21
∞, {0ι,..., θχ} — {0, ±4, ±8,...}. For reasons of identifiability, we suggest fixing
σ = 1. This choice implicitly restricts cell probabilities π2,... , 7⅛∙-ι to be at most
0.95. The first and last cell probabilities, π1 and ιrκ, are unrestricted. This is impor-
tant in the context of the later application to adverse event rates, when the first level
of the ordinal outcome corresponds to no toxicity, which is often greater than 0.95. If
larger cell probabilities are desired for intermediate outcomes, the widths between θ∣t
and θk+ι can be increased or decreased accordingly. Figure 2.2 illustrates the model
for one toxicity with three possible outcomes, i.e., ¾∙ ∈ {0,1,2} and K3 = 2. The
figure shows how the proposed model with G = 2 fits the cell probabilities πj⅛. The
figure shows the mixture of normal distribution of the latent random variable, vn,
under two values of the covariate xi = — 1,1. In both mixtures, the darkly shaded,
lightly shaded and white areas correspond to the probability of the ordinal outcome
taking the values 0, 1 and 2, respectively. Notice that this particular set of cell
probabilities does not satisfy the parallel regression assumption. Therefore, these
probabilities cannot be represented by a model with a unimodal distributed latent
random variable and random cutpoints. Finally, we fix at σ% — 4, implying non-
negligible prior probability for random effects ri to be in a range that covers several
cutpoints θ∣ς-
2.5 Priors, Posterior and Simulation Scheme
We use conjugate priors for the probit regression parameters, centering the prior to
represent the prior judgment about the marginal prevalence of the outcomes and the
effects of the covariates. We use
f0j(βj)≡N(m0j,σ0). (2.10)
As default choice for G, the size of the mixture model in (2.4), we suggest G = K — 1.
Our recommendation is based on empirical evidence. On one hand, small values
of G create faster mixing Markov chains, but may not be sufficient to fit the data.