Inj rating & Gs missed per season SOLVED AT LAST!

Inj rating & Gs missed per season SOLVED AT LAST!

Postby ClowntimeIsOver » Sun Aug 22, 2010 9:01 pm

I've calculated the exact number of games per season that a player will miss, depending on whether his injury rating is 1, 2, 3, 4, 5, or 6 (for less than 600 AB+W) or 1 or 2 (for 600 or more AB+W).

By "exact number," I mean the average over an infinite number of 162 game seasons (no post-season games), where a player would have 700 PAs if played in every inning of all 162 games.

Of course, that 700 PAs number is an estimate, so the "exact" figures would have to be adjusted slightly up or down if someone knew the actual average of full-season PAs (ignoring bunts, H&Rs, and other SOM exceptions); until someone figures out that actual average over a big sample of every-inning-every-game players, 700 will have to do. With a slight error (because the formula doesn't allow a simple pro-rated substitution for 700), a different number can be pro-rated to 700.

The "exact" number reflects the fact that a player who gets injured will have fewer PAs than the 700 figure. In other words, if it were calculated as just "per 700 PAs," it would show the number of missed games over MORE than one season, because it would take more than one season for a player who ever got injured to have that many PAs. This problem was fixed by "solving for x", where x is the number of missed games and is also the number of games subtracted from 162 to adjust the 700 PA number downwards.

One problem: the table below does not account for injuries that are shortened because the season ends. For example, a 6-game injury in game 158 ends up being a 4-game injury. Therefore, the average numbers have to be slightly lowered. I'll try to fix this eventually. (The problem is minor for 600+ PA guys, small for injury-1 ratings, and increases in importance up to injury-6 ratings.)

-------------------------
INJURY TABLE (ASSUMING 700 PAs = 162 FULL GAMES)

inj rating = average number of missed starts AND missed games (respectively) per season

(As explained below, "remainder of game" is counted as "one-half missed game," so e.g. a 3-game injury is 3.5 missed games but only 3 missed starts.)

The result for "less than 600 AB+W" players:

inj 1 = 10.883 missed starts and 12.003 missed games
inj 2 = 20.396 and 22.349
inj 3 = 28.782 and 31.361
inj 4 = 36.230 and 39.280
inj 5 = 42.890 and 46.293
inj 6 = 48.879 and 52.549

The result for "600 or more AB+W" players:

inj 1 = 5.932 and 7.126
inj 2 = 11.445 and 13.651
--------------------------------

EXPLANATION:

The 20-side roll for "number of games missed" works out to an average of 3.6 starts missed and 4 games missed per injury (for "less than 600 AB+W" players) or 1.9 starts missed and 2.3 games missed per injury (for "600 or more AB+W" players).

This is a simple thing to figure: you just add up the number of games or starts missed for each 20-side roll (anywhere from zero to 15.5), and divide by 20. The result is the average number of games missed per injury. NOTE: I counted "remainder of game" as equaling one-half of one game, which in fact in the long run is the correct average for non-platoon starters. For example, a 15-game injury is counted as 15.5 missed games (but only 15 missed starts) because it's "remainder of game plus 15." However, this means that "missed games" is slightly exaggerated for bench players and platoon players, because they'll get injured in pinch-hits and late-game substitutions where they miss only an inning or two in the "remainder of game," rather than an average of half a game (but, again, "half a game" is accurate in the long run for non-platoon starters, slightly excepting guys who get pulled for defensive subs). Note that the "bench and platoon" difference is really negligible for longer injuries, because the "remainder of game" part is such a small proportion.

This is how I got the results:

Since injury ratings are per 216 PAs, the relevant numbers are 216; 700 (estimated average full-year PAs for an every-inning 162-game player); 700/162 (average PAs per game); the injury rating (1 through 6); and 4 (average number of games missed per injury). The last number is changed to 3.6 for missed starts; for 600+ guys, the number is 2.3 games or 1.9 starts.

700 minus (x times (700/162)) = number of PAs per season, where "x" is the number of missed games.

Divide this number by 216 to get the number of times a "one" chance will occur (i.e., a roll that occurs 1/216 times).

Multiply the result by the injury rating (to change 1/216 to 2/216 or 3/216 or etc.).

Multiply the result by 4 (games per injury), or one of the other options.

The result is "x" (number of games missed per season). But because "x" is already part of the formula (see the beginning, above), it has to be moved out of one side of the equation to the other.

I may have done something wrong, but I think it ends up being this:

-----------------
x equals

4 times the injury rating times 700/216

divided by

1 plus (700/(216 times 162) times 4 times the injury rating)
-----------------

That last part involves the number 700/(216 times 162), which is .020004572, so the formula is easier to read as

-----------------
x equals

4 times the injury rating times 700/216

divided by

1 plus (.020004572 times 4 times the injury rating)
-------------------

[Again, "4" is changed to 3.6, 1.9, or 2.3, depending on which part of the above table is being solved.]
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Postby the splinter » Mon Aug 23, 2010 11:10 am

dude....get a date. :lol:
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Postby junkfood johnny » Mon Aug 23, 2010 11:27 am

i wholeheartedly agree with the splinter on this one.
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Postby visick » Mon Aug 23, 2010 11:42 am

Now I remember why I became a chiropractor. The math part of my grey matter broke years ago.
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Postby MARCPELLETIER » Mon Aug 23, 2010 1:19 pm

This looks solid. Best work so far on the topic.

But I would say that you slightly overestimate the number of games/at-bats injured, and I formulate this opinion based as much on observation than on theoretical reflexion.

On the mathematical side, the reason why the numbers are slightly overestimating injury time are the following:

1- (I know you mention something about this, but I thought I should start with this anyway...) 700 PA seems a bit high for an estimation of at-bats in one full season for our strat leagues. I found out that taking 672 AB for a full season is a better estimate (maybe the seasons I play in are biased towards pitching, though). Because of the way you made your calculations, this increased value impacts on the value of "x" and on the value of injured time, which should be slightly reduced consequently.

2- As you mention, injuries might be cut off because of a season ending. But also, by default, players start the season all healthy. So injury-risk is truncated at both sides of the seasonal playing time, which reduces the real impact of injury time.

3- Also, you made a fair estimation that, on average, rolls for an injury will happen once every 216 AB (for someone with an injury-risk at 2 or 12), but in reality, it's more complicated. At one extreme, a player may be injured right off the first at-bat. And at the other extreme, a player may never be injured at all during the whole 700 PAs or 672 PAs that the season lasts. So there is an unbalance, towards the possibility of not being injured. While such events are rare, their weight on the overall average is larger, so the overall impact of such rare events are not negligeable, and they favour the probability of not being injured. I don't know the precise effect of this, but it is likely that it slightly reduces the average of injury frequency.

To illustrate this last point (with a player with 1 chance of injury): if the true average of any player to be injured was once every 216 PAs, then the risk of being injured right off the first at-bat should be equal to or greater than the risk of NOT being injured after 432 PAs. But it is not. The risk of being injured at the first at-bat of the season is 1/216, or 0.46%. The risk of NOT being injured after 432 PAs is: (215/216)^432, or 13.47%. In fact, the risk of NOT being injured during the whole season is 4.42%, a small but not insignificant effect.

This logic applies for any situation where you play with very small percentages. The overall impact is that it shiftes the "real" mean injury risk to some number smaller than 1/216 (perhaps 1/234, I would say from a very rough estimate)

So, to express my point differently, your numbers correctly estimate the number of games lost for injuries for roughly 87% of players with a 2-12 injury risk (after correcting for points 1 and 2 above; these are the 87% of players who will get injured between the first and the 432th at-bat), but overestimate the risk of injuries by a good margin for 13% of players (those who will still not have been injured by the 432th at-bat).

4- On average, players have more impact than half-a-game before they get injured, because they complete the at-bat during which they get injured (if it were 50%, then the risk of being injured before the at-bat would also have to be 50%; in fact it's 0%--injuries always occur after the at-bat). The exact percentage of injury-time for any game a player gets injured is somewhere around 38%.

5- In Strat, injuries don't happen during bunts, squeeze, and hit-and-runs.

All five effects described here are small, not to say very small. But their overall interplay explains why I observe, on an empirical level, an average of 8-9 missed games per season (for 2-12 injury-risk with less than 600 AB) rather than numbers closer to 11.
Last edited by MARCPELLETIER on Mon Aug 23, 2010 6:27 pm, edited 2 times in total.
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Postby MARCPELLETIER » Mon Aug 23, 2010 1:40 pm

BTW, what I call "emperical observation" is what I got through simming, when I did last year. I simmed something like 50 seasons of my team, and number of games played by injured players was one stat I was looking at. Going from memory, the average was 8.6 games per season for 2,12-injury risk players with less than 600 PAs.

Perhaps I should get a date too!!

(Perhaps it was 9.6 games. I would have to check.)
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Postby dharmabums » Mon Aug 23, 2010 4:53 pm

This thread just sends a chill down my geeky spine. Very cool stuff guys. Thanks. :D
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Postby mesquiton » Mon Aug 23, 2010 4:56 pm

Hey marcus, looks like you've got a date just for the asking!
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Postby dharmabums » Mon Aug 23, 2010 5:11 pm

[quote:a759f22195="mesquiton"]Hey marcus, looks like you've got a date just for the asking![/quote:a759f22195]

Or the beginnings of the geek convention. Atlantic City here we come!!! :P
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Postby ClowntimeIsOver » Mon Aug 23, 2010 6:22 pm

[quote:9510abbca9="the splinter"]dude....get a date. :lol:[/quote:9510abbca9]

Seriously? A guy who's had TWO HUNDRED AND EIGHTY NINE SOM TEAMS is telling ME (35 teams) to get a date?

In any case, it only took an hour or two on a Sunday afternoon. Today, being Monday, I had to get back to my nerdy analysis of my substantial stock portfolio (which is beating the S&P by 12 percentage points, year-to-date).
Last edited by ClowntimeIsOver on Mon Aug 23, 2010 7:12 pm, edited 1 time in total.
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