‘ind.decomp’ estimates how much initial disparities would persist under two scenarios: (i) Everyone follows the optimal treatment regime (OTR), yielding the Individualized Controlled Direct Effect (ICDE); (ii) The proportion of individuals following the OTR is equalized across groups, yielding the Individualized Interventional Effect (IIE). This function implements the method proposed by Park, Kang, and Lee (2025+).
ind.decomp(outcome, group, group.ref, risk.factor, intermediates, moderators,
covariates, data, B, cluster = NULL)a matrix containing the point estimates for:
the initial disparity and the proportion recommended for treatment,
the disparity remaining and percent reduction based on individualized conditional effects,
the disparity remaining, disparity reduction, and percent reduction based on individualized intervention effects.
It also returns the nonparametric bootstrap confidence intervals for each estimate.
a character string indicating the name of the outcome.
a character string indicating the name of the social group indicator such as race or gender.
reference group of the social group indicator.
a character string indicating the name of the risk factor.
vector containing the name of intermediate confounders.
a character string indicating the name of variables that have heterogeneous effects on the outcome based on the risk factor.
a vector containing the name of baseline covariate variable(s).
The data set for analysis (data.frame).
Number of bootstrapped samples in the causal decomposition analysis.
a vector of cluster indicators for the bootstrap. If provided, the cluster bootstrap is used. Default is 'NULL'.
Soojin Park, University of California, Riverside, soojinp@ucr.edu; Suyeon Kang, University of Central Florida, suyeon.kang@ucf.edu; Karen Xu, University of California, Riverside, karenxu@ucr.edu.
This function first estimates the initial disparity, defined as the average difference in an outcome between groups within the same levels of outcome-allowable covariates. Formally, $$\tau_{c} \equiv E[Y \mid R = 1, c] - E[Y \mid R = 0, c], \quad \text{for } c \in \mathcal{C}$$ where \(R = 1\) is the comparison group and \(R = 0\) is the reference group. The term \(c \in \mathcal{C}\) represents baseline covariates.
For individualized conditional effects, disparity remaining is defined as $$ \zeta_{c}^{\mathrm{ICDE}}(d^{opt}) \equiv E[Y(d^{opt}) \mid R = 1, c] - E[Y(d^{opt}) \mid R = 0, c], \quad \text{for } c \in \mathcal{C} $$ where \(d^{opt}\) is an optimal value for risk factor \(M\). This definition of disparity remaining states the difference in an outcome between the comparison and reference groups after setting their risk factor \(M\) to the optimal value obtained from the OTRs.
For individualized interventional effects, disparity reduction and disparity remaining due to equalizing compliance rates across groups can be expressed as follows: $$ \delta_{c}^{\mathrm{IIE}}(1) \equiv E[Y \mid R = 1, c] - E[Y(K) \mid R = 1, c] $$ $$ \zeta_{c}^{\mathrm{IIE}}(0) \equiv E[Y(K) \mid R = 1, c] - E[Y \mid R = 0, c] $$ where \(Y(K)\) represents a potential outcome under the value of \(M\) that is determined by a random draw from the compliance distribution of the reference group among individuals with the same levels of target-factor-allowable covariates. Disparity reduction (\(\delta_{c}^{\mathrm{IIE}}(1)\)) represents the disparity in outcomes among a comparison group after intervening to setting the compliance rate equal to that of a reference group within the same target-factor-allowable covariate levels. Disparity remaining (\(\zeta_{c}^{\mathrm{IIE}}(0)\)) quantifies the outcome difference that persists between a comparison and reference group even after the intervention.
Park, S., Kang, S., & Lee, C. (2025). Simulation-Based Sensitivity Analysis in Optimal Treatment Regimes and Causal Decomposition with Individualized Interventions. arXiv preprint arXiv:2506.19010.
ind.sens
data(idata)
# NOTE: We recommend using at least B = 500 boostrap replications.
results_unadj <- ind.decomp(outcome = "Y", group = "R", group.ref = "0", risk.factor = "M",
intermediates = c("X1", "X2", "X3"), moderators = c("X1", "X2"),
covariates = "C", data = idata, B = 50)
results_unadj
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