Question

i) Show how you would create two dummy variables, d₁ and d₂, for when a person started smoking, if we wanted never smokers to be the referent level.

D1={ 1; if smoking as an adult, 0; if otherwise

D2={ 1; if smoking as a child, 0; if otherwise

ii) Using the dummy variables from part (i), suppose you decide to do the analysis using a multiple regression model:

         Y  =  b₀ + b₁d₁ + b₂d₂ + b₃age + b₄(d₁)(age) + b₅(d₂)(age) + e

Simplify the model for each of the three levels of when the person started smoking.  For each level, combine terms so that there is one intercept (which might be a single coefficient, or the sum of two) and one slope (this may also be a single coefficient or the sum of two).

Answer 1

a)

those who never smoke :

D1 = D2 = 0

those who started as adult

D1 = 1 ,D2 = 0

those who started as child

D1 = 0 , D2 = 1

b)

those who never smoke :

D1 = D2 = 0

Y  =  b0 + b1d1 + b2d2 + b3age + b4(d1)(age) + b5(d2)(age) + e

Y  =  b0 + b3age + e

those who started as adult

D1 = 1 ,D2 = 0

Y  =  b0 + b1d1 + b2d2 + b3age + b4(d1)(age) + b5(d2)(age) + e

hence

Y  =  b0 + b1d1 + b3age + b4(d1)(age) + e

those who started as child

D1 = 0 , D2 = 1

Y  =  b0 + b1d1 + b2d2 + b3age + b4(d1)(age) + b5(d2)(age) + e

Y  =  b0 + b2d2 + b3age + b5(d2)(age) + e