Exposure-Response Modeling

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