R: Sensitivity analysis for individual causal association

sensitivity_analysis_SurvSurv_copula {Surrogate}

R Documentation

Sensitivity analysis for individual causal association

Description

The sensitivity_analysis_SurvSurv_copula() function performs the sensitivity analysis for the individual causal association (ICA) as described by Stijven et al. (2024).

Usage

sensitivity_analysis_SurvSurv_copula(
  fitted_model,
  composite = TRUE,
  n_sim,
  eq_cond_association = TRUE,
  lower = c(-1, -1, -1, -1),
  upper = c(1, 1, 1, 1),
  degrees = c(0, 90, 180, 270),
  marg_association = TRUE,
  copula_family2 = fitted_model$copula_family[1],
  n_prec = 5000,
  ncores = 1,
  sample_plots = NULL,
  mutinfo_estimator = NULL,
  restr_time = +Inf
)

Arguments

`fitted_model`	Returned value from `fit_model_SurvSurv()`. This object contains the estimated identifiable part of the joint distribution for the potential outcomes.
`composite`	(boolean) If `composite` is `TRUE`, then the surrogate endpoint is a composite of both a "pure" surrogate endpoint and the true endpoint, e.g., progression-free survival is the minimum of time-to-progression and time-to-death.
`n_sim`	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).
`eq_cond_association`	Boolean. `TRUE` (default): Assume that the association in `(\tilde{S}_1, T_0)' \| \tilde{S}_0` and `(\tilde{S}_0, T_1)' \| \tilde{S}_1` are the same. `FALSE`: There is not specific a priori relationship between the above two associations.
`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)`.
`degrees`	(numeric) vector with copula rotation degrees. Defaults to `c(0, 90, 180, 270)`. This argument is not used for the Gaussian and Frank copulas since they already allow for positive and negative associations.
`marg_association`	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.
`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_prec`	Number of Monte-Carlo samples for the numerical approximation of the ICA in each replication of the sensitivity analysis.
`ncores`	Number of cores used in the sensitivity analysis. The computations are computationally heavy, and this option can speed things up considerably.
`sample_plots`	Indices for replicates in the sensitivity analysis for which the sampled individual treatment effects are plotted. Defaults to `NULL`: no plots are displayed.
`mutinfo_estimator`	Function that estimates the mutual information between the first two arguments which are numeric vectors. Defaults to `FNN::mutinfo()` with default arguments. @param plot_deltas (logical) Plot the sampled individual treatment effects?
`restr_time`	Restriction time for the potential outcomes. Defaults to `+Inf` which means no restriction. Otherwise, the sampled potential outcomes are replace by `pmin(S0, restr_time)` (and similarly for the other potential outcomes).

Value

A data frame is returned. Each row represents one replication in the sensitivity analysis. The returned data frame always contains the following columns:

ICA, sp_rho: ICA as quantified by R^2_h(\Delta S^*, \Delta T^*) and \rho_s(\Delta S, \Delta T).
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. Also note that sp_s1t1 is constant whereas sp_s0t0 is not. This is a particularity of the MC procedure to calculate both measures and thus not a bug.
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).

Information-Theoretic Causal Inference Framework

The information-theoretic causal inference (ITCI) is a general framework to evaluate surrogate endpoints in the single-trial setting (Alonso et al., 2015). In this framework, we focus on the individual causal effects, \Delta S = S_1 - S_0 and \Delta T = T_1 - T_0 where S_z and T_z are the potential surrogate end true endpoint under treatment Z = z.

In the ITCI framework, we say that S is a good surrogate for T if \Delta S conveys a substantial amount of information on \Delta T (Alonso, 2018). This amount of shared information can generally be quantified by the mutual information between \Delta S and \Delta T, denoted by I(\Delta S; \Delta T). However, the mutual information lies in [0, + \infty] which complicates the interpretation. In addition, the mutual information may not be defined in specific scenarios where absolute continuity of certain probability measures fails. Therefore, the mutual information is transformed, and possibly modified, to enable a simple interpretation in light of the definition of surrogacy. The resulting measure is termed the individual causal association (ICA). This is explained in the next sections.

While the definition of surrogacy in the ITCI framework rests on information theory, shared information is closely related to statistical association. Hence, we can also define the ICA in terms of statistical association measures, like Spearman's rho and Kendall's tau. The advantage of the latter are that they are well-known, simple and rank-based measures of association.

Surrogacy in The Survival-Survival Setting

General Introduction

Stijven et al. (2024) proposed to quantify the ICA through the squared informational coefficient of correlation (SICC or R^2_H), which is a transformation of the mutaul information to the unit interval:

R^2_H = 1 - e^{-2 \cdot I(\Delta S; \Delta T)}

where 0 indicates independence, and 1 a functional relationship between \Delta S and \Delta T. The ICA (or a modified version, see next) is returned by sensitivity_analysis_SurvSurv_copula(). Concurrently, the Spearman's correlation between \Delta S and \Delta T is also returned.

Issues with Composite Endpoints

In the survival-survival setting where the surrogate is a composite endpoint, care should be taken when defining the mutual information. Indeed, when S_z is progression-free survival and T_z is overall survival, there is a probability atom in the joint distribution of (S_z, T_z)' because P(S_z = T_z) > 0. In other words, there are patient that die before progressing. While this probability atom is correctly taken into account in the models fitted by fit_model_SurvSurv(), this probability atom reappears when considering the distribution of (\Delta S, \Delta T)' because P(\Delta S = \Delta T) > 0 if we are considering PFS and OS.

Because of the atom in the distribution of (\Delta S, \Delta T)', the corresponding mutual information is not defined. To solve this, the mutual information is computed excluding the patients for which \Delta S = \Delta T when composite = TRUE. The proportion of excluded patients is, among other things, returned when marginal_association = TRUE. This is the proportion of "never" patients following the classification of Nevo and Gorfine (2022). See also Additional Assumptions.

This modified version of the ICA quantifies the surrogacy of S when "adjusted for the composite nature of S". Indeed, we exclude patients where \Delta S perfectly predicts \Delta T *just because S is a composite of T (and other variables).

Other (rank-based) statistical measures of association, however, remain well-defined and are thus computed without excluding any patients.

Sensitivity Analysis

Monte Carlo Approach

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.

Intervals of Ignorance and Uncertainty

The results of the sensitivity analysis can be formalized (and summarized) in intervals of ignorance and uncertainty using sensitivity_intervals_Dvine().

Additional Assumptions

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:

\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. (2024).

References

Alonso, A. (2018). An information-theoretic approach for the evaluation of surrogate endpoints. In Wiley StatsRef: Statistics Reference Online. John Wiley & Sons, Ltd.

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., and Burzykowski, T. (2015). On the relationship between the causal-inference and meta-analytic paradigms for the validation of surrogate endpoints. Biometrics 71, 15–24.

Stijven, F., Alonso, a., Molenberghs, G., Van Der Elst, W., Van Keilegom, I. (2024). 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

Examples


# Load Ovarian data
data("Ovarian")
# Recode the Ovarian data in the semi-competing risks format.
data_scr = data.frame(
  ttp = Ovarian$Pfs,
  os = Ovarian$Surv,
  treat = Ovarian$Treat,
  ttp_ind = ifelse(
    Ovarian$Pfs == Ovarian$Surv &
      Ovarian$SurvInd == 1,
    0,
    Ovarian$PfsInd
  ),
  os_ind = Ovarian$SurvInd
)
# Fit copula model.
fitted_model = fit_model_SurvSurv(data = data_scr,
                                  copula_family = "clayton",
                                  n_knots = 1)
# Illustration with small number of replications and low precision
sens_results = sensitivity_analysis_SurvSurv_copula(fitted_model,
                  n_sim = 5,
                  n_prec = 2000,
                  copula_family2 = "clayton",
                  eq_cond_association = TRUE)
# Compute intervals of ignorance and uncertainty. Again, the number of
# bootstrap replications should be larger in practice.
sensitivity_intervals_Dvine(fitted_model, sens_results, B = 10)

[Package Surrogate version 3.2.5 Index]