130 Exposure-Response Modeling: Methods and Practical Implementation
can add a latent subject effect known as frailty in TTE models, in the hazard
function, so that conditional on u
i
:
h
i
(t|u
i
) = h
0
(t)u
i
exp(X
T
i
β). (5.45)
A common choice for the distribution of u
i
is gamma parameterized so that
u
i
∼ gamma(θ) with E(u
i
) = 1 and var(u
i
) = θ and frailty increases with the
increase of θ. Since u
i
is unobserved, one needs to de rive the marginal model
in order to use the ML approach. To this end one can derive conditional
distribution from the hazard function, (5.45) then the marginal distribution
can be derived by integrating the conditional distribution over the gamma
distribution for u
i
. The resulting marginal likelihood is complex (Klein, 1992),
hence is