Generalising Heterogeneous Treatment Effects Under Runtime Confounding

Greek Stochastics 2024

Author

Vanessa Rodrgiuez

1 Simulation Settings Table

Simulation # Runtime XSX_S XμX_\mu τ\tau Sample Size
1 -1 -1 -1 -1 1000
2 -1 -1 -1 1 1000
3 1 -1 -1 -1 1000
4 -1 1 -1 -1 1000
5 1 1 -1 1 1000
6 -1 -1 1 -1 1000
7 1 -1 1 1 1000
8 -1 1 1 1 1000
9 1 1 1 -1 1000
10 -1 -1 -1 -1 10000
11 1 -1 -1 1 10000
12 -1 1 -1 1 10000
13 1 1 -1 -1 10000
14 -1 -1 1 1 10000
15 1 -1 1 -1 10000
16 -1 1 1 -1 10000
17 1 1 1 1 10000

2 Simulation Data Generating Process

Xi∼N(0,1),∀Xi∈XX_i \sim N(0,1), \quad \forall X_i \in X

Outcome model −1:μ0∼Xμ,1+Xμ,2+Xμ,1×Xμ,21:μ0∼∑i=15Xμ,𝑖+Xμ,1×Xμ,2 \begin{align} -1: \qquad \mu_0 &\sim X_{\mu,1}+X_{\mu,2}+X_{\mu, 1}×X_{\mu,2} \\ 1: \qquad\mu_0&∼\sum_{i=1}^5X_{\mu,𝑖} +X_{\mu, 1}×X_{\mu,2} \end{align} Y0=μ0+ϵ,ϵ∼N(0,1)Y^0 = \mu_0 + \epsilon, \quad \epsilon \sim N(0,1)

Tau model −1:𝜏∼−1+Xτ,1+Xτ,12+Xτ,2+Xτ,221:𝜏∼−1+Xτ,12+sin(Xτ,1)+Xτ,22+sin(Xτ,2) \begin{align} -1: \qquad 𝜏&∼−1+X_{\tau,1}+X_{\tau,1}^2+ X_{\tau,2}+X_{\tau,2}^2 \\ 1: \qquad 𝜏&∼−1+X_{\tau,1}^2+\sin(X_{\tau,1})+X_{\tau,2}^2+ \sin(X_{\tau,2}) \end{align} Y1=Y0+τY^1 = Y^0+\tau

Selection model

ℙ(S=1)=1/(1+exp(−mS(Xτ,XS)))\mathbb P(S=1) = 1/(1 + \exp(-m_S(X_\tau, X_S))) −1:mS=Xτ,1+XS−12𝔼[Xτ,1+XS]1:mS=Xτ,1+Xτ,2+XS−𝔼[Xτ,1+Xτ,2+XS] \begin{align} -1: \qquad m_S & = X_{\tau,1} + X_S - \frac 12 \mathbb E[ X_{\tau,1} + X_S]\\ 1: \qquad m_S & = X_{\tau,1} + X_{\tau,2} + X_S - \mathbb E[ X_{\tau,1} + X_{\tau,2} + X_S] \end{align}

S∼Bernoulli(ℙ(S=1))S\sim \text{Bernoulli}(\mathbb P(S=1))

3 References

(Robertson, Steingrimsson, and Dahabreh 2023), (Künzel et al. 2019), (Jacob 2020), (Coston, Kennedy, and Chouldechova 2021)

References

Coston, Amanda, Edward H. Kennedy, and Alexandra Chouldechova. 2021. “Counterfactual Predictions Under Runtime Confounding,” April. https://doi.org/10.48550/arXiv.2006.16916.
Jacob, Daniel. 2020. “Cross-Fitting and Averaging for Machine Learning Estimation of Heterogeneous Treatment Effects,” August.
Künzel, Sören R., Jasjeet S. Sekhon, Peter J. Bickel, and Bin Yu. 2019. “Metalearners for Estimating Heterogeneous Treatment Effects Using Machine Learning.” Proceedings of the National Academy of Sciences 116 (10): 4156–65. https://doi.org/10.1073/pnas.1804597116.
Robertson, Sarah E., Jon A. Steingrimsson, and Issa J. Dahabreh. 2023. “Regression-Based Estimation of Heterogeneous Treatment Effects When Extending Inferences from a Randomized Trial to a Target Population.” European Journal of Epidemiology 38 (2): 123–33. https://doi.org/10.1007/s10654-022-00901-5.