## 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.

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