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Which tells us more: The last 7 at bats or 7 at bats against this pitcher?

Suppose that a batter is 5-for-7 over the course of his last seven plate appearances.  Pretty good two-game stretch, eh?  Now, suppose that another player has a career 5-for-7 mark against the day's starter, going back a few years.  (You could substitute "the reliever who was just brought in" if you like.)  Which is the more meaningful, the last seven PA's chronologically, or those seven PA's against the pitcher?

The proper answer to this is "Who cares?  Seven plate appearances isn't enough of a sample to tell us much of anything."  Let's leave that aside for a moment.  Let's enter the magical world inhabited by magical gnomes, Joe Morgan, and more MLB managers than I care to mention.  Let's pretend that you can actually draw serious inferences about future performance from seven plate app... I almost got through that with a straight face.

The bigger question is which is the better source of information: performance in the recent past against a different set of pitchers or performance against this specific pitcher over an array of years, given an equal number of plate apperances in each case.  I suppose it's an empirical question and one that I haven't really seen a study of directly.  I've seen a few things that lead me to believe that neither is a particularly good predictor, and so the "winner" between those two would be the winner of the "tallest grasshopper" award.

I don't care so much about which really is the better predictor right now.  What amazes me is that managers, I presume in the absence of systematic study, have very clearly made up their minds.  Consider how often you see managers make strategic personnel decisions (whether pinch hitting or moving someone around in a lineup or starting a guy who's normally on the bench) because "he has had success against Smith in the past."  It's rare-er to see a manager do something like move a player up in the lineup because he's had a couple of good days in a row.  (It does happen... just not as much.)  Managers seem to value pitcher vs. hitter matchup stats.  Now, if the reason that our batter had such great success against Smith is that Smith is a right-handed fastball guy and our batter eats those guys for breakfast, then that's fine.  Of course, the problem is that over 7 PA's, anyone can get lucky against anyone, and to interpret that as anything meaningful is just silly.

But if it's just a matter of not understanding small sample sizes, we should see a great deal more re-shuffling of lineups based on recent performance.  Managers seem to know, whether explicitly or intuitively, that the last seven PA's don't tell you too much about a player and so there's no need to change your strategy based on that.  Why is it that managers understand small sample sizes in one situation, but not in another?  There has to be another force at work.

I think there's something special in the manager's mind when it's batter vs. pitcher, and I think it's something that's a systematic error in thought across a lot of what passes for baseball analysis in the public sphere, something I like to call the virtus fallacy.  Virtus is a Latin word which is hard to translate properly.  It is the immediate predecessor of the English word "virtue", but in Latin, there's an added dimension.  Consider that the Latin word for man is vir.  Virtus was a mixture of courage, power, and honor that a Roman male citizen strove for.  It was a characteristic that was "demonstrated" by some noteworthy act, often athletic or military.  It might even be best translated as "manliness."  The fact that you did well showed that you had virtus and if you lost, well... you just weren't the virtus kind of guy.

Following this logic, if a batter goes 5-for-7 against a pitcher, it must be because the batter is more of a man than the pitcher.  And while it might be silly to believe that a plate appearance from five years ago when both pitcher and batter were five years younger in body and skill level would have a bearing on what's about to happen here, people generally assume that virtue is unchanging.  If you were more virtuous five years ago, you're probably still the better man today.  So, I should pinch hit you in this key situation.  If a batter is 5-for-7 over his last few at bats, he probably did it against four different pitchers, so there's no chance for him to "prove" that he is more virtuous than any of those pitchers.  He might be (and probably is) a good player or maybe he just had a good couple of days.  But he's not proven that he's better than a specific other man.

So, the next time you see a manager base a key decision clearly on batter vs. pitcher matchup data (and make a hilarious strategic blunder), it might not be evidence that he is statistically challenged.  Well, it is... just not in the way that you think.  He's shown in the past by what he doesn't do that he doesn't make decisions based on micro sample sizes.  It's just that his threshhold for accepting a "big enough to be relevant" sample size goes way down when the information is framed as a measure of who is somehow a "better man."

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