## Functional form

\[\begin{eqnarray} \\
P(Y <= y') &=& \Lambda(Z\gamma^T + X\beta^T + \theta_{y'}), y \in \{-1, 0\} \\
\mathbf{\gamma} &\sim& \mathrm{N}(0, \sigma) \\
G &\sim& \mathrm{BetaBinom}(82, \Lambda(\alpha + \delta \gamma), \phi)
\end{eqnarray}\]
## Definitions

### Functions

```
- $\Lambda(x)$: the logistic CDF $\frac{exp(x)}{1 + exp(x)}$ (i.e. inverse log odds function).
- $\mathrm{BetaBinom}$: 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.
- See "X matrix structure" below for details.
- $Y$: $N \times 1$ 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

```
- $\beta$: 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.
```