The name is absent

19

where,

G

r_i~N(O,σf), ξ_ij ~∑PjgN(μ_jg,σf)             (2.4)

9=1

and

(Pji,Pj2, ∙ ■ ∙ ,PjG) ~ Dirichlet(a_jι,a_j2, ■ ■ ■ ,a_jc), j = l,...,J.       (2.5)

Here, β_j parameterize the ordinal probit model for z_ij. Notice that β_j does not
include an intercept parameter. An intercept is already implicitly included in μ_j∙₉.
Consider the implied model for each v_ij after marginalizing with respect to r_i. The
marginal distribution for each uʊ, j = 1,..., J, is a mixture of normal distributions
sharing the scale parameter {∖J^σ'r ÷ ^σf)> with distinct location parameters (x'_iβ_j +
μ_jg), fixed number of components (G) and weights (p₉∙₉). It can be shown that without
loss of generality we can fix the cutpoints θ_jk 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 G_j 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 w_ij. Specifically, conditional on w_ij = g, β_j and
r_i, the latent variable t⅛∙ follows a normal distribution: ¾∙ ∣ w_ij = g,β_rιτ_l,μ_3g ~
N(χTβj + Ti + μ_jg, σξ). The prior probability for u⅛ = g is Pr(w_ij = g) = p_jg. Let
Φ(∙) denote the standard normal cdf. Marginalizing with respect to both r_i and the
latent variable ¾∙, we have:

Pr(z_ij = к I w_ij = g, β_j,μ_j9) = Φ f ^gfe+^'~ ¾ ~     - Φ ( ⅛      ~

∖ ∖]^σr^+σl J ∖ ∖]^σr+^σl

(2.6)

Pr(w_ij =g) =p_jg.                          (2.7)

For each category j the probability of a response at level к is

G

π_jk ≡ Pr{z_ij = k) =    -^pr(¾ = ^k I ^wij = 9)Pjg             (2-8)

9=1

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