I got TBIII, and its a pain to change all the formulas in the spreadsheet, so I just left it the way it was. (It also appears that the TBIII weights are smoothed out to make each XB worth a standard increment above a single, which seems fishy to me).

You may have noticed that for the DPs, I used T&P's "Outs on base" weight of 0.5 runs. I don't know if this is a legitimate weight or if it is one of T&P's famous made up numbers (like for the fielding runs - the roundness of this number makes me suspicious), but I couldn't find a better one, so I decided to just use it. If anybody has a better idea, let me know.

Note that I have lumped errors, hits, and what not into "NorH" (which really should be NorNO for "non-outs", but its too late now). I don't know how to weight errors, so I decided to pretend they are singles. The number of real hits given up dwarfs the number of fielding errors anyway, so I'm not too worried about this assumption.