Relievers' Run Expectation Report Glossary

by Michael Wolverton

ARP -- Adjusted Runs Prevented.

The number of runs that the reliever prevented over an average pitcher, given the bases/outs situation when he entered and left each game, adjusted for league and park. The exact formula for a reliever's ARP for a game is

(ER(s_S,P) - ER(s_F,P) + IF*ER(s₀,P) - R) / pe(P)

where

• ER(s,P) is the expected number of runs that will score in the remainder of an inning starting in bases/outs state s in park P,
• s_S is the bases/outs state when the reliever entered the game,
• s_F is the bases/outs state when the reliever left the game,
• IF is the number of innings the reliever finished,
• s₀ is a special state for the beginning of an inning (distinct from the state for no outs, none on),
• R is the number of runs that scored while the reliever was in the game, and
• pe(P) is the park effect for park P.

RRA -- Runs Responsible Average.

A rate stat indicating how well the reliever prevented runs per inning, comparable to park-adjusted RA. It's derived by using a formula that gives RA from Adjusted Pitching Runs (APR),except putting ARP in place of APR:

RRA = LgRA - 9 * ARP / IP

One major way this stat differs from RA: in extreme situations, it can be negative.

IRP -- Inherited Runs Prevented.

The expected number of inherited runners the reliever prevented from scoring. If this number is positive, the reliever is chopping points off his teammates' RAs; if it's negative, he's adding points to his teammates' RAs. It is calculated by looking at the runners on base when the reliever enters the game, figuring out how many would be expected to score given their location on the bases and the number of outs, and comparing that number to (1) the number of inherited runners who actually did score, plus (2) if any of those inherited runners were still on base when the reliever left the game, how many would be expected to score from that state. The formula for a reliever's IRP for a game is

(EIR(s_S,P)-EIR(s_F, P)-IR) / pe(P)

where

• EIR(s,P) is the number of runners that (1) the reliever inherited and (2) are currently on base in state s, who would be expected to score starting from bases/outs state s in park P,
• s_S is the bases/outs state when the reliever entered the game,
• s_F is the bases/outs state when the reliever left the game (EIR(s_F) is
• automatically 0 if the reliever doesn't leave in the same inning he started),
• IR is the number of inherited runners that scored while the reliever was in the game, and
• pe(P) is the park effect for P.

BRS -- Bequeathed Runners Saved.

The expected number of the reliever's Own Bequeathed Runners that were prevented from scoring by the reliever's successors. If this number is positive, then the reliever's bullpen-mates helped him by chopping points off his RA; if it's negative, the reliever's bullpen-mates added points to his RA. A reliever's own bequeathed runners are the runners who are on base when the reliever leaves the game, and who are the reliever's responsibility. The formula for a reliever's BRS for a game is

(EBR(s_F,P)-BR) / pe(P)

where

• EBR(s_F,P) is the number of the reliever's own runners on base when he leaves the game who would be expected to score starting from bases/outs state s_F in park P,
• BR is the number of the reliever's own bequeathed runners who actually did score (i.e., the number of runs charged to the reliever which scored after the reliever left the game
• pe(P) is the park effect for P.

APR -- Adjusted Pitching Runs.

Pete Palmer's method for calculating a pitcher's value in runs above average. Included for comparison with ARP. The exact formula we use differs in a few ways from the one used in Total Baseball. Our version:

APR = L * IP - R/pf(P)

where L is the average number of runs per inning pitched in the league, and pf(P) is the park factor for the player's home park P.

ARA -- Adjusted Runs Allowed Average.

The number of runs charged to the reliever per nine innings pitched, adjusted for league and park. The formula:

ARA = R * 9 / (IP * pf(P))

where pf(P) is the park factor for the player's home park P.

R -- Runs.

The number of runs the reliever was charged with.

IP -- Innings Pitched.

The number of innings the reliever pitched.

Diff. -- ARP - APR.

IRnr -- Inherited Runners.

The total number of runners who were on base when the reliever entered his games.

EIR_s -- Expected Inherited Runs, Start.

The number of runners inherited by the reliever over the course of the year who would be expected to score, given their position on the bases and number of outs when the reliever entered.

EIR_f -- Expected Inherited Runs, Finish.

Of those runners inherited by the reliever who were still on base when he left his games, the number who would be expected to score, given their position on the bases and number of outs when the reliever left.

IR -- Inherited Runs.

The number of runners inherited by the reliever who scored while the reliever was in the game.

OBRnr -- Own Bequeathed Runners.

The number of runners who were on base when the reliever left his games, and who were the reliever's responsibility.

EBR -- Expected Bequeathed Runs.

Of the reliever's Own Bequeathed Runners, the number who would be expected to score, given their position on the bases and number of outs when the reliever left.

BR -- Bequeathed Runs.

Of the reliever's Own Bequeathed Runners, the number who scored. Put another way: the number of runs charged to the reliever who crossed the plate after he left the game.

G -- Games.

The number of games the reliever appeared in. For a team, this is the number of pitching changes made during the year.

IRnr/G -- Inherited Runners per Game.

The number of runners inherited by the reliever per appearance he made.

EIR_s/G -- Expected Inherited Runs per Game.

The number of runners inherited by the reliever who would be expected to score, divided by the number of appearances by the reliever. This number gives a sense of the difficulty (in terms of preventing runs) of the average appearance the reliever made.

Comments and questions welcome. Copyright © 1999 Michael Wolverton.