The name is absent

25

(d) The latent indicator variables w_ij in equation (2.7) are sampled from the complete
conditional posterior distribution (marginalized with respect to ¾∙):

Pr(w_v = g I β_j,r_i,μ_jg,p_jg) <xp_jg [Φ_μ^,_σ2‰₊₁j - xf β_j ~ r_i)

^-$∕⅛σ∣(¾_j,J ^{- X}i βj ~ ^ri)] ∙

(e) For each j, we update (p₇ι,... ,p_ja) from

f(p_jl,.. .,p_jtG I wι_j,..., w_nj) = Dirichlet(a'_jl,..., a'_jG),
with a'_jg = a_jg + #{z : w_ij = g}.

(f) The latent variables v_ij are updated by draws from the truncated normal distri-
bution

f(υ_ij I β_j,r_i,w_ij -g,μ_jg,z_ij = k) <x N(υ_ij ∣ χTβ_j+r_i+μ_jg,σf)I{θ_kj ≤ z_ij < 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 π_j∙⅛ ≡ Pr(zf_j — 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 η^m 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(w_fj = 9 I z)

~ ɪ        V'^g ʃ<τ> (^θk+¹∙i~^x'∕9j^l-μ^₁∖ _ j, (^θki~^x}9f-μ^,j^lg∖ 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

<<   34   35   36   37   38   39   40   >>

More intriguing information