## Pinch hitting: David Roher

This morning’s addition to the Pinch Hitters series comes from David Roher, who used his understanding of numbers and statistics to analyze the postseason decision to occasionally start Jose Molina ahead of Jorge Posada.

David lives in Westchester and is a history major in the middle of his sophomore year at Harvard. He’s a coxswain on varsity lightweight crew and the co-President of the Harvard Sports Analysis Collective, a club on campus. HSAC revamped its blog late last year and provides content for the Huffington Post. A fan of the Bombers since age 6, David wrote that he spends much of his time in Boston getting yelled at from passing cars on account of his Yankees sweatshirt.

———

Jorge or José? If you followed the Yanks through last October, that question was probably on your mind. In playoff games that A.J. Burnett started, Joe Girardi sacrificed Posada’s offense for Molina’s defense, particularly his ability to coax a “Good A.J.” performance out of the Jekyll-and-Hyde pitcher.

Part of what made the decision so controversial was that while we had no clue whether Burnett’s better splits with Molina were the real deal, we had a precise knowledge of the offensive impact… right?

Not so much. As I was analyzing the decision on the offensive side, I realized that none of the statistical tools available gave me an adequate answer. This study was the best anyone could do: give a runs-per-plate-appearance estimate. So I decided to create a new stat, Result-Change Probability, which I’m debuting on LoHud today. I don’t want to find whether the risk was worth taking – just precisely how big a risk it was.

When deciding which player should start over an entire season, I ask how many more games the team will win with one player over the other. In a statistical sense, this is usually one number – something like Wins above Replacement. But in reality, it’s two numbers, and two questions: how many games would the team win with Player A that it would have lost with player B? And how many games would the team lose with A that it would have won with B?

My idea is to apply that thinking to a single game, and to the Posada/Molina question in this case. We have to change the questions a bit: what is the probability that Molina’s offensive contribution would lose a Posada win, and vice versa? We still wouldn’t know about Molina’s effect on Burnett, but we’d have a much better idea of what it would take to overcome the offensive loss.

There are a lot of different ways to answer the question, just like there are many different ways to compute the number of wins a player is worth. The calculations for some of these methods are pretty intensive, and I’ll be developing them at our blog over the next few months.

To answer it here, I assumed that we were talking about one or two plate appearances per game, as Molina would be pinch hit for after that point. I created three models based on 2009 totals:

• The Yankees’ winning percentage based on the number of runs they score.

• The Yankees’ chance of scoring a certain number of runs in a game with only Posada, based on work by Baseball Prospectus and Baseball Musings.

• The Yankees’ chance of scoring a certain number of runs in a game with Molina starting.

As a comparison, here are the last two together:

After a couple (hundred million) simulations, here are the probabilities we’re interested in:

Yankees win with either: 52.02%

Yankees win with Posada’s offense, lose with Molina’s: 15.94%

Yankees win with Molina’s offense, lose with Posada’s: 14.19%

Yankees lose with either: 17.85%

That third figure is really bizarre, isn’t it? Molina is indisputably much, much worse on offense, yet his presence on that side of the ball alone sees an extra win roughly one out of seven times. It’s just a product of random chance – sometimes the inferior team wins. To get the final product, Result-Change Probability, subtract the third number from the second: Posada’s offensive presence increases win probability by 1.75%.

Contrast that small number with the potential effect of the starting pitcher (plus Molina’s conventional defensive skills), and Girardi’s decision might make more sense. Even if he thought that Molina had only a tiny effect on Burnett, it may have very well been worth benching Posada.