Efficient Causal Mediation Analysis with Intermediate Confounders

**Authors:** Nima Hejazi, Iván Díaz, and Kara Rudolph

`medoutcon`

?The `medoutcon`

R package provides facilities for efficient estimation of stochastic (in)direct effects that measure the impact of a treatment variable *A* on an outcome variable *Y*, through a direct path (through *A* only) and an indirect path (through a set of mediators *M* only), in the presence of an intermediate **med**iator-**out**come **con**founder *Z*, itself affected by the treatment *A*. While the proposed approach is similar to those appearing in VanderWeele, Vansteelandt, and Robins (2014), Rudolph et al. (2017), and Zheng and van der Laan (2017), `medoutcon`

is designed as a software implementation to accompany the methodology proposed in Díaz et al. (2020). Both an efficient one-step bias-corrected estimator with cross-fitting (Pfanzagl and Wefelmeyer 1985; Zheng and van der Laan 2011; Chernozhukov et al. 2018) and a cross-validated targeted minimum loss estimator (TMLE) (van der Laan and Rose 2011; Zheng and van der Laan 2011) are made available. `medoutcon`

integrates with the `sl3`

R package (Coyle et al. 2020) to leverage statistical machine learning in the estimation procedure.

Install the most recent *stable release* from GitHub via `remotes`

:

remotes::install_github("nhejazi/medoutcon")

To illustrate how `medoutcon`

may be used to estimate stochastic interventional (in)direct effects of the exposure (`A`

) on the outcome (`Y`

) in the presence of mediator(s) (`M`

) and a mediator-outcome confounder (`Z`

), consider the following example:

library(data.table) library(tidyverse) library(medoutcon) set.seed(1584) # produces a simple data set based on ca causal model with mediation make_example_data <- function(n_obs = 1000) { ## baseline covariates w_1 <- rbinom(n_obs, 1, prob = 0.6) w_2 <- rbinom(n_obs, 1, prob = 0.3) w_3 <- rbinom(n_obs, 1, prob = pmin(0.2 + (w_1 + w_2) / 3, 1)) w <- cbind(w_1, w_2, w_3) w_names <- paste("W", seq_len(ncol(w)), sep = "_") ## exposure a <- as.numeric(rbinom(n_obs, 1, plogis(rowSums(w) - 2))) ## mediator-outcome confounder affected by treatment z <- rbinom(n_obs, 1, plogis(rowMeans(-log(2) + w - a) + 0.2)) ## mediator -- could be multivariate m <- rbinom(n_obs, 1, plogis(rowSums(log(3) * w[, -3] + a - z))) m_names <- "M" ## outcome y <- rbinom(n_obs, 1, plogis(1 / (rowSums(w) - z + a + m))) ## construct output dat <- as.data.table(cbind(w = w, a = a, z = z, m = m, y = y)) setnames(dat, c(w_names, "A", "Z", m_names, "Y")) return(dat) } # set seed and simulate example data example_data <- make_example_data() w_names <- str_subset(colnames(example_data), "W") m_names <- str_subset(colnames(example_data), "M") # quick look at the data head(example_data) #> W_1 W_2 W_3 A Z M Y #> 1: 1 0 1 0 0 0 1 #> 2: 0 1 0 0 0 1 0 #> 3: 1 1 1 1 0 1 1 #> 4: 0 1 1 0 0 1 0 #> 5: 0 0 0 0 0 1 1 #> 6: 1 0 1 1 0 1 0 # compute one-step estimate of the interventional direct effect os_de <- medoutcon(W = example_data[, ..w_names], A = example_data$A, Z = example_data$Z, M = example_data[, ..m_names], Y = example_data$Y, effect = "direct", estimator = "onestep") summary(os_de) #> lwr_ci param_est upr_ci param_var eif_mean #> -0.1884 -0.0726 0.0433 0.0035 -4.4100e-17 #> estimator param #> onestep direct_effect # compute targeted minimum loss estimate of the interventional direct effect tmle_de <- medoutcon(W = example_data[, ..w_names], A = example_data$A, Z = example_data$Z, M = example_data[, ..m_names], Y = example_data$Y, effect = "direct", estimator = "tmle") summary(tmle_de) #> lwr_ci param_est upr_ci param_var eif_mean #> -0.203 -0.0859 0.0311 0.0036 4.4084e-03 #> estimator param #> tmle direct_effect

For details on how to use data adaptive regression (machine learning) techniques in the estimation of nuisance parameters, consider consulting the vignette that accompanies the package.

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

After using the `medoutcon`

R package, please cite the following:

```
@article{diaz2020nonparametric,
title={Non-parametric efficient causal mediation with intermediate
confounders},
author={D{\'\i}az, Iv{\'a}n and Hejazi, Nima S and Rudolph, Kara E
and {van der Laan}, Mark J},
year={2020},
url = {https://arxiv.org/abs/1912.09936},
doi = {},
journal={},
volume={},
number={},
pages={},
publisher={}
}
@manual{hejazi2020medoutcon,
author={Hejazi, Nima S and D{\'\i}az, Iv{\'a}n and Rudolph, Kara E},
title = {{medoutcon}: Efficient causal mediation analysis under
intermediate confounding},
year = {2020},
url = {https://github.com/nhejazi/medoutcon},
note = {R package version 0.1.0}
}
```

© 2020 Nima S. Hejazi

The contents of this repository are distributed under the MIT license. See below for details:

```
MIT License
Copyright (c) 2020 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,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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.
```

Chernozhukov, Victor, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. 2018. “Double/Debiased Machine Learning for Treatment and Structural Parameters.” *The Econometrics Journal* 21 (1). https://doi.org/10.1111/ectj.12097.

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.

Díaz, Iván, Nima S Hejazi, Kara E Rudolph, and Mark J van der Laan.

- “Non-Parametric Efficient Causal Mediation with Intermediate Confounders.” https://arxiv.org/abs/1912.09936.

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

Rudolph, Kara E, Oleg Sofrygin, Wenjing Zheng, and Mark J van der Laan.

- “Robust and Flexible Estimation of Stochastic Mediation Effects: A Proposed Method and Example in a Randomized Trial Setting.”
*Epidemiologic Methods*7 (1). De Gruyter.

van der Laan, Mark J, and Sherri Rose. 2011. *Targeted Learning: Causal Inference for Observational and Experimental Data*. Springer Science & Business Media.

VanderWeele, Tyler J, Stijn Vansteelandt, and James M Robins. 2014. “Effect Decomposition in the Presence of an Exposure-Induced Mediator-Outcome Confounder.” *Epidemiology (Cambridge, Mass.)* 25 (2). NIH Public Access: 300.

Zheng, Wenjing, and Mark J van der Laan. 2011. “Cross-Validated Targeted Minimum-Loss-Based Estimation.” In *Targeted Learning*, 459–74. Springer.

———. 2017. “Longitudinal Mediation Analysis with Time-Varying Mediators and Exposures, with Application to Survival Outcomes.” *Journal of Causal Inference* 5 (2). De Gruyter.