Efficient Estimation of the Causal Effects of Stochastic Interventions

Authors: Nima Hejazi and David Benkeser

## What’s txshift?

The txshift R package is designed to provide facilities for the construction of efficient estimators of the counterfactual mean of an outcome under stochastic interventions that depend on the natural value of treatment (Dı́az and van der Laan 2012; Haneuse and Rotnitzky 2013). txshiftimplements and builds upon a simplified algorithm for the targeted maximum likelihood (TML) estimator of such a causal parameter, originally proposed by Dı́az and van der Laan (2018), and makes use of analogous machinery to compute an efficient one-step estimator (Pfanzagl and Wefelmeyer 1985). txshift integrates with the sl3 package (Coyle, Hejazi, Malenica, et al. 2022) to allow for ensemble machine learning to be leveraged in the estimation procedure.

For many practical applications (e.g., vaccine efficacy trials), observed data is often subject to a two-phase sampling mechanism (i.e., through the use of a two-stage design). In such cases, efficient estimators (of both varieties) must be augmented to construct unbiased estimates of the population-level causal parameter. Rose and van der Laan (2011) first introduced an augmentation procedure that relies on introducing inverse probability of censoring (IPC) weights directly to an appropriate loss function or to the efficient influence function estimating equation. txshift extends this approach to compute IPC-weighted one-step and TML estimators of the counterfactual mean outcome under a shift stochastic treatment regime. The package is designed to implement the statistical methodology described in Hejazi et al. (2020) and extensions thereof.

## Installation

For standard use, we recommend installing the package from CRAN via

install.packages("txshift")

Note: If txshift is installed from CRAN, the sl3, an enhancing dependency that allows ensemble machine learning to be used for nuisance parameter estimation, won’t be included. We highly recommend additionally installing sl3 from GitHub via remotes:

remotes::install_github("tlverse/sl3@master")

For the latest features, install the most recent stable version of txshift from GitHub via remotes:

remotes::install_github("nhejazi/txshift@master")

To contribute, install the development version of txshift from GitHub via remotes:

remotes::install_github("nhejazi/txshift@devel")

## Example

To illustrate how txshift may be used to ascertain the effect of a treatment, consider the following example:

library(txshift)
#> txshift v0.3.8: Efficient Estimation of the Causal Effects of Stochastic
#> Interventions
library(sl3)
set.seed(429153)

# simulate simple data
n_obs <- 500
W <- replicate(2, rbinom(n_obs, 1, 0.5))
A <- rnorm(n_obs, mean = 2 * W, sd = 1)
Y <- rbinom(n_obs, 1, plogis(A + W + rnorm(n_obs, mean = 0, sd = 1)))

# now, let's introduce a a two-stage sampling process
C_samp <- rbinom(n_obs, 1, plogis(W + Y))

# fit the full-data TMLE (ignoring two-phase sampling)
tmle <- txshift(
W = W, A = A, Y = Y, delta = 0.5,
estimator = "tmle",
g_exp_fit_args = list(
fit_type = "sl",
sl_learners_density = Lrnr_density_hse$new(Lrnr_hal9001$new())
),
Q_fit_args = list(fit_type = "glm", glm_formula = "Y ~ .")
)
tmle
#> Counterfactual Mean of Shifted Treatment
#> Intervention: Treatment + 0.5
#> txshift Estimator: tmle
#> Estimate: 0.7685
#> Std. Error: 0.019
#> 95% CI: [0.7292, 0.8037]

# fit a full-data one-step estimator for comparison (again, no sampling)
os <- txshift(
W = W, A = A, Y = Y, delta = 0.5,
estimator = "onestep",
g_exp_fit_args = list(
fit_type = "sl",
sl_learners_density = Lrnr_density_hse$new(Lrnr_hal9001$new())
),
Q_fit_args = list(fit_type = "glm", glm_formula = "Y ~ .")
)
os
#> Counterfactual Mean of Shifted Treatment
#> Intervention: Treatment + 0.5
#> txshift Estimator: onestep
#> Estimate: 0.7685
#> Std. Error: 0.019
#> 95% CI: [0.7292, 0.8037]

# fit an IPCW-TMLE to account for the two-phase sampling process
tmle_ipcw <- txshift(
W = W, A = A, Y = Y, delta = 0.5, C_samp = C_samp, V = c("W", "Y"),
estimator = "tmle", max_iter = 5, eif_reg_type = "glm",
samp_fit_args = list(fit_type = "glm"),
g_exp_fit_args = list(
fit_type = "sl",
sl_learners_density = Lrnr_density_hse$new(Lrnr_hal9001$new())
),
Q_fit_args = list(fit_type = "glm", glm_formula = "Y ~ .")
)
tmle_ipcw
#> Counterfactual Mean of Shifted Treatment
#> Intervention: Treatment + 0.5
#> txshift Estimator: tmle
#> Estimate: 0.7603
#> Std. Error: 0.0204
#> 95% CI: [0.718, 0.798]

# compare with an IPCW-agumented one-step estimator under two-phase sampling
os_ipcw <- txshift(
W = W, A = A, Y = Y, delta = 0.5, C_samp = C_samp, V = c("W", "Y"),
estimator = "onestep", eif_reg_type = "glm",
samp_fit_args = list(fit_type = "glm"),
g_exp_fit_args = list(
fit_type = "sl",
sl_learners_density = Lrnr_density_hse$new(Lrnr_hal9001$new())
),
Q_fit_args = list(fit_type = "glm", glm_formula = "Y ~ .")
)
os_ipcw
#> Counterfactual Mean of Shifted Treatment
#> Intervention: Treatment + 0.5
#> txshift Estimator: onestep
#> Estimate: 0.7601
#> Std. Error: 0.0204
#> 95% CI: [0.7178, 0.7979]

## Issues

If you encounter any bugs or have any specific feature requests, please file an issue. Further details on filing issues are provided in our contribution guidelines.

## Contributions

Contributions are very welcome. Interested contributors should consult our contribution guidelines prior to submitting a pull request.

## Citation

After using the txshift R package, please cite the following:

    @article{hejazi2020efficient,
author = {Hejazi, Nima S and {van der Laan}, Mark J and Janes, Holly
E and Gilbert, Peter B and Benkeser, David C},
title = {Efficient nonparametric inference on the effects of
stochastic interventions under two-phase sampling, with
applications to vaccine efficacy trials},
year = {2020},
doi = {10.1111/biom.13375},
url = {https://doi.org/10.1111/biom.13375},
journal = {Biometrics},
publisher = {Wiley Online Library}
}

@article{hejazi2020txshift-joss,
author = {Hejazi, Nima S and Benkeser, David C},
title = {{txshift}: Efficient estimation of the causal effects of
stochastic interventions in {R}},
year  = {2020},
doi = {10.21105/joss.02447},
url = {https://doi.org/10.21105/joss.02447},
journal = {Journal of Open Source Software},
publisher = {The Open Journal}
}

@software{hejazi2022txshift-rpkg,
author = {Hejazi, Nima S and Benkeser, David C},
title = {{txshift}: Efficient Estimation of the Causal Effects of
Stochastic Interventions},
year  = {2022},
doi = {10.5281/zenodo.4070042},
url = {https://CRAN.R-project.org/package=txshift},
note = {R package version 0.3.7}
}

• R/tmle3shift - An R package providing an independent implementation of the same core routines for the TML estimation procedure and statistical methodology as is made available here, through reliance on a unified interface for Targeted Learning provided by the tmle3 engine of the tlverse ecosystem.

• R/medshift - An R package providing facilities to estimate the causal effect of stochastic treatment regimes in the mediation setting, including classical (IPW) and augmented double robust (one-step) estimators. This is an implementation of the methodology explored by Dı́az and Hejazi (2020).

• R/haldensify - A minimal package for estimating the conditional density treatment mechanism component of this parameter based on using the highly adaptive lasso (Coyle, Hejazi, Phillips, et al. 2022; Hejazi, Coyle, and van der Laan 2020) in combination with a pooled hazard regression. This package implements a variant of the approach advocated by Dı́az and van der Laan (2011).

## Funding

The development of this software was supported in part through grants from the National Library of Medicine (award no. T32 LM012417) and the National Institute of Allergy and Infectious Diseases (award no. R01 AI074345) of the National Institutes of Health, as well as by the National Science Foundation (award no. DMS 2102840).

MIT License

Copyright (c) 2017-2022 Nima S. Hejazi

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.

## References

Coyle, Jeremy R, Nima S Hejazi, Ivana Malenica, Rachael V Phillips, and Oleg Sofrygin. 2022. sl3: Modern Machine Learning Pipelines for Super Learning. https://doi.org/10.5281/zenodo.1342293.

Coyle, Jeremy R, Nima S Hejazi, Rachael V Phillips, Lars W van der Laan, and Mark J van der Laan. 2022. hal9001: The Scalable Highly Adaptive Lasso. https://doi.org/10.5281/zenodo.3558313.

Dı́az, Iván, and Nima S Hejazi. 2020. “Causal Mediation Analysis for Stochastic Interventions.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 82 (3): 661–83. https://doi.org/10.1111/rssb.12362.

Dı́az, Iván, and Mark J van der Laan. 2011. “Super Learner Based Conditional Density Estimation with Application to Marginal Structural Models.” International Journal of Biostatistics 7 (1): 1–20.

———. 2012. “Population Intervention Causal Effects Based on Stochastic Interventions.” Biometrics 68 (2): 541–49.

———. 2018. “Stochastic Treatment Regimes.” In Targeted Learning in Data Science: Causal Inference for Complex Longitudinal Studies, 167–80. Springer Science & Business Media.

Haneuse, Sebastian, and Andrea Rotnitzky. 2013. “Estimation of the Effect of Interventions That Modify the Received Treatment.” Statistics in Medicine 32 (30): 5260–77.

Hejazi, Nima S, Jeremy R Coyle, and Mark J van der Laan. 2020. “hal9001: Scalable Highly Adaptive Lasso Regression in R.” Journal of Open Source Software 5 (53): 2526. https://doi.org/10.21105/joss.02526.

Hejazi, Nima S, Mark J van der Laan, Holly E Janes, Peter B Gilbert, and David C Benkeser. 2020. “Efficient Nonparametric Inference on the Effects of Stochastic Interventions Under Two-Phase Sampling, with Applications to Vaccine Efficacy Trials.” Biometrics 77 (4): 1241–53. https://doi.org/10.1111/biom.13375.

Pfanzagl, J, and W Wefelmeyer. 1985. “Contributions to a General Asymptotic Statistical Theory.” Statistics & Risk Modeling 3 (3-4): 379–88.

Rose, Sherri, and Mark J van der Laan. 2011. “A Targeted Maximum Likelihood Estimator for Two-Stage Designs.” International Journal of Biostatistics 7 (1): 1–21.