After some R coding to count the necessary information, I fit a multiple logistic regression model with:
- I.run: binary response, the indicator of whether the run scored,
- I.PR: an indicator of whether there was a pinch runner,
- lead: the score difference between the two teams (either -1, 0, or 1). Maybe somewhat surprisingly this makes a difference,
- outs: the number of outs when the runner first got on base (this is categorical because there's no reason outs would be related linearly to the log odds of scoring),
- I.2nd: an indicator of whether the runner was on 2nd base (the alternative is the reference level, 1st base),
- player: identity of the original runner (for parsimony I'm lumping all pinch runners together as being fast guys),
I'm making the assumption that there are only three outcomes stemming from possibly pinch running: the run won't score either way, it will score either way, or the PR would score when the original player would've been stranded. I'm ignoring the possible effects - that is, on subsequent runs in this inning that are less likely because of the extra out - of the slow runner being more susceptible to getting thrown out on the bases (by a force play or otherwise). I imagine the frequency of being thrown out when the PR wouldn't have been is low compared to the frequency of times this slow runner just gets held up by the 3rd base coach when the PR wouldn't have been. Regardless, it might get too complicated figuring out how much this extra out affects the distribution of run scoring in the inning. However if I fail to find that pinch running is beneficial (in part 3), I may have to revisit this assumption. I can think of two different ways to weaken it:
- instead of the binary response, the response could be a count of the number of runs scored in the inning subsequent to this runner reaching base. I balked at that for this entry because in an entry last year, I had trouble getting R to do multinomial regression with covariance matrices provided - but that data set was bigger, so it may be worth a shot in my next post,
- instead of separating steps 1 and 2, just fit a binary logistic regression with win/loss as the response. This gets at the question directly, but I'm worried that the response will be pretty noisy.
There are a number of interactions that could potentially enter the logistic model, but only two interactions were close to significant (for more on the I.PR:player interaction see the end of this post); they are shown below. I used the default treatment contrasts in R for I.PR, lead, outs, and I.2nd, but sum contrasts for the players, so the intercept corresponds to the log odds of scoring for the average player, running for himself, after a lead-off single when trailing by one run. The parameter estimates and their p-values are listed below; the residual deviance is 32836 on 27742 degrees of freedom, making the overdispersion parameter estimate less than 1.09, so I didn't bother allowing for overdispersion. It would not have changed any of the conclusions anyway. I've included David Ortiz's estimate below because he will come up again in a moment.
| lead0||0.104||0.004 **|
| lead1||0.216||2.48e-07 ***|
| I.PR:lead0||0.161||0.046 *|
| I.PR:lead1|| 0.146||0.209|
|Ortiz|| 0.674||0.010 *|
The coefficient estimates of interest are the ones concerning I.PR. The main effect for I.PR says that exp(0.086) is the multiplicative change in the odds of scoring created by pinch running when lead=-1, and similarly adding the main effect to each interaction term allows estimation of the multiplicative change in the odds when lead=0 or 1. I expect the more meaningful number in part 3 will be the additive change in probability - this will be different depending on the situation. The following table compares the probability of scoring a run with no PR to the probability of scoring a run with a PR in all 18 situations. This is done for an average player. A Taylor series expansion was used to estimate the variance of the difference of the probability estimates, and normality was assumed to calculate the p-values for the one-sided tests of those differences.
|situation||P(run) no PR||P(run) PR||p-value|
The following table is the same as the previous one, except it's for David Ortiz instead of the average player. He often hit prior to Manny Ramirez or Kevin Youkilis (good hitters).
|situation||P(run) no PR||P(run) PR||p-value|
The Ortiz coefficient's standard error was 0.263, so these probability estimates may be larger than the true probabilities: and surely they are because even Manny Ramirez cannot hit about .400, which is what the two-out numbers above are suggesting, with RISP in the long term.
We can see from the low p-values in both tables that pinch running helps increase the chance of scoring, at least when the leading or tied. The PR effect is actually much bigger than I imagined it would be. Other than the large probability increases from the 1st table to the 2nd, there are a couple of interesting things. First, as the I.PR*lead interaction suggested, the PR effect is much smaller when the team is trailing; the raw probability of scoring is also smaller when trailing. I think the latter is easily explained by the fact that the opposing team's best pitchers tend to be their set-up man and closer, and these are the pitchers you face when trailing by one run in the 8th or 9th inning. Another less likely explanation might be that if you are losing by one run, you are not hitting as well on the day, and hence less likely to score. This gets into the "hot hand effect" though, and so I think the likely explanation is the opposing pitching.
I don't think the smaller PR effect in games where the team is trailing is as intuitive, but it probably also has a lot to do with facing the opposing team's best pitching: the speed of your base runner is not important if the subsequent hitters are not getting hits. I actually fit a poisson regression with expected runs scored subsequent to the runner reaching base as the response, and interestingly the expected runs scored with the PR when trailing by one were lower than the expected runs with no PR. The effect was extremely small and not statistically significant, but with the size of my sample I wouldn't expect to see that. I wonder if it has to do with a different strategy being employed when trailing - the PR tries to steal 2nd base, or the manager tries to bunt him over to 2nd base. Either of these moves certainly decrease the expected runs scored in the inning (although possibly increasing the chance of scoring at least one run).
By not allowing for an interaction between I.PR and player, which would measure how fast the original base runner is, I've basically assumed that David Ortiz has the same speed as the average player (at least, average among those who have ever pinch run for - which is almost everybody at one time or another). But the variation in each of these interaction estimates would be even bigger than the variation we're now seeing in the player estimates, so I didn't bother with this here; I will have to include this is my final analysis in part 3 though, perhaps averaging several players together to more precisely estimate an overall effect of pinch running for good hitters. Anyway, because Ortiz is unusually slow, you can probably safely take away a little bit from Ortiz's probabilities and add a little bit to his PR's.
This wraps up step 1. I've had to make some simplifications and assumptions to fit this logistic model (some of which I might have to try to relax in my next post), but at least the data have allowed me to show that pinch running significantly increases the chance of scoring.
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