Question

Has homework got you down? It could be worse. Depression, like other illnesses, is more prevalent among adults with less education than you have. R. A. Miech and M. J. Shanahan investigated the association of depression with age and education, based on a 1990 nationwide telephone survey of 2,031 adults aged 18 to 90. Of particular interest was their finding that the association of depression with education strengthens with increasing age – a phenomenon they called “divergence hypothesis.” They constructed a depression score from responses to several related questions. Education was categorized as (i) college degree, (ii) high school degree plus some college, or (iii) high school degree only.

(a) Write down a linear regression model in which the mean depression score changes linearly with age in all three categories, with possibly unequal slopes and intercepts. Identify a single parameter that measures the diverging gap between categories (iii) and (i) with age. If you use indicator variables, make sure you define them.

(b) Modify the model to specify that the slopes of the regression lines with age are equal in categories (i) and (ii) but possibly different in category (iii). Again identify a single parameter measuring divergence.

Answer 1

(a)

The linear regression model is,

$y = \beta_0 + \beta_1 ~Age + \delta_1 ~HC + \delta_2~H + \delta_3 HC * Age + \delta_4 H*Age$

where y is mean depression score

H is categorical variable with value 1 for High School degree only (iii) and 0 for College degree (i) and High School degree plus some college (ii)

HC is categorical variable with value 1 for High School degree plus some college (ii) and 0 for College degree (i) and  High School degree only (iii)

For category (i), HC = H = 0, the linear model becomes,

$y = \beta_0 + \beta_1 ~Age$

For category (iii), HC = 0 and H = 1, the linear model becomes,

$y = \beta_0 + \beta_1 ~Age + \delta_2 + \delta_4 Age$

$\Rightarrow y = (\beta_0 +\delta_2) + (\beta_1+ \delta_4) ~Age$

Thus, $\delta_4$ is the parameter that measure between categories (iii) and (i) with age.

(b)

Now, if the slope of the regression lines with age are equal in categories (i) and (ii), then $\delta_3$ HC * age parameter in the previous linear model should be omitted.

$y = \beta_0 + \beta_1 ~Age + \delta_1 ~HC + \delta_2~H + \delta_4 H*Age$

Then for category (ii), HC = 1, H = 0

$y = \beta_0 + \beta_1 ~Age + \delta_1$

$\Rightarrow y = (\beta_0+ \delta_1) + \beta_1 ~Age$

thus, the slopes for both categories (i) and (ii) is $\beta_1$

Again here,  $\delta_4$ is the parameter that measure between categories (iii) and (i) with age.

For category (i), HC = H = 0, the linear model becomes,

$y = \beta_0 + \beta_1 ~Age$

For category (iii), HC = 0 and H = 1, the linear model becomes,

$y = \beta_0 + \beta_1 ~Age + \delta_2 + \delta_4 Age$

$\Rightarrow y = (\beta_0 +\delta_2) + (\beta_1+ \delta_4) ~Age$

Thus, $\delta_4$ is the parameter that measure between categories (iii) and (i) or (ii) with age.