CAS Exam MAS-II Fall 2019, Question 7
Study guide: Exam MAS-II
The heights of 20 schoolgirls are measured on a yearly basis from age 6 to 10. The girls were classified into three groups. The three groups are based on the height of their mother (SHORT, MEDIUM, or TALL).
The table below illustrates the data:
| Child ID | Height of Mother | Child Age | Child Height |
|---|---|---|---|
| 1 | SHORT | 6 | 111 |
| 1 | SHORT | 7 | 116.4 |
| 1 | SHORT | 8 | 121.7 |
| 1 | SHORT | 9 | 126.3 |
| 1 | SHORT | 10 | 130.5 |
| 2 | SHORT | 6 | 110 |
| … | … | … | … |
| 7 | MEDIUM | 6 | 116 |
| … | … | … | … |
| 20 | TALL | 10 | 152.3 |
The hierarchical Linear Mixed Model given below is used to model HEIGHT_{ti}, where HEIGHT_{ti} is the t^{th} annual height measurement of the i^{th} schoolgirl.
Level 1 Model (Time):
HEIGHT_{ti} = b_{0i} + b_{1i} \times AGE_{ti} + \varepsilon_{ti}, \qquad \varepsilon_{ti} \sim N(0, \sigma^2)Level 2 Model (individual schoolgirl):
\begin{align*}
b_{0i} &= \beta_0 + \beta_2 \times SHORT_i + \beta_3 \times TALL_i + u_{0i} \\
b_{1i} &= \beta_1 + u_{1i}
\end{align*}where u_i = \begin{pmatrix} u_{0i} \\ u_{1i} \end{pmatrix} \sim N(0, \mathbf{D}).
Determine which of the following Linear Mixed Models is equivalent to the above hierarchical model.
- A)
HEIGHT_{ti} = \beta_0 + \beta_1 \times AGE_{ti} + \beta_2 \times SHORT_i + \beta_3 \times TALL_i + u_{0i} + \varepsilon_{ti} - B)
HEIGHT_{ti} = \beta_0 + \beta_1 \times AGE_{ti} + \beta_2 \times SHORT_i + \beta_3 \times TALL_i + u_{0i} + u_{1i} + \varepsilon_{ti} - C)
HEIGHT_{ti} = \beta_0 + \beta_1 \times AGE_{ti} + \beta_2 \times SHORT_i + \beta_3 \times TALL_i + u_{0i} + u_{1i} \times AGE_{ti} + \varepsilon_{ti} - D)
HEIGHT_{ti} = \beta_0 + \beta_1 \times SHORT_i + \beta_2 \times TALL_i + \beta_3 \times AGE_i + u_{0i} + u_{1i} \times AGE_{ti} + \varepsilon_{ti} - E)
HEIGHT_{ti} = \beta_0 + \beta_1 \times AGE_{ti} + \beta_2 \times SHORT_i + \beta_3 \times TALL_i + u_{0i} \times AGE_{ti} + u_{1i} + \varepsilon_{ti}
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