Last post, we illustrated fitting an exchangeable model on home run hitting probabilities for all players (non pitchers) in the 2007 season.
But actually, teams aren’t really interested in estimating hitting probabilities. They are primarily interested in predicting a player’s home run performance the following 2008 baseball season.
We already have the posterior distribution for the 2007 home run probabilities. Assuming that the probabilities stay the same for the following season and that the players will have the same number of at-bats, we can easily predict their 2008 home run counts from the posterior predictive distribution.
For each player, I simulated the predictive distribution of the number of home runs
In the first graph, for all players I plot the number of 2007 home runs (horizontal) against the predicted number of 2008 home runs – the number of 2007 home runs (vertical). Given the 2007 home run count, I’m interested in my prediction of the change to the following season. (I jittered the points so you can see the individual observations.) For the weak home run hitters (on the left side of the graph), I predict they will perform a little worse or better the following season. But for the heavy hitters, I predict they will hit 5-7 home runs worse in the 2008 season. Generally, there is a negative slope in the graph that illustrates the “regression to the mean” effect.
Since we have the actual 2008 home run counts, we can actually compare the 2007 and 2008 counts to see if there is a similar effect. In the following graph, I plot the 2007 counts against the change (2008 count – 2007 count). What do we see? There is a negative correlation as we suspected. Hitters who did well in 2007 tended to get worse in 2008, and a number of light hitters in 2007 tended to do better in 2008. The biggest difference between the actual data and our predictions is the magnitude of the changes. Three of the top hitters, for example, hit 15-20 home runs in 2008.