sensitivity_intervals_Dvine {Surrogate}

Compute Sensitivity Intervals

Description

sensitivity_intervals_Dvine() computes the estimated intervals of ignorance and uncertainty within the information-theoretic causal inference framework when the data are modeled with a D-vine copula model.

Usage

sensitivity_intervals_Dvine(
  fitted_model,
  sens_results,
  measure = "ICA",
  B = 200,
  alpha = 0.05,
  n_prec = 5000,
  mutinfo_estimator = NULL,
  restr_time = +Inf,
  ncores = 1
)

Arguments

fitted_model

Returned value from fit_model_SurvSurv(). This object contains the estimated identifiable part of the joint distribution for the potential outcomes.

sens_results

Dataframe returned by sensitivity_analysis_SurvSurv_copula(). If additional assumptions need to be incorporated, this dataframe can first be filtered.

measure

Compute intervals for which measure of surrogacy? Defaults to "ICA". See first column names of sens_results for other possibilities.

B

Number of bootstrap replications

alpha

(numeric) 1 - alpha is the level of the confidence interval

n_prec

Number of Monte-Carlo samples for the numerical approximation of the ICA in each replication of the sensitivity analysis.

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).

ncores

Number of cores used in the sensitivity analysis. The computations are computationally heavy, and this option can speed things up considerably.

Value

An S3 object of the class sensitivity_intervals_Dvine which can be printed.

Intervals of Ignorance and Uncertainty

Vansteelandt et al. (2006) formalized sensitivity analysis for partly identifiable parameters in the context of missing data and MNAR. These concepts can be applied to the estimation of the ICA. Indeed, the ICA is also partly identifiable because 50% if the potential outcomes are missing.

Vansteelandt et al. (2006) replace a point estimate with a interval estimate: the estimated interval of ignorance. In addition, they proposed several extension of the classic confidence interval together with appropriate definitions of coverage; these are termed intervals of uncertainty.

sensitivity_intervals_Dvine() implements the estimated interval of ignorance and the pointwise and strong intervals of uncertainty. Let \boldsymbol{\nu}_l and \boldsymbol{\nu}_u be the values for the sensitivity parameter that lead to the lowest and largest ICA, respectively, while fixing the identifiable parameter at its estimated value \hat{\boldsymbol{\beta}}. See also summary_level_bootstrap_ICA(). The following intervals are implemented:

1. Estimated interval of ignorance. This interval is defined as [ICA(\hat{\boldsymbol{\beta}}, \boldsymbol{\nu}_l), ICA(\hat{\boldsymbol{\beta}}, \boldsymbol{\nu}_u)].

2. Pointiwse interval of uncertainty. Let C_l (and C_u) be the lower (and upper) limit of a one-sided 1 - \alpha CI for ICA(\boldsymbol{\beta_0}, \boldsymbol{\nu}_l) (and ICA(\boldsymbol{\beta_0}, \boldsymbol{\nu}_l)). This interval is then defined as [C_l, C_u] when the ignorance is much larger than the statistical imprecision.

3. Strong interval of uncertainty. Let C_l (and C_u) be the lower (and upper) limit of a two-sided 1 - \alpha CI for ICA(\boldsymbol{\beta_0}, \boldsymbol{\nu}_l) (and ICA(\boldsymbol{\beta_0}, \boldsymbol{\nu}_l)). This interval is then defined as [C_l, C_u].

The CIs, which are need for the intervals of uncertainty, are based on percentile bootstrap confidence intervals, as documented in summary_level_bootstrap_ICA(). In addition, \boldsymbol{\nu}_l is not known. Therefore, it is estimated as

\arg \min_{\boldsymbol{\nu} \in \Gamma} ICA(\hat{\boldsymbol{\beta}}, \boldsymbol{\nu}),

and similarly for \boldsymbol{\nu}_u.

References

Vansteelandt, Stijn, et al. "Ignorance and uncertainty regions as inferential tools in a sensitivity analysis." Statistica Sinica (2006): 953-979.

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)