The name is absent

23

essentially identical terms) make the posterior distribution on μ_gg and pj_g meaningless
to interpret.

We assume that w_ij takes on discrete values 1,2,..., G with prior probability
Pji, ∙ ∙ ■ ,PjG, respectively. For the location parameter (μ_j∙_s) in the components of
the mixture of normal model (2.4), we use independent normal priors f_μ(μj_g∖φj) ≡
N{^φj,σ²_μ) with a conjugate hyperprior fφ_j(φ) ≡ 7V(0, σ^).

Keeping in mind the default suggestion for the cutpoints 0⅛ we recommend σ_μ ≈
i.e., half the span from the first to last cutpoint, averaging across all
toxicities.

An investigator might be interested in assessing how different dose levels affect
toxicity grade. Our model may be used in this context. For a cytotoxic agent, it
is usually assumed that a higher dose incurs worse toxicity. The parameter β_g, the
dose effect on the toxicity grade, may be restricted to be positive when there are only
two dose levels and the lower dose group is the reference group. When there are M
dose groups, β_g becomes an (M-l)-dimensional vector. In this case one could enforce
monotonicity of toxicity with increasing dose by introducing an order constraint on βj
as follows: assuming that the lowest dose is the reference group and the highest dose
group is group M, monotonicity can be represented as β(M-i)j > β(M-2)j > ∙ ∙ ∙ > βij∙
This assumption guarantees a priori that a higher dose incurs worse toxicity.

All full conditional posterior distributions are derived from the joint probability
model (2.9). Since we adopt conjugate priors for all parameters, all full conditional
posterior distributions have tractable closed forms, allowing straightforward imple-
mentation of Markov Chain Monte Carlo (MCMC) posterior simulation using a Gibbs
sampler. We start with initial values for the latent variables v and w. The values for
V need to comply with (2.2). One iteration of the Gibbs sampler is described by the
following transition probabilities:

(a) For each j — 1,..., J, we first marginalize with respect to φj (recall that φ_j is
the prior mean for μ_gg). Then, sample (∕3₇∙, μ_jι,..., M_jg) from the joint complete

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