25
(d) The latent indicator variables wij in equation (2.7) are sampled from the complete
conditional posterior distribution (marginalized with respect to ¾∙):
Pr(wv = g I βj,ri,μjg,pjg) <xpjg [Φμ^,σ2‰+1j - xf βj ~ ri)
-$∕⅛σ∣(¾j,J - Xi βj ~ ri)] ∙
(e) For each j, we update (p7ι,... ,pja) from
f(pjl,.. .,pjtG I wιj,..., wnj) = Dirichlet(a'jl,..., a'jG),
with a'jg = ajg + #{z : wij = g}.
(f) The latent variables vij are updated by draws from the truncated normal distri-
bution
f(υij I βj,ri,wij -g,μjg,zij = k) <x N(υij ∣ χTβj+ri+μjg,σf)I{θkj ≤ zij < ⅛+lj∙).
Finally, for each toxicity type and grade, we evaluate the posterior probability of
toxicity for a future patient. For each type of toxicity, using the equations (2.6) and
(2.8), the posterior probability πj∙⅛ ≡ Pr(zfj — fc∣z), f = n + 1, that a future patient
with covariate vector xf exhibits the toxicity j at level к is estimated from the Gibbs
sampler outputs. Let ηm denote the imputed value of the generic parameter η after
m iterations of the Gibbs sampler. We report
7rJfc = ∑f=ι pΦfj = k I wfj = 9, z)pr(wfj = 9 I z)
~ ɪ V'g ʃ<τ> (θk+1∙i~x'∕9jl-μ^1∖ _ j, (θki~x}9f-μ,jlg∖ ɪ
~ M Z-∕m=l 2^3=1 ɪ y∕91+9J J ∖ v∕σ≡+σf J ʃ
where M is the total number of MCMC iterations retained after an initial burn-in.
2.6 Applications
2.6.1 A Simulated Dataset
We use a simulated dataset to validate the model. A total of n = 1000 subjects were
assigned into two groups, A and B, of equal size. For each subject i, there were four
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