CAS Exam MAS-II Fall 2019, Question 11
Study guide: Exam MAS-II
You are building a Linear Mixed Model to estimate the number of polices written by sales representatives in various agencies in the company. You are given the following information:
- The number of polices written (
Y_{ij}) is measured for a sales representativeiwho works in agencyj. L_jandM_jrepresent characteristics of individual agencies.N_{ij}represents a characteristic of an individual sales representative.\mu_j(as printed; the model writesu_j) is a random effect for each agency that is normally distributed with mean 0 and variance\sigma^2_{agency}.\varepsilon_{ij}is a residual that is normally distributed with mean 0 and variance\sigma^2.- The general model specification is in the following form:
Y_{ij} = \beta_0 + \beta_1 \times L_j + \beta_2 \times M_j + \beta_3 \times N_{ij} + \beta_4 \times M_j \times N_{ij} + u_j + \varepsilon_{ij}Determine the hierarchical model specification.
- A) Level 1:
Y_{ij} = b_{0j} + b_{1j} \times M_j + \varepsilon_{ij}; Level 2:b_{0j} = \beta_0 + \beta_1 \times L_j + \beta_3 \times N_{ij} + u_j,b_{1j} = \beta_2 + \beta_4 \times N_{ij} - B) Level 1:
Y_{ij} = b_{0j} + b_{1j} \times M_j; Level 2:b_{0j} = \beta_0 + \beta_1 \times L_j + \beta_3 \times N_{ij} + u_j + \varepsilon_{ij},b_{1j} = \beta_2 + \beta_4 \times N_{ij} - C) Level 1:
Y_{ij} = b_{0j} + b_{1j} \times N_{ij} + \varepsilon_{ij}; Level 2:b_{0j} = \beta_0 + \beta_1 \times L_j + \beta_2 \times M_j + u_j,b_{1j} = \beta_3 + \beta_4 \times M_j - D) Level 1:
Y_{ij} = b_{0j} + b_{1j} \times N_{ij}; Level 2:b_{0j} = \beta_0 + \beta_1 \times L_j + \beta_2 \times M_j + u_j + \varepsilon_{ij},b_{1j} = \beta_3 + \beta_4 \times M_j - E) Level 1:
Y_{ij} = b_{0j} + b_{1j} \times N_{ij} + \varepsilon_{ij}; Level 2:b_{0j} = \beta_0 + \beta_1 \times L_j + \beta_3 \times M_j + u_j,b_{1j} = \beta_2 + \beta_4 \times M_j
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