Question

In class we talked about the effects of cigarette smoking on birthweight. We defined a model E ( O | C = c ) = (β0)^1 + ((β1)^1 x c) where O is birthweight in ounces and C is cigarettes smoked per day. Suppose instead we used the model E(L|P =p)=(β0)^2 +((β1)^2 x p) where L is birthweight in pounds and P is packs of cigarettes smoked per day (there are 20 cigarettes in a pack). What is the relationship between (β0)^1 and (β0)^2 and between (β1)^1 and (β1)^2? Give a descriptive interpretation of both (β1)^1 and (β1)^2.

Answer 1

Solution

Back-up Theory

CHANGE OF SCALE

If a variable X is transformed by multiplying or dividing it by a constant, the transformation is called Change of Scale.

Effect of Change of Scale on various statistical measures

If U = bX and V = dY, ………………………………………………………………………. (1)

then

Ubar = bXbar, Vbar = dYbar ………………………………………………………………..(2)

Var (U) = σ²_U = b²Var(X) = c²σ²_X and Var (V) = σ²_V = d²Var(Y) = d²σ²_Y ………………….. (3)

Cov(U, V) = bdCov(X, Y) ……………………………………………………………………. (4)

Now to work out the solution,

Since a pound is 16 ounces and one pack = 20 cigarettes,

L = (1/16)o and P = 20c, here b = 20 and d = 1/16

Part (a)

β₁² = Cov(P, L)/σ²_P = (20/16)Cov(c, o)/400σ²_c [vide (3) and (4)]

= (1/320){Cov(c, o)/σ²_c} [vide (3) and (4)]

= (1/320)β₁

Thus, the relationship between β₁² and β₁ is: β₁² = β₁/320 Answer 1

β ₀² = Lbar - β₁²Pbar

= (1/16)obar – (β₁/320)20cbar [vide (1) and Answer 1]

= (1/16)(obar – β₁cbar)

= β ₀/16.

Thus, the relationship between β₀² and β₀ is: β₀² = β₀/16 Answer 2

Part (b)

β₁² represents the slope and its interpretation is that for every unit change in the number packs of cigarette smoked, the birth-weight would change by β₁² pounds. If β₁² is positive, an increase/decrease in the number packs of cigarette smoked will be accompanied by increase/decrease in the birth-weight. If β₁² is negative, an increase/decrease in the number packs of cigarette smoked will be accompanied by decrease/increase in the birth-weight.

β₀² represents the y-intercept and its interpretation is that even when no smoking is there, the birth-weight would be β₀² pounds. Answer

DONE