Perform Sensitivity Analysis for the Individual Causal Association with a Continuous Surrogate and Binary True Endpoint
sensitivity_analysis_BinCont_copula(
fitted_model,
n_sim,
cond_ind,
lower = c(-1, -1, -1, -1),
upper = c(1, 1, 1, 1),
marg_association = TRUE,
n_prec = 10000,
ncores = 1
)A data frame is returned. Each row represents one replication in the sensitivity analysis. The returned data frame always contains the following columns:
R2H, sp_rho, minfo: ICA as quantified by \(R^2_H\), Spearman's rho, and
Kendall's tau, respectively.
c12, c34: estimated copula parameters.
c23, c13_2, c24_3, c14_23: sampled copula parameters of the
unidentifiable copulas in the D-vine copula. The parameters correspond to
the parameterization of the copula_family2 copula as in the copula
R-package.
r12, r34: Fixed rotation parameters for the two identifiable copulas.
r23, r13_2, r24_3, r14_23: Sampled rotation parameters of the
unidentifiable copulas in the D-vine copula. These values are constant for
the Gaussian copula family since that copula is invariant to rotations.
The returned data frame also contains the following columns when
marg_association is TRUE:
sp_s0s1, sp_s0t0, sp_s0t1, sp_s1t0, sp_s1t1, sp_t0t1:
Spearman's rho between the corresponding potential outcomes. Note that these
associations refer to the observable potential outcomes. In contrary, the
estimated association parameters from fit_copula_model_BinCont() refer to
associations on a latent scale.
Returned value from fit_copula_model_BinCont(). This object
contains the estimated identifiable part of the joint distribution for the
potential outcomes.
Number of replications in the sensitivity analysis. This value should be large enough to sufficiently explore all possible values of the ICA. The minimally sufficient number depends to a large extent on which inequality assumptions are subsequently imposed (see Additional Assumptions).
Boolean.
TRUE: Assume conditional independence (see Additional Assumptions).
FALSE (default): Conditional independence is not assumed.
(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).
Boolean.
TRUE: Return marginal association measures
in each replication in terms of Spearman's rho. The proportion of harmed,
protected, never diseased, and always diseased is also returned. See also
Value.
FALSE (default): No additional measures are returned.
Number of Monte-Carlo samples for the numerical approximation of the ICA in each replication of the sensitivity analysis.
Number of cores used in the sensitivity analysis. The computations are computationally heavy, and this option can speed things up considerably.
In the information-theoretic causal-inference (ITCI) framework to evaluate surrogate endpoints, the ICA is the measure of primary interest. This measure quantifies how much information the individual causal treatment effect on the surrogate (\(\Delta S\)) provides on the individual causal treatment effect on the true endpoint (\(\Delta T\)). The mutual information between \(\Delta S\) and \(\Delta T\), denoted by \(I(\Delta S; \Delta T)\) is a natural candidate to quantify this amount of shared information. However, the mutual information is difficult to interpret as there does not exist a general upper bound. Alonso et al. (na) therfore proposed to quantify the ICA thorugh a transformation of the mutual information that is guaranteed to lie in the unit interval. It is the following measure, $$R^2_H = \frac{I(\Delta S; \Delta T)}{H(\Delta T)}$$ where \(H(\Delta T)\) is the entropy of \(\Delta T\). By token of that transformation of the mutual information, \(R^2_H\) is restricted to the unit interval where 0 indicates independence, and 1 a functional relationship between \(\Delta S\) and \(\Delta T\).
The association between \(\Delta S\) and \(\Delta T\) can also be quantified by Spearman's \(\rho\) (or Kendall's \(\tau\)). This quantity requires appreciably less computing time than the mutual information. This quantity is therefore always returned for every replication of the sensitivity analysis.
Because \(S_0\) and \(S_1\) are never simultaneously observed in the same
patient, \(\Delta S\) is not observable, and analogously for \(\Delta
T\). Consequently, the ICA is unidentifiable. This is solved by considering a
(partly identifiable) model for the full vector of potential outcomes,
\((T_0, S_0, S_1, T_1)'\). The identifiable parameters are estimated. The
unidentifiable parameters are sampled from their parameters space in each
replication of a sensitivity analysis. If the number of replications
(n_sim) is sufficiently large, the entire parameter space for the
unidentifiable parameters will be explored/sampled. In each replication, all
model parameters are "known" (either estimated or sampled). Consequently, the
ICA can be computed in each replication of the sensitivity analysis.
The sensitivity analysis thus results in a set of values for the ICA. This set can be interpreted as all values for the ICA that are compatible with the observed data. However, the range of this set is often quite broad; this means there remains too much uncertainty to make judgements regarding the worth of the surrogate. To address this unwieldy uncertainty, additional assumptions can be used that restrict the parameter space of the unidentifiable parameters. This in turn reduces the uncertainty regarding the ICA.
There are two possible types of assumptions that restrict the parameter space of the unidentifiable parameters: (i) equality type of assumptions, and (ii) inequality type of assumptions. These are discussed in turn in the next two paragraphs.
The equality assumptions have to be incorporated into the sensitivity
analysis itself. Only one type of equality assumption has been implemented;
this is the conditional independence assumption which can be specified
through the cond_ind argument:
$$\tilde{S}_0 \perp T_1 | \tilde{S}_1 \; \text{and} \;
\tilde{S}_1 \perp T_0 | \tilde{S}_0 .$$ This can informally be
interpreted as ``what the control treatment does to the surrogate does not
provide information on the true endpoint under experimental treatment if we
already know what the experimental treatment does to the surrogate", and
analogously when control and experimental treatment are interchanged. Note
that \(\tilde{S}_z\) refers to either the actual potential surrogate
outcome, or a latent version. This depends on the content of fitted_model.
The inequality type of assumptions have to be imposed on the data frame that
is returned by the current function; those assumptions are thus imposed
after running the sensitivity analysis. If marginal_association is set to
TRUE, the returned data frame contains additional unverifiable quantities
that differ across replications of the sensitivity analysis: (i) the
unconditional Spearman's \(\rho\) for all pairs of (observable/non-latent)
potential outcomes, and (ii) the proportions of the population strata as
defined by Nevo and Gorfine (2022) if semi-competing risks are present. More
details on the interpretation and use of these assumptions can be found in
Stijven et al. (2022).