CAS Exam MAS-II Fall 2019, Question 28
Study guide: Exam MAS-II
A varying-intercepts model is built to model the test scores of students in eight schools:
\begin{align*}
score_i &\sim \text{Normal}(\mu_i, p_i) \\
\mu_i &= \alpha_{SCHOOL[i]} + \beta_1 x_i \\
\alpha_{SCHOOL} &\sim \text{Normal}(\alpha, \sigma) \\
\beta_1 &\sim \text{Normal}(0, 2) \\
\alpha &\sim \text{Normal}(0, 1) \\
\sigma &\sim \text{HalfCauchy}(0, 1)
\end{align*}Three additional models (Model I, Model II, and Model III) are proposed:
Model I
\begin{align*}
score_i &\sim \text{Normal}(\mu_i, p_i) \\
\mu_i &= \alpha + \alpha_{SCHOOL[i]} + \beta_1 x_i \\
\alpha &\sim \text{Normal}(0, 1) \\
\alpha_{SCHOOL} &\sim \text{Normal}(0, \sigma) \\
\beta_1 &\sim \text{Normal}(0, 2) \\
\sigma &\sim \text{HalfCauchy}(0, 1)
\end{align*}Model II
\begin{align*}
score_i &\sim \text{Normal}(\mu_i, p_i) \\
\mu_i &= \alpha_{SCHOOL[i]} + \beta_1 x_i \\
\alpha_{SCHOOL} &\sim \text{Normal}(\alpha, 2\sigma) \\
\beta_1 &\sim \text{Normal}(0, 2) \\
\alpha &\sim \text{Normal}(0, 1) \\
\sigma &\sim \text{HalfCauchy}(0, 1)
\end{align*}Model III
\begin{align*}
score_i &\sim \text{Normal}(\mu_i, p_i) \\
\mu_i &= \alpha_{SCHOOL[i]} + \beta_1 x_i \\
\alpha_{SCHOOL} &\sim \text{Normal}(0, 1) \\
\beta_1 &\sim \text{Normal}(0, 2)
\end{align*}Determine which of the three proposed models imposes additional model shrinkage:
- A) None
- B) Model I and Model II only
- C) Model I and Model III only
- D) Model II and Model III only
- E) The answer is not given by (A), (B), (C), or (D)
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