THE BOOK--Playing The Percentages In Baseball

I should also note that from 1959-2006, for all nonpitchers (25,000 seasons): while the aggregate OBP was .330, the simple average OBP was .303.

I e-mailed the author of the article and wrote this:

Mr. Katz,

The explanation for your “conundrum” in your article in this month’s SABR BTN is really quite simple and in fact you articulated it yourself and then dismissed it (I don’t know why you dismissed it because I could not follow your explanation and logic).

In fact, the “linear relationship” you found between AB’s per season and BA is quite expected.

The number of AB’s a player gets in a season, in general, tells us two things:

One, it tells us how well a batter performed in those AB’s irrespective of his true talent or true BA.  As you say in the article, if certain batters get a hit in one AB they are more likely to get a second (or more) AB.  If certain batters do well in 2 AB’s, they are more likely to get a third (or more) AB.  Etc.  This will automatically create somewhat of a nice linear relationship between BA and number of AB in a season (and in a career as well, BTW).

Two, how many AB’s a player gets, independent from the first reason above, tells us something about a player’s true talent level (true BA) as known or perceived by scouts, coaches, managers and GM’s.  For example, as you say in the article, a large proportion of low AB players are pitchers.  They are also defensive specialists, pinch hitter, reserve players, and otherwise poorer than average hitters.  If a player is perceived as a good hitter (e.g., if he had a high BA in the minors), then he is likely to get more AB’s in the majors in any one season regardless of how he performs in those AB’s as compared to a player who is perceived as a poor hitter.  For example, say you have two rookie players, A and B.  Player A was a good minor league hitter and scouts think he is a good hitter.  Player B was a poor minor league hitter and scouts think he will be a poor major league hitter (maybe he is a good defensive SS).  If both player A and player B get 100 AB’s at the start of the season and they both hit .200, player A is more likely to get more AB’s because they “know” that he is probably a better hitter than that.  With player A, they were either hoping he would hit better (he did not and they are done with him) or he was only going to play part time in the first place.

These two factors above (one is selective sampling and the other is an inference about true talent garnered from number of AB’s given to or gotten by a player) will conspire to give us exactly the type of relationship that you observed between AB per season and BA.

Sincerely,

Mitchel G. Lichtman (MGL)

In the Aug issue of BTN, one of the writers said:

The data shows that the best coefficient to use when weighting OBP is 1.8. This was also confirmed by Tom Tango though I am unawarewhere his study is located. In fact, The Hardball Times currently uses a stat called “GPA,” which adjusts OPS using a 1.8 coefficient for OBP and divides by 4 to make the stat on a similar scale to batting average.

http://www.tangotiger.net/ops.html
http://www.tangotiger.net/ops2.html

http://www.insidethebook.com/ee/index.php/site/comments/ops_and_clutchiness/

http://www.baseball-fever.com/showpost.php?p=643668&postcount=2
with 1.8*OBP+SLG having the best correlation

MGL:
The third factor is that a high-BA hitter will hit higher in the lineup than a low-BA hitter, thus giving him more PA/ABs.  The fact that the relationship continues even from 500 to 700 ABs presumably owes a lot to this third factor.  In fact, about the only way to get 700 ABs—which means you draw almost no walks yet still bat high in the order—is to have a very high BA.

I would say Katz’s conundrum is fairly clear:
the separate analytic prescriptions to discount small sample size and to regress to the league mean do not explain what is going on with these rarely used players. And who would have expected their performance to be so far below replacement level? [ I’ve summarized AL hitters at 1-24 PA from ‘73-’96 (effectively excluding pitchers) in this google spreadsheet. ] I’d think you’d have to press Mickey’s first cause very hard indeed to explain these poor results. But that is a testable explanation…

BTW, I once observed a related phenomenon with batting splits using data from the late 80s and early 90s; usually the more PA a player got, the more often he had to hit with the platoon disadvantage. Conversely as PA decreased, the % of PA with the platoon advantage increased; and the observed platoon differential became very wide, especially for LHB. That seemed to suggest “rustiness” due to infrequency of opportunity.

Tango (#3),

My bad, I should have written you for the link and passed it on to the author.

Guy (#4),

Good catch, I never thought of that when corresponding with Mr. Katz.  I assumed it was a positive correlation between good hitting and walks (and thus lower AB), but your explanation is probably the right one.

Yes, Guy, that is definitely true.  Since we know there is no “magic” here, the only explanation is my two points plus Guy’s (and perhaps something that has not been thought of yet).

The only really interesting thing is that the relationship is so linear, so steep, and so narrow (the correlation is so high).  I doubt that anyone would have “predicted” that would be the case, but it is not shocking.

The author wrote me back with the following (I hope he does not mind me re-printing it without permission):

With respect to your view that
>As you say in the article, if certain batters
>get a hit in one AB they are more likely to get a second (or more) AB. 
>If certain batters do well in 2 AB’s, they are more likely to get a
>third (or
>more) AB.  Etc.:

“It’s difficult for me to believe that managers will refuse to accord additional at-bats to players who fail at their first time at bat. Would a player be benched permanently after going 0-for-1 or 2?  Yet even among players with minute at-bat totals the correlation remains in force; between ABs and BAs for players with from 1 to 10 ABs in a season the figure is .92. Why should players with 8 ABs in a year nevertheless have a higher aggregate BA than players with 6, and why should players with 10 ABs have a higher aggregate BA than those with 8, 13 more than 10, etc.?  How does the linearity conduct itself with such precision at this magnitude? Do managers track these fine-grained differential that closely? Just one thought.”

What the author fails to realize, perhaps because he does not do a lot of baseball or similar research, is that all it takes to get a very selective sample is “an occasional bias.” In fact, most selective samples occur that way and the overall result can be quite significant.  What I mean is that in order to get the result we are talking about (a steep linear relationship) all it takes is a “tendency” for managers to bench players when they are doing poorly - not literally for a manager to bench all or even most players if they fail in the first or second AB.

What is happening is that there are lots of players who were likely only going to get one or a few AB’s in the major leagues for whatever reasons.  Of those players, and of the ones who got a hit an their first AB, some of them got another chance simply because they got a hit in that first AB.  Of the players who did not get a hit in the first AB, less of them were given another AB.  This bias continues almost ad infinitum throughout the spectrum of number of AB’s.  For example, if a marginal player is doing poorly after 200 AB’s, he is more likely to get benched than if he were doing well after 200 AB.

As I said, all it takes is a little bias, which clearly there is going to be, to create a selective sample that has the kind of impact that this author found.

In any case, there certainly is no “connundrum” (a question without an answer) here.  Because of the author’s cryptic and colorful style of writing, I don’t really know what he is suggesting in the article.  As I also said, I think that he mentioned the selective sampling issue, which is certainly true, and then for some reason, which I could not follow, he dismissed it.

Yes, we have to throw in “rustiness” into the mix.  Remember we found a large pinch hitting penalty in “The Book” (not literally IN the book). We have to assume that lots of those AB’s in low AB seasons were pinch hitting appearances.  It also may be that simply having few AB’s in a season, even if they were an occasional start, is like a pinch hitting appearance in that you are “rusty” from not playing much.

Remember that 27 of the 103 hitters with exactly 20 PA in season x had no PA in x-1 or x+1.  They were the Jamie Bubela.  That’s a heckavalot of players.

In order to analyze what mgl is saying about “if he got 1 hit, he’ll get another chance” is to look at these guys.  If you look at all guys with 0 to 30 PA in year x, and had no other PA in their careers (essentially guys that were pre-scouted that they will get September callups and will never play in the majors again), we can test mgl’s idea about the managers giving a chance to guys who have no shot at the majors who did something.

It’s alot of work, since now you have to go to PBP data.  I ain’t doing it, but if someone else wants to try…

Using ‘73 - ‘96 AL data (to eliminate pitchers’ batting), I do not see a linear relationship of PA to BA. I see a curve with inflection points around 60 PA and 600 PA. (The same relationship appears for OBP and SLG as well as BA.) The relationship does approach a linear one for the interval between 200 and 600 PA, but also almost flattens out between 450 and 600 PA. The selection bias identified by Guy probably kicks in significantly starting around 600 PA. [However, I guess what I did doesn’t inevitably contradict Katz’s claim of linearity, since his is based on AB, not PA].

I think the below-replacement-level performance of players who get few PA requires some explanation beyond selection bias because of extrinsic estimates of their talent. I have taken a very limited look at 17 position players with exactly 17 AB in a season from 2002-2006, and there may be a position bias: 7 of the 17 were catchers.  Only 2 of the 17 were established veterans at the end of their career; most were prospects or career minor leaguers, split roughly evenly between September callups and temporary mid-season promotions (or both), presumably as fill-ins for injured players.