Tue Apr 23, 2013 11:16 pm

"starts missed" is different from "games missed" -- guys get injured while in the game, which is a partial game missed

defining "game" as "9 innings" (so being injured for rest of game at the mid=point would equal "one game" if it happened twice), the short-hand rule for fewer than 600 AB+W is this:

(23-inj)x(inj/2)

a 5 inj is thus 18x2.5= 45 (i.e., 45 x 9 innings missed), with a small margin of error the higher the rating

of course, the higher the rating, the greater the standard deviation -- the above formula is the average, without regard to standard deviation

it varies a lot if the guy plays every day and leads off, compared to a platoon guy who bats ninth

here's the math (from an old post):

FORMULA FOR AVERAGE NUMBER OF STARTS MISSED, DEPENDING ON INJURY RATING:

(BIA/C) / (1 + (BIA/EC) + 0.9(DB/EC))

"/" means "divided by"; unseparated letters are multiplied

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FORMULA FOR AVERAGE NUMBER OF GAMES MISSED, DEPENDING ON INJURY RATING:

This accounts for the "remainder of game" amount that's tacked on to every injury.

It's easier to think of "games missed" this way:

GAMES MISSED = "STARTS MISSED" TIMES (1 + (D/A))

The full formula (which just inserts the formula instead of the words "starts missed") is:

((BIA/C) / (1 + (BIA/EC) + 0.9(DB/EC)))(1 + (D/A))

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For the meaning of the letters in the formula, scroll down. Except for "A" and "I," the values can be modified based on your preferences. However, "E" (games in season) must be 162 for the formula to work perfectly; other numbers will introduce a slight error. (Note: Personally, I think "C" should always be 216.)

One remaining problem, or maybe not a problem: In any 162 game season, the actual limit of starts missed due to injury (though it would never happen) is 151, not 162. This fact (i.e., that a range of zero to 151 isn't the same as zero to 162) probably means that all results of the formula should be slightly lowered, perhaps by three percent, give or take. I don't know -- it's also possible that the effect is already baked into the formula.

A AVERAGE NUMBER OF STARTS MISSED PER INJURY

B AVERAGE NUMBER OF FULL-SEASON PAs ("EVERY-INNING" PLAYER)

C 216 (OR PREFERRED SLIGHTLY HIGHER NUMBER)

D "REMAINDER OF GAME" AVERAGE VALUE (MAXIMUM 0.5)

E GAMES IN SEASON

I INJURY RATING

A = 3.45432062 FOR SUB-600 AB+W PLAYERS

A = 1.82932095 FOR 600-PLUS AB+W PLAYERS

(These weird numbers account for the fact that end-of-season injuries are shortened if the season ends before the injury "runs out," thus lowering the overall average. I give the full numbers for spread-sheets; obviously, you can remove the last five decimals otherwise.)

B = your estimate of how many PAs an every-inning full-season player "would" have, on average (if preferred, note that full-time lead-off hitters have more than average, full-time 9th hitters fewer, etc.; also, slugging teams have more PAs on average, while low-scoring teams have fewer; and non-pitching starters will average slightly more in no-DH leagues). This number is for a mythical "ideal" player who is in every single moment of every game. This number excludes sacrifice bunts and H&Rs.

C = 216 if you think the long-term probability of any roll is a multiple of 1/216; use a slightly higher number if you think injuries are proportionally less likely than other types of "plays"

D = on average, how many "games" does "remainder of game" equal? The maximum possible is 0.5; the actual number is 0.47 or lower, depending on how you define the value of "remainder of game" immediately after a plate appearance has occurred. It would be irrational to use a number lower than, say, 0.25.

E = 162 for an opening-day-roster regular season; 174 for the maximum through game 7 of the finals. The number can also reflect a mid-season replacement. NOTE: any number different from 162 will introduce a slight error in "average starts missed per injury," the "A" in the formula: "A" will slightly increase for numbers between 163-174, somwewhat decrease for numbers lower than 162, and dramatically decrease for numbers substantially lower than 162. (That's due to end-of-season injuries factoring into the average, because sometimes the season ends before the injury "runs out"; this lowers the overall regular-season average in general, but especially for mid-season replacements.)

I = injury rating, from 1 to 6.

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Here's an example, using 670 for "B" (average full-season PAs) and 0.42 for "D" (average value of "remainder of game"). I left "C" as 216 and "E" (length of season) as 162. ("A" and "I" are not variable.) You can use other numbers if you prefer, with no change in the formula's strength except if you vary "E" substantially.

(If any of the below varies from what the formula "should" produce, then it just means I've made a number-crunching error -- I didn't double-check.)

Average starts and games missed per season for "sub-600 AB+W" players:

rating 1 -- 9.98 starts missed and 11.20 games missed

2 -- 18.81 and 21.09

3 -- 26.66 and 29.90

4 -- 33.70 and 37.80

5 -- 40.04 and 44.91

6 -- 45.79 and 51.35

Average starts and games missed per season for "600 or more AB+W" players:

1 -- 5.44 and 6.69

2 -- 10.53 and 12.95