sample_copula_parameters {Surrogate}

Sample Unidentifiable Copula Parameters

Description

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.

Usage

sample_copula_parameters(
  copula_family2,
  n_sim,
  eq_cond_association = FALSE,
  lower = c(-1, -1, -1, -1),
  upper = c(1, 1, 1, 1)
)

Arguments

copula_family2

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}).

n_sim

Number of copula parameter vectors to be sampled.

eq_cond_association

(boolean) Indicates whether \rho_{13;2} and \rho_{24;3} are set equal.

lower

(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.

upper

(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).

Value

A n_sim by 4 numeric matrix where each row corresponds to a sample for \boldsymbol{\theta}_{unid}.

Sampling

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:

1. 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}).

2. Transform the sampled Spearman's rho parameters to the copula parameter scale, \boldsymbol{\theta}_{unid}.

These two steps are repeated n_sim times.

Conditional Independence

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.