First, shoulder injuries. In order of strength of prediction, the best predictors of whether or not you will have a shoulder injury in the coming year are
whether you had a shoulder injury last year, how many pitches you threw last year,
whether you had a shoulder injury two years ago, how many extra batters you faced last year from the year before (with a greater increase meaning that you were
less likely to be injured), and the two-strike foul rate (just barely). It's clear that guys with pre-existing conditions are a risk. This shouldn't be too big a surprise. But if you were entrusted to face more batters last year, it might be a sign that the team thinks your shoulder is okay. It’s hard to tell whether the two-strike fouls issue is cause or effect. If you're not able to blow that fastball by hitters, it might be because there is some shoulder damage that's really the beginnings of an injury.
For elbows (in order): Home run rate (lower
HR rate guys have elbow injuries more often),
whether you had an elbow injury last year, the number of batters you faced last year, the change in the number of innings you pitched last year (again, a bigger increase leads to a lower rate of injury), and ERA (the higher the ERA, the more likely you are to get hurt).
For any injury at all, there were two factors: You are more likely to get injured if you threw more pitches last year, and
if you had an injury last year.
For spending time on the disabled list, we see a similar pattern: the number of pitches thrown in the last year,
spending time on the DL last year, and the change in the number of batters faced (once again, a big increase meant a drop in injury chances.)