80 Exposure-Response Modeling: Methods and Practical Implementation
we examine the idea behind the RC approach. Since only c
i
is observed, we
would like to fit y
i
with its conditional mean E(y
i
|c
i
) = βE(c
∗
i
|c
i
), i.e., we fit
y
i
= βE(c
∗
i
|c
i
) + ε
∗
i
, (4.24)
where ε
∗
i
= ε
i
+ β(c
∗
i
−E(c
∗
i
|c
i
)). It can be verified that as long as E(c
∗
i
|c
i
) is
sp ecified correctly, fitting this model gives a consistent estimate for β, although
ε
∗
i
depe nds on β. To this end, note that the RC estimate
ˆ
β
RC
is the solution
to
S
RC
(β) =
n
X
i=1
E(c
∗
i
|c
i
)(y
i
− βE(c
∗
i
|c
i
))/n = 0, (4.25)
we check E(S
RC
(β)) = 0, which is the key condition for consistency of
ˆ
β
RC
(A.6). E(S
RC
(β)) can be calculated by first conditioning on c
i
:
E(S
RC
(β)) = E
c
(E(c
∗
i
|c
i
)E(y
i
− βE(c
∗
i
|c
i
)|c
i
)) (4.26)
in which E(y
i
− βE(c
∗
i
|c
i