The name is absent

18

2.4 A Hierarchical Model for Ordinal Data Nested

within Categories

For each recorded adverse event, the data report one variable. This variable is an
ordinal outcome, ¾∙, which reports the grade at which the j^th categorical outcome was
observed on the i^th individual, i ∈ {1,..., n} and j ∈ {1,..., J}, where, respectively,
n and J denote the total number of patients and the total number of different toxicity
types recorded in the study. The variable ¾ takes values к = 0,1,...,Kj. The
observation ¾ = к indicates that the i^th patient exhibited the toxicity of type j
at grade k. The additional grade к = 0 is used to indicate that toxicity j was not
recorded for patient i.

Let X be a (n × H) matrix of possible regressors, with the i^th row, x_i, recording
H covariates for the i^th patient, i = 1,..., n. For inference on a dose effect one could
use x_i↑ — 0 when the drug is not present, хц = 1 for the lowest dose of the drug,
Xn = 2 for the second lowest dose and so on. In our specific example, we have a
dichotomous covariate. We use x_iι — 1 when patient i is treated with isotretinoin,
and Xu — —1 for placebo. Considering just the treatment effect we have H = I and
the i^th row of the covariate matrix, X, is just x_il. In general, the covariates could be
occasion-specific and indexed by patient i and toxicity j. We only use patient-specific
covariates in the application and proceed therefore for simplicity with patient-specific
covariates only.

We set up an ordinal probit regression for ¾∙ on covariates x_i. The cell probability
Pr(zij = k) is represented as the probability that a continuous latent variable ¾ falls
into the interval (f⅛∙, 0fc+ι_j∙). A patient specific random effect r_i induces correlation
across all toxicity observations for the same patient. Multiple cutpoints are required
for the Kj ordinal outcomes:

z_ij — к if θkj ≤ Vij < θk+ι,j for к = 0,1,..., Kj.           (2.2)

v_ij = x{β_j + n + ξ_ij,                          (2.3)

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