Pitcher or batter ?

• Are home runs about the hitter or pitcher?

• One way to address: use random effects model.

\[ \log \left(\frac{p}{1-p}\right) = \mu + \beta_i + \gamma_j \]

• \(\mu\) is the overall effect
• \(\beta_i\) is the effect due to the \(i\)th hitter
• \(\gamma_j\) is the effect due to the \(j\)th pitcher.
• \(\beta\) and \(\gamma\) are both assumed to be normally distributed:
  • \(\beta_i \sim N(0,\sigma_h)\)
  • \(\gamma_j \sim N(0,\sigma_p)\)