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 a causal parameter defined as the counterfactual mean of an outcome under stochastic mechanisms for treatment assignment (Díaz and van der Laan 2012). txshift implements and builds upon a simplified algorithm for the targeted maximum likelihood (TML) estimator of such a causal parameter, originally proposed by Díaz 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 et al. 2020) 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 under a stochastic treatment regime.

## Installation

Install the most recent stable release from GitHub via devtools:

devtools::install_github("nhejazi/txshift", build_vignettes = FALSE)

## Example

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

library(txshift)
library(haldensify)
set.seed(429153)

# simulate simple data
n_obs <- 1000
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 <- 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_fit_args = list(fit_type = "hal",
n_bins = 5,
grid_type = "equal_mass",
lambda_seq = exp(seq(-1, -9, length = 300))),
Q_fit_args = list(fit_type = "glm",
glm_formula = "Y ~ .")
)
summary(tmle)
#>     lwr_ci  param_est     upr_ci  param_var   eif_mean  estimator     n_iter
#>     0.7474     0.7782     0.8061      2e-04 7.0199e-11       tmle          0

# 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_fit_args = list(fit_type = "hal",
n_bins = 5,
grid_type = "equal_mass",
lambda_seq = exp(seq(-1, -9, length = 300))),
Q_fit_args = list(fit_type = "glm",
glm_formula = "Y ~ .")
)
summary(os)
#>      lwr_ci   param_est      upr_ci   param_var    eif_mean   estimator
#>      0.7472      0.7779      0.8059       2e-04 -1.6704e-03     onestep
#>      n_iter
#>           0

# fit an IPCW-TMLE to account for the two-phase sampling process
ipcw_tmle <- txshift(W = W, A = A, Y = Y, delta = 0.5,
C = C, V = c("W", "Y"),
estimator = "tmle",
max_iter = 5,
ipcw_fit_args = list(fit_type = "glm"),
g_fit_args = list(fit_type = "hal",
n_bins = 5,
grid_type = "equal_mass",
lambda_seq =
exp(seq(-1, -9, length = 300))),
Q_fit_args = list(fit_type = "glm",
glm_formula = "Y ~ ."),
eif_reg_type = "glm"
)
summary(ipcw_tmle)
#>      lwr_ci   param_est      upr_ci   param_var    eif_mean   estimator
#>      0.7435      0.7765      0.8063       3e-04 -4.0365e-05        tmle
#>      n_iter
#>           1

# compare with an IPCW-agumented one-step estimator under two-phase sampling
ipcw_os <- txshift(W = W, A = A, Y = Y, delta = 0.5,
C = C, V = c("W", "Y"),
estimator = "onestep",
ipcw_fit_args = list(fit_type = "glm"),
g_fit_args = list(fit_type = "hal",
n_bins = 5,
grid_type = "equal_mass",
lambda_seq =
exp(seq(-1, -9, length = 300))),
Q_fit_args = list(fit_type = "glm",
glm_formula = "Y ~ ."),
eif_reg_type = "glm"
)
summary(ipcw_os)
#>      lwr_ci   param_est      upr_ci   param_var    eif_mean   estimator
#>      0.7427      0.7758      0.8058       3e-04 -2.0555e-03     onestep
#>      n_iter
#>           0

## Issues

If you encounter any bugs or have any specific feature requests, please file an issue.

## 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},
url = {http://arxiv.org/abs/2003.13771}
}

@manual{hejazi2020txshift,
author = {Hejazi, Nima S and Benkeser, David C},
title = {{txshift}: Efficient Estimation of the Causal Effects of
Stochastic Interventions},
year  = {2020},
url = {https://github.com/nhejazi/txshift},
note = {R package version 0.3.4}
}

## Funding

The development of this software was supported in part through a grant from the National Institutes of Health: T32 LM012417-02.

MIT License

Copyright (c) 2017-2020 Nima S. Hejazi

Permission is hereby granted, free of charge, to any person obtaining a copy
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in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
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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, and Oleg Sofrygin. 2020. sl3: Modern Pipelines for Machine Learning and Super Learning. https://github.com/tlverse/sl3. https://doi.org/10.5281/zenodo.1342293.

Coyle, Jeremy R, Nima S Hejazi, and Mark J van der Laan. 2019. hal9001: The Scalable Highly Adaptive Lasso. https://github.com/tlverse/hal9001. https://doi.org/10.5281/zenodo.3558313.

Díaz, Iván, and Nima S Hejazi. 2020. “Causal Mediation Analysis for Stochastic Interventions.” Journal of the Royal Statistical Society: Series B (Statistical Methodology). Wiley Online Library. https://doi.org/10.1111/rssb.12362.

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

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

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

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.” The International Journal of Biostatistics 7 (1): 1–21.