## Definitions

### Functions

- \(\Lambda(x)\): the logistic CDF \(\frac{exp(x)}{1 + exp(x)}\) (i.e. inverse log odds function).
- \(\text{BB}\): the beta-binomial distribution
- parameterized by size (82 games), probability, and precision

### Data

- \(N\): the number of
*half-seconds* of regular-season play in the data set.
- \(K\): the number of
*player-years* in the data set.
- \(Z\): \(N \times K\) matrix of dummies for each player being on ice.
- \(X\): \(N\)-row matrix of dummies for lead (by period) and skater advantage.
- \(Y\): \(N \times 1\) matrix of dummies for shots on goal each half-second.
- \(G\): \(K \times 1\) matrix of regular-season games played by player and year.
- All dummies take values \(-1\) for visitors, 0, or \(1\) for home.

### Parameters

- \(\alpha\): baseline log odds of an event for either team.
- \(\beta\): a vector of score-period and skater-advantage effects.
- \(\gamma\): a vector of length \(K\); rating for each player-year.
- \(\Theta\): vector of two ordered logit intercepts.
- i.e. log of baseline visitor and visitor-or-none SOG odds.

- \(\alpha\): expected intercept of \(\Lambda^{-1}(G / 82)\) versus \(\gamma\).
- i.e. expected log odds of starting for an average NHLer.

- \(\delta\): expected slope of \(\Lambda^{-1}(G / 82)\) versus \(\gamma\).
- i.e. log odds of starting for an NHLer with \(\gamma = 1\).

- \(\phi\): precision parameter for beta-binomial distribution.
- \(\sigma\): overall standard deviation of NHL player ability levels.