19
where,
G
ri~N(O,σf), ξij ~∑PjgN(μjg,σf) (2.4)
9=1
and
(Pji,Pj2, ∙ ■ ∙ ,PjG) ~ Dirichlet(ajι,aj2, ■ ■ ■ ,ajc), j = l,...,J. (2.5)
Here, βj parameterize the ordinal probit model for zij. Notice that βj does not
include an intercept parameter. An intercept is already implicitly included in μj∙9.
Consider the implied model for each vij after marginalizing with respect to ri. The
marginal distribution for each uʊ, j = 1,..., J, is a mixture of normal distributions
sharing the scale parameter {∖Jσ'r ÷ σf)> with distinct location parameters (x'iβj +
μjg), fixed number of components (G) and weights (p9∙9). It can be shown that without
loss of generality we can fix the cutpoints θjk when working with the mixture of normal
model in (2.4) instead of a single normal. See also the discussion below. While we
assume that G is fixed, for j = 1,..., J, in (2.4), using different Gj for each toxicity
is possible without additional complications.
The mixture model can alternatively be written as a hierarchical model by intro-
ducing a latent indicator variable wij. Specifically, conditional on wij = g, βj and
ri, the latent variable t⅛∙ follows a normal distribution: ¾∙ ∣ wij = g,βrιτl,μ3g ~
N(χTβj + Ti + μjg, σξ). The prior probability for u⅛ = g is Pr(wij = g) = pjg. Let
Φ(∙) denote the standard normal cdf. Marginalizing with respect to both ri and the
latent variable ¾∙, we have:
Pr(zij = к I wij = g, βj,μj9) = Φ f gfe+^'~ ¾ ~ - Φ ( ⅛ ~
∖ ∖]σr+σl J ∖ ∖]σr+σl
(2.6)
Pr(wij =g) =pjg. (2.7)
For each category j the probability of a response at level к is
G
πjk ≡ Pr{zij = k) = -pr(¾ = k I wij = 9)Pjg (2-8)
9=1