Functional form

\[\begin{eqnarray} \\ \begin{aligned} \text{Pr(away shot)} &= \Lambda(\alpha_{sa} - X\beta_s - Z\gamma_s) & \\ \text{Pr(home shot)} &= \Lambda(\alpha_{sh} + X\beta_s + Z\gamma_s) & \\ \text{Pr(away pen)} &= \Lambda(\alpha_{pa} - X\beta_p - Z\gamma_p) & \\ \text{Pr(home pen)} &= \Lambda(\alpha_{ph} + X\beta_p + Z\gamma_p) & \end{aligned} \tag{likelihood} \\ \end{eqnarray}\] \ \[\begin{align} \tag{prior} \gamma_s &\sim N(0, \sigma_s) & \gamma_p &\sim N(0, \sigma_p) & \end{align}\] \ \[\begin{eqnarray} \tag{[name?]} G \sim \text{BB}(82, \alpha_G + \gamma_s\beta_G, \phi) \end{eqnarray}\]

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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.