The ica_SurvSurv_sens() function performs the sensitivity analysis
for the individual causal association (ICA) as described by Stijven et
al. (2022).
ica_SurvSurv_sens(
fitted_model,
n_sim,
n_prec,
minfo_prec = 0,
restr = TRUE,
copula_family2,
ncores = 1,
get_marg_tau = FALSE,
cond_ind = FALSE
)A data frame is returned. Each row represents one replication in the sensitivity analysis. The returned data frame always contains the following columns:
kendall, sp_rho, minfo: ICA as quantified by Kendall's \(\tau\),
Spearman's \(\rho\), and the mutual information, respectively.
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.
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 get_marg_tau
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 potential time-to-composite events
and/or time-to-true endpoint event. In contrary, the estimated association
parameters from fit_model_SurvSurv() refer to associations between the
time-to-surrogate event and time-to true endpoint event.
prop_harmed, prop_protected, prop_always, prop_never: proportions
of the corresponding population strata in each replication. These are defined
in Nevo and Gorfine (2022).
Returned value from fit_model_SurvSurv(). 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).
Number of Monte-Carlo samples for the numerical approximation of the ICA in each replication of the sensitivity analysis.
Number of quasi Monte-Carlo samples for the numerical
integration to obtain the mutual information. If this value is 0 (default),
the mutual information is not computed and NA is returned for that column.
Default value should not be modified by the user.
Parametric family of the unidentifiable copulas in the
D-vine copula. One of the following parametric copula families:
"clayton", "frank", "gaussian", or "gumbel".
Number of cores used in the sensitivity analysis. The computations are computationally heavy, and this option can speed things up considerably.
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.
Boolean.
TRUE: Assume conditional independence (see Additional Assumptions).
FALSE (default): Conditional independence is not assumed.
In the causal-inference framework to evaluate surrogate endpoints, the ICA is
the measure of primary interest. This measure quantifies the association
between the individual causal treatment effects on the surrogate (\(\Delta
S\)) and on the true endpoint (\(\Delta T\)). Stijven et al. (2022) proposed
to quantify this association through the squared informational coefficient of
correlation (SICC or \(R^2_H\)), which is based on information-theoretic
principles. Indeed, \(R^2_H\) is a transformation of the mutual information
between \(\Delta S\) and \(\Delta T\), $$R^2_H = 1 - e^{-2 \cdot
I(\Delta S; \Delta T)}.$$ By token of that transformation, \(R^2_H\) is
restricted to the unit interval where 0 indicates independence, and 1 a
functional relationship between \(\Delta S\) and \(\Delta T\). The mutual
information is returned by ica_SurvSurv_sens() if a non-zero value is
specified for minfo_prec (see Arguments).
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 to ica_SurvSurv_sens() through the cond_ind argument:
$$\tilde{S}_0 \perp \!\!\! \perp T_1 | \tilde{S}_1 \; \text{and} \;
\tilde{S}_1 \perp \!\!\! \perp T_0 | \tilde{S}_0 .$$ This can informally be
interpreted as ``what the control treatment does to the surrogate does not
provide information on the survival time under experimental treatment if we
already know what the experimental treatment does to the surrogate", and
analogously when control and experimental treatment are interchanged.
The inequality type of assumptions have to be imposed on the data frame that
is returned by the ica_SurvSurv_sens() function; those assumptions are thus
imposed after running the sensitivity analysis. If get_marg_tau is set to
TRUE, the returned data frame contains two types of additional unverifiable
quantities that differ across replications of the sensitivity analysis: (i)
the unconditional Spearman's \(\rho\) for all pairs of potential outcomes,
and (ii) the proportions of the population strata as defined by Nevo and
Gorfine (2022). More details on the interpretation and use of these
assumptions can be found in Stijven et al. (2022).
Stijven, F., Alonso, a., Molenberghs, G., Van Der Elst, W., Van Keilegom, I. (2022). An information-theoretic approach to the evaluation of time-to-event surrogates for time-to-event true endpoints based on causal inference.
Nevo, D., & Gorfine, M. (2022). Causal inference for semi-competing risks data. Biostatistics, 23 (4), 1115-1132
library(Surrogate)
data("Ovarian")
# For simplicity, data is not recoded to semi-competing risks format, but the
# data are left in the composite event format.
data = data.frame(
Ovarian$Pfs,
Ovarian$Surv,
Ovarian$Treat,
Ovarian$PfsInd,
Ovarian$SurvInd
)
ovarian_fitted =
fit_model_SurvSurv(data = data,
copula_family = "clayton",
nknots = 1)
# Illustration with small number of replications and low precision
ica_SurvSurv_sens(ovarian_fitted,
n_sim = 5,
n_prec = 2000,
copula_family2 = "clayton")
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