The sample_copula_parameters() function samples the unidentifiable copula
parameters for the partly identifiable D-vine copula model, see for example
fit_copula_model_BinCont() and fit_model_SurvSurv() for more information
regarding the D-vine copula model.
sample_copula_parameters(
copula_family2,
n_sim,
eq_cond_association = FALSE,
lower = c(-1, -1, -1, -1),
upper = c(1, 1, 1, 1)
)A n_sim by 4 numeric matrix where each row corresponds to a
sample for \(\boldsymbol{\theta}_{unid}\).
Copula family of the other bivariate copulas. For the
possible options, see loglik_copula_scale(). The elements of
copula_family2 correspond to \((c_{23}, c_{13;2}, c_{24;3}, c_{14;23})\).
Number of copula parameter vectors to be sampled.
(boolean) Indicates whether \(\rho_{13;2}\) and \(\rho_{24;3}\) are set equal.
(numeric) Vector of length 4 that provides the lower limit,
\(\boldsymbol{a} = (a_{23}, a_{13;2}, a_{24;3},
a_{14;23})'\). Defaults to c(-1, -1, -1, -1). If the provided lower limit
is smaller than what is allowed for a particular copula family, then the
copula family's lowest possible value is used instead.
(numeric) Vector of length 4 that provides the upper limit,
\(\boldsymbol{b} = (b_{23}, b_{13;2}, b_{24;3},
b_{14;23})'\). Defaults to c(1, 1, 1, 1).
In the D-vine copula model in the Information-Theoretic Causal Inference (ITCI) framework, the following copulas are not identifiable: \(c_{23}\), \(c_{13;2}\), \(c_{24;3}\), \(c_{14;23}\). Let the corresponding copula parameters be $$\boldsymbol{\theta}_{unid} = (\theta_{23}, \theta_{13;2}, \theta_{24;3}, \theta_{14;23})'.$$ The allowable range for this parameter vector depends on the corresponding copula families. For parsimony and comparability across different copula families, the sampling procedure consists of two steps:
Sample Spearman's rho parameters from a uniform distribution, $$\boldsymbol{\rho}_{unid} = (\rho_{23}, \rho_{13;2}, \rho_{24;3}, \rho_{14;23})' \sim U(\boldsymbol{a}, \boldsymbol{b}).$$
Transform the sampled Spearman's rho parameters to the copula parameter scale, \(\boldsymbol{\theta}_{unid}\).
These two steps are repeated n_sim times.
In addition to range restrictions through the lower and upper arguments,
we allow for so-called conditional independence assumptions.
These assumptions entail that \(\rho_{13;2} = 0\) and \(\rho_{24;3} =
0\). Or in other words, \(U_1 \perp U_3 \, | \, U_2\) and \(U_2 \perp U_4 \, | \, U_3\).
In the context of a surrogate evaluation trial (where \((U_1, U_2, U_3,
U_4)'\) corresponds to the probability integral transformation of \((T_0,
S_0, S_1, T_1)'\)) this assumption could be justified by subject-matter knowledge.