17.2 Mathematical Models

The three models considered set \(y_n=log(radon)\), and \(x_n\) records floor (0=basement, 1=first floor) for homes \(n=1, \ldots, N\).

17.2.1 Complete Pooling

The complete pooling model pools all counties together to give them one single estimate of the \(log(radon)\) level, \(\hat{\alpha}\).

  • The error term \(\epsilon_n\) may represent variation due to measurement error, within-house variation, and/or within-county variation.
  • Fans of the random intercept model think that \(\epsilon_n\), here, captures too many sources of error into one term, and think that this is a fault of the completely pooled model.

\[\begin{equation*} \begin{split} y_n = \alpha & + \epsilon_n \\ & \epsilon_n \sim N(0, \sigma_{\epsilon}^{2}) \end{split} \end{equation*}\]

17.2.2 No Pooling

  • The no pooling model gives each county an independent estimate of \(log(radon\)), \(\hat{\alpha}_{j[n]}\).
  • Read the subscript \(j[n]\) as home \(n\) is nested within county \(j\). Hence, all homes in each county get their own independent estimate of \(log(radon)\).
  • This is equivalent to the fixed effects model
  • Here again, one might argue that the error term captures too much noise.

\[\begin{equation*} \begin{split} y_n = \alpha_{j[n]} & + \epsilon_n \\ \epsilon_n & \sim N(0, \sigma_{\epsilon}^{2}) \end{split} \end{equation*}\]

17.2.3 Partial Pooling (RI)

  • The random intercept model, better known as the partial pooling model, gives each county an intercept term \(\alpha_j[n]\) that varies according to its own error term, \(\sigma_{\alpha}^2\).
  • This error term measures within-county variation
    • Separating measurement error (\(\sigma_{\epsilon}^{2}\)) from county level error (\(\sigma_{\alpha}^{2}\)) .
  • This multi-level modeling shares information among the counties to the effect that the estimates \(\alpha_{j[n]}\) are a compromise between the completely pooled and not pooled estimates.
  • When a county has a relatively smaller sample size and/or the variance \(\sigma^{2}_{\epsilon}\) is larger than the variance \(\sigma^2_{\alpha}\), estimates are shrunk more from the not pooled estimates towards to completely pooled estimate.

\[\begin{equation*} \begin{split} y_n = \alpha_{j[n]} & + \epsilon_n \\ \epsilon_n & \sim N(0, \sigma_{\epsilon}^{2}) \\ \alpha_j[n] & \sim N(\mu_{\alpha}, \sigma_{\alpha}^2) \end{split} \end{equation*}\]