
256 Exposure-Response Modeling: Methods and Practical Implementation
For linear models
y
i
= β
0
+ βc
i
+ ε
i
c
i
= θ
0
+ θd
i
+ e
i
, (9.1 4)
some interesting analy tical results can be derived. For the Dose–response re-
lationship, c onditional on d
i
we can write the marginal model as
y
i
= β
0
+ β(θ
0
+ θd
i
) + βe
i
+ ε
i
= β
∗
0
+ β
∗
d
i
+ ε
∗
i
(9.15)
where β
∗
0
≡ β
0
+ βθ
0
, β
∗
= βθ, and ε
∗
i
∼ N (0, β
2
σ
2
e
+ σ
2
ε
). Here the key
parameter for the dose–exposure relationship, and also for dose finding, is β
∗
.
Hence we examine the accuracy of its estimates either based on fitting the
dose–exposure data to model (9.15) or the joint modeling approach fitting
both models in (9.14) as
ˆ
β
ˆ
θ. In this simple ...