19.        Depression and Education. Has homework got you depressed? 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 edu- cation, based on a 1990 nationwide (U.S.) 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 the “divergence hypothesis.”

 

 

 

 

 

 

 

Year		Winner		Starters		NetToWinner		Time		Speed		Track		Conditions
1896		Ben Brush		8		4,850		127.75		35.23		Dusty		Fast
1897		Typhoon II		6		4,850		132.50		33.96		Heavy		Slow
1898		Plaudit		4		4,850		129.00		34.88		Good		Fast
1899		Manuel		5		4,850		132.00		34.09		Fast		Fast
1900 . . .		Lieut. Gibson		7		4,850		126.25		35.64		Fast		Fast
2011		Animal Kingdom		20		2,000,000		122.04		36.87		Fast		Fast

 

 

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. (See “Socioeconomic Status and Depression over the Life Course,” Journal of Health and Social Behaviour 41(2) (June, 2000): 162–74.)

(a)     Construct a multiple linear regression model in which the mean depression score changes linearly with age in all three education categories, with possibly unequal slopes and inter- cepts. Identify a single parameter that measures the diverging gap between categories (iii) and (i) with age.

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

This and other studies found evidence that the mean depression is high in the late teens, declines toward middle age, and then increases towards old age. Construct a multiple linear regression model in which the association has these characteristics, with possibly different structures in the three edu- cation categories. Can this be done in such a way that a single parameter characterizes the divergence hypothesis?