# Concluding Thoughts

\[ \renewcommand{\P}{\mathsf{P}} \newcommand{\m}{\mathsf{m}} \newcommand{\p}{\mathsf{p}} \newcommand{\q}{\mathsf{q}} \newcommand{\bb}{\mathsf{b}} \newcommand{\g}{\mathsf{g}} \newcommand{\rr}{\mathsf{r}} \newcommand{\IF}{\mathbb{IF}} \newcommand{\dd}{\mathsf{d}} \newcommand{\Pn}{$\mathsf{P}_n$} \newcommand{\E}{\mathsf{E}} \]

### Thank you!

Thank you for attending this workshop! We encourage you to reach out to us with any questions or feedback.

### Including in a manuscript

If you decide to use `lmtp`

in your research, here’s a summary that can be included for explaining the methodology. It is based on the explanation in Rudolph et al. (2024):

We estimated the effect of [exposure] on [outcome], adjusting for covariates. This effect can be written: \(\E(Y^{\dd, C=1} - Y^{C=1}),\) [change the preceding to correspond to your effect] where \(\E(Y^{C=1})\) [change as applicable] denotes the expected value of the counterfactual outcome had the exposure not been intervened on (i.e., remained as observed) and had no one been censored, and where \(\E(Y^{\dd, C=1})\) [change as applicable] denotes the expected value of the counterfactual outcome had the exposure been intervened on as dictated by the function \(\dd(A)\) and had no one been censored. We defined \(\dd(A)\) as a hypothetical intervention that increased the value of the exposure by 20% (i.e., multiplied each person’s value by 1.2). [change the preceding to correspond to your intervention, \(\dd\)]

The above statistical estimand is a type of ``modified treatment policy’’ (Haneuse and Rotnitzky (2013), Muñoz and Van Der Laan (2012), Young, Hernán, and Robins (2014), Dı́az et al. (2023)). The statistical estimand can be interpreted causally under the identifying assumptions of: 1) conditional exchangeability, meaning that there is no unobserved/unmeasured confounding of the relationship between the set of treatments and outcome conditional on covariates and that there is no unobserved/unmeasured confounding between censoring and the outcome conditional on the covariates and treatment; 2) positivity, and 3) consistency.

We estimated this statistical estimand using a doubly robust, nonparametric targeted minimum loss-based estimator (Dı́az et al. (2023), Williams and Dı́az (2023)). A cross-fitted version of this estimator was used with [fill in number]-folds. This estimator fits regressions for the outcome mechanism, treatment mechanism, and censoring mechanism. These regressions were fit using an ensemble of machine learning algorithms (van der Laan, Polley, and Hubbard (2007)) consisting of [fill in the algorithms included in your superlearner library].

You can cite the package using the following BibTeX entries:

```
@article{diaz2023nonparametric,
title={Nonparametric causal effects based on longitudinal modified treatment policies},
author={D{\'\i}az, Iv{\'a}n and Williams, Nicholas and Hoffman, Katherine L and Schenck, Edward J},
journal={Journal of the American Statistical Association},
volume={118},
number={542},
pages={846--857},
year={2023},
publisher={Taylor \& Francis}
}
```

```
@article{williams2023lmtp,
title={lmtp: An {R} package for estimating the causal effects of modified treatment policies},
author={Williams, Nicholas and D{\'\i}az, Iv{\'a}n},
journal={Observational Studies},
volume={9},
number={2},
pages={103--122},
year={2023},
publisher={University of Pennsylvania Press}
}
```

You can reference this workshop with:

```
@inproceedings{lmtpSER,
author = {Williams, Nicholas and D{\'\i}az, Iv{\'a}n and Rudolph, Kara E},
title = {Beyond the Average Treatment Effect},
booktitle = {Society for Epidemiologic Research Conference},
year = {2024},
address = {Austin, TX},
month = {June},
organization = {Society for Epidemiologic Research},
note = {Additional Note},
url = {https://nt-williams.github.io/lmtp-workshop/}
}
```

## References

*Journal of the American Statistical Association*118 (542): 846–57.

*Statistics in Medicine*32 (30): 5260–77.

*Biometrics*68 (2): 541–49.

*arXiv Preprint arXiv:2404.11802*.

*Stat Appl Genet Mol Biol*6(1) (25, 1): Article 25. doi: 10.2202/1544–6115.1309.

*Observational Studies*9 (2): 103–22.

*Epidemiologic Methods*3 (1): 1–19.