MATH235 Week 3 - Moodle Quiz-assessed problems

We look at how to model the relationship between FEV and gender and smoking status. Define the following indicator variables for gender,

x_{i,1}=\left\{\begin{array}[]{cc}1&\mbox{if individual $i$ is male;}\\ 0&\mbox{if individual $i$ is female,}\end{array}\right.~{}~{}~{}\mbox{and}~{}~% {}~{}x_{i,2}=\left\{\begin{array}[]{cc}1&\mbox{if individual $i$ is female;}\\ 0&\mbox{if individual $i$ is male,}\end{array}\right.

and for smoking status,

x_{i,3}=\left\{\begin{array}[]{cc}1&\mbox{if individual $i$ smokes;}\\ 0&\mbox{if individual $i$ does not smoke,}\end{array}\right.~{}~{}~{}\mbox{and% }~{}~{}~{}x_{i,4}=\left\{\begin{array}[]{cc}1&\mbox{if individual $i$ does not% smoke;}\\ 0&\mbox{if individual $i$ smokes.}\end{array}\right.

Two linear regression models are proposed for log FEV ( $Y_{i}$ ),

\mathbb{E}[\log Y_{i}]=\beta_{1}+\beta_{2}x_{i,1}+\beta_{3}x_{i,3}{}

and

\mathbb{E}[\log Y_{i}]=\gamma_{1}x_{i,1}+\gamma_{2}x_{i,2}+\gamma_{3}x_{i,3}{}

Q3.1

Intercept
Which of the following is correct? In model F1, the coefficient $\beta_{1}$ is

[marks: 2]

Equivalent coefficients
Compare the coefficients in the two models. Which of the following is correct?

[marks: 2]

Male smoker
Which of the following is the expected log FEV for a male smoker?

[marks: 2]

Linear dependence
Consider the design matrix for a further model

\mathbb{E}[\log Y_{i}]=\zeta_{1}x_{i,1}+\zeta_{2}x_{i,2}+\zeta_{3}x_{i,3}+% \zeta_{4}x_{i,4}{}

Is this design matrix linearly independent?

[marks: 2]

Explanatory variables
Which of the following is correct? The explanatory variables for model F2 consist of

[marks: 2]