Exposure-Response Modeling

Confounding bias and caus al inference in exposure–response modeling 243

to adjust the confounding bias. But u

i

may not be entirely confounded. It is

the component correlated to v

i

that causes bias and has to be adjusted for.

The CF method uses the IV mo de l to predict this component, then includes

its prediction to adjust the y-part model. To this end, we need to assume that

u

i

can be written as

u

i

= av

i

+ ε

i

(8.49)

where ε

i

is a ra ndom term independent of v

i

. Therefore, u

i

has two compo-

nents; av

i

is the source of confounding and the error term ε

i

is independent

of c

i

. The re lationship (8.49) holds when u

i

and v

i

are jointly normally dis-

tributed. But it