correct formula: injury ratings = starts&games missed

[ClowntimeIsOver]

[edited 8/27/10 to correct for mid-season replacement formula]

This has been corrected from the original attempt, which was posted Sunday. Also, now it's stated in parameters, so that it can be used in a spreadsheet and different numbers can be entered for "average PAs," etc. Thanks to "Marcus" for suggestions.

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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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Approximate mid-season replacement correction (results are approximate for the decreased value of "A"):

1) For both formulas, decrease "A" by 1% for every 20 games that have already been played (e.g., if 100 games have been played, decrease "A" by 5%).

2) After deriving the full result from either formula, multiply THAT result by E/162, to pro-rate for the remainder of season (as explained below, E = number of games left in season).

If figuring 174-game average (to include full post-season for opening-day roster), increase "A" by 0.5%; multiply formula's result by 174/162.

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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 600-679 AB+W PLAYERS
(680 or more AB+W players are ignored and don't figure into the results)

(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, somewhat 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.) [NOTE: This problem can be virtually eliminated by applying the method in the above edit for "Approximate mid-season replacement correction."]

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

[The Biomechanical Man]

Thank you, wachovia (and Marcus) for your work on this. It seems to me that this is a very exact equation to give the correct averages, but there will be a lot variation player-to-player, season-to-season. In other words, if I build a team with a stud player with less than 600 PA and a 1 injury rating, he is predicted to miss 9.98 starts. I might get lucky and the player might miss 5 starts all season. Then again, he might miss 18. So, personally, I don't need an exact formula. Based upon this math, I'm going to use the following when I build my teams:
[color=blue:b5660bc8de][b:b5660bc8de]
If a <600 PA guy has
a 1 injury, he'll miss about 10 games.
A 2 injury will miss about 20 games.
A 3 injury will miss about 30 games.

A >600 PA guy will about 6 games.[/b:b5660bc8de][/color:b5660bc8de]

[MARCPELLETIER]

sounds pretty exact.

Glenn,

if you were to use a simple formula, I would rather use the second number, the one with the "remainder" game, especially since the 680PA+ seem to never be injured, not even for the rest of the game. Also, I would a declining step-by-step, like the following:

For players with < 600 PAs
1. +11 games
2. +10 games (21 games)
3. +9 games (30 games)
4. +8 games (38 games)
5. +7 games (45 games)
6. +6 games (51 games)

[ClowntimeIsOver]

The 680+ players don't figure at all into either formula (deliberately -- they equal "zero games missed"). The numbers for "A" are the relevant ones for this point: one is for less than 600, and the other is for 600-679 (incidentally, those specific numbers don't figure into the formulas, except in choosing which "A" number to plug in). I'll edit my post above to reflect this.

Glenn: thanks. I'm sure you understood this, but I want to reiterate that the numbers at the bottom of my post are just an example, using "670 PAs" as the average for a full-time player, and "0.42 games" as the average value of "remainder of game." Different inputs would produce slightly different results. A poster on this board (I think it was in mid-February) gave numbers which indicate 700 full-season PAs on average for DH teams, and 712 PAs on average (excluding pitchers) for no-DH teams. Marcus suggested these numbers were a little high. Also, I don't know if these numbers include bunts and H&Rs, which should be subtracted.

Also, what you said about variations in different seasons (sometimes a guy never gets injured, sometimes he gets injured over and over again, sometimes the injuries are short, sometimes long) doesn't affect anything. The formulas reflect the AVERAGE, and so they fully take into account every possible season, normal or weird or very weird. In the long run (which is what player values are based upon), the formulas will be almost completely correct, relative to the accuracy of the inputs (most important, the value of "B").

A really interesting point is that an injury-1 player with less than 600 AB+W is worth only 3% less (in starts played on average) than the same player would be worth with 600-679 AB+W. Joe Mauer, etc. etc. Also, this suggests that 600-679 players may not be as cost-effective as sub-600 injury-1 guys; the usual discount seems to be much more than a mere 3%, meaning you probably get bargains almost all the time, leaving more $$ for the bench or whatever.

A general point: the numbers should be reduced for catchers, who periodically become "bullet-proof." This wouldn't much affect teams whose back-up catcher is only used if absolutely necessary (because only the cheap back-up catcher would gain most of the benefit), but it would have a significant effect on teams that use both catchers a lot (i.e. it would reduce the total number of starts missed for the two catchers, both of whom are equally valuable).

[ClowntimeIsOver]

[quote:961af787dc="marcus wilby"]For players with < 600 PAs
1. +11 games
2. +10 games (21 games)
3. +9 games (30 games)
4. +8 games (38 games)
5. +7 games (45 games)
6. +6 games (51 games)[/quote:961af787dc]

Wow! Cool -- I didn't notice that at all.

So this makes an easy rule of thumb:

for a sub-600 player, to figure how many games will be missed (on average) in a 162-game season:

"23 minus the injury rating" times "half the injury rating"

e.g., inj-4 = 19 times 2 = 38
inj-6 = 17 times 3 = 51

or (which is the same thing) "the injury rating" times "23 minus the injury rating," cut in half

inj-5 = 5 times 18 = 90, and half = 45
inj-1 = 1 times 22 = 22, and half = 11

[The Biomechanical Man]

Approximate number of games missed
[quote:1292fe7053="marcus wilby"]
For players with < 600 PAs
1: 11 games
2: 21 games
3: 30 games
4: 38 games
5: 45 games
6: 51 games[/quote:1292fe7053]

I like this. Simple and useful. :)

[MARCPELLETIER]

[quote:bed292e78d]
1. +11 games
2. +10 games (21 games)
3. +9 games (30 games)
4. +8 games (38 games)
5. +7 games (45 games)
6. +6 games (51 games)[/quote:bed292e78d]

One mnemonic: numbers always make up 12
(when you add up chances of injury with the increased number of games compared to the preceding level).

[ClowntimeIsOver]

bump

(just once -- because the preliminary thread keeps getting bumped, but this thread gives the correct solution)