  Evaluating the overall ability of players in the National Hockey League (NHL)
is a difficult task. Existing methods such as the famous "plus/minus" statistic
have many shortcomings. Standard linear regression methods work well when
player substitutions are relatively uncommon and scoring events are relatively
common, such as in basketball, but as neither of these conditions exists for
hockey, we use an approach that embraces the unique characteristics of the
sport. We model the scoring rate for each team as its own semi-Markov process,
with hazard functions for each process that depend on the players on the ice.
This method yields offensive and defensive player ability ratings which take
into account quality of teammates and opponents, the game situation, and other
desired factors, that themselves have a meaningful interpretation in terms of
game outcomes. Additionally, since the number of parameters in this model can
be quite large, we make use of two different shrinkage methods depending on the
question of interest: full Bayesian hierarchical models that partially pool
parameters according to player position, and penalized maximum likelihood
estimation to select a smaller number of parameters that stand out as being
substantially different from average. We apply the model to all five-on-five
(full-strength) situations for games in five NHL seasons.
