class: center, top, .title-slide, title-slide .title[ # Estimation and Testing of Direct Effects in Randomized Experiments With Non-Smooth General Interference ] .subtitle[ ## Preliminary Exam Presentation<br>.small[Slides online at <a href="https://rpubs.com/rmtrane/prelim-presentation">https://rpubs.com/rmtrane/prelim-presentation</a>] ] .author[ ### Ralph Møller Trane ] .institute[ ### University of Wisconsin–Madison<br><br> ] .date[ ### 2022-05-17</br>.small[(last compiled: 2022-05-17)] ] --- # Introductions * Ralph Trane * 4th year PhD student in the Department of Statistics * Born and raised in Denmark, alumni of the University of Copenhagen. * Previously worked as an Assistant Researcher in the Department of Ophthalmology and Visual Sciences here at UW--Madison under Karen Cruickshank. * Interested in estimation of causal effects under interference, the use of interactive visuals to communicate ideas, and more... * Previous project: Nonparametric Bounds in Two-Sample Summary-Data Mendelian Randomization Studies. * Published earlier this year in Statistics in Medicine [(Trane and Kang, 2022)](#referencescont) --- <!-- # Nonparametric Bounds in Two-Sample Summary-Data MR --> <!-- Elevator pitch: --> <!-- * Mendelian Randomization studies often take advantage of two-sample data --> <!-- * one on (instrument, exposure), another on (instrument, outcome) --> <!-- * We wanted to learn about the behavior of non-parametric bounds for the ATE based on three-leveled IV in two-sample data --> <!-- * for binary IV in one-sample data, bounds thoroughly studied by, for example, [(Manski, 1990; Balke and Pearl, 1997)](#references) --> <!-- * General observations: --> <!-- * two-sample bounds wider than one-sample bounds --> <!-- * in the kind of data often seen in two-sample MR studies, bounds so wide they offer little to no information --> <!-- Published earlier this year in Statistics in Medicine [(Trane and Kang, 2022)](#referencescont) --> <!-- --- --> # Overview Part I: Background 1. Setup of Potential Outcomes in the Presence of Interference * general potential outcomes, interference graph * estimand and estimation 2. Random Graph Asymptotics * introduction of the graphon, potential outcomes model as per [Li and Wager (2021)](#referencescont) * asymptotic normality Part II: Original Work 3. Improving/building on [Li and Wager (2021)](#referencescont) * more efficient estimator * more inclusive model 4. Summary/Future Work --- layout: true # Part I: General Setup --- **Potential Outcomes in the Presence of General Interference** We are interested in a *population* of subjects `\(i = 1, ..., n\)` Each subject is randomly assigned a *treatment* `\(W_i \in \{0,1\}\)` as `\(W_i \overset{\text{iid}}{\sim} \text{Bernoulli}(\pi)\)` `\((\pi \in (0,1))\)` For each subject, an *outcome* `\(Y_i\)` is observed. This observed outcome is one of many *potential outcomes* `\(Y_i(\boldsymbol{w}) \in \mathbb{R}\)`, which we assume exist and are **fixed** for all possible treatment assignment vectors `\(\boldsymbol{w} \in \{0,1\}^n\)`. Assume a *generalized SUTVA*, i.e. `\(Y_i = \sum_{\boldsymbol{w'} \in \{0,1\}^n} 1[\boldsymbol{W} = \boldsymbol{w}'] Y_i(\boldsymbol{w}')\)`. We use `\(Y_i(w_i, \boldsymbol{w}_{-i})\)` to indicate potential outcome for subject `\(i\)` when subject `\(i\)` assigned `\(w_i\)`, and others assigned `\(\boldsymbol{w}_{-i} \in \{0,1\}^{n-1}\)`. -- **Interference Graph** In parallel to the potential outcomes, we introduce the notion of an *interference graph*. An interference graph `\(\mathcal{G}\)` consists of nodes or vertices at the `\(n\)` subjects, and an edge or adjacency matrix `\(\boldsymbol{E} = \{E_{ij}\}_{i,j=1}^n\)`, where `\(E_{ij} = 1\)` if `\(W_j\)` influences `\(Y_i(\boldsymbol{W})\)`, and `\(E_{ij} = 0\)` otherwise. (Formally, `\(Y_i(w, \boldsymbol{w}_{-i}) = Y_i(w, \boldsymbol{w}_{-i}')\)` for all `\(\boldsymbol{w}_{-i}, \boldsymbol{w}_{-i}' \in \{0,1\}^{n-1}\)` where `\(w_j = w_j'\)` for all `\(j\)` with `\(E_{ij} = 1\)`.) --- **No Interference Example**: the effect of a drug (for example, aspirin) on disease status (headache). No interference seems reasonable, so `\(E_{ij} = 0\)` Two potential outcomes for each individual because `\(Y_i(w_i, \boldsymbol{w}_{-i}) = Y_i(w_i)\)`. An intuitive estimand: Average Treatment Effect (ATE) = `\(\frac{1}{n} \sum_{i=1}^n Y_i(1) - Y_i(0)\)`. --- **Interference Example**: the effect of vaccination status on disease status. No interference unlikely. `\(E_{ij} = 1\)` if `\(i\)` and `\(j\)` often spend time together. Here, the previous definition of the ATE not exactly useful. [Hudgens and Halloran (2008)](#references) provide a nice generalization of the ATE: `\begin{align} \bar{\tau}_\text{DIR} = \frac{1}{n} \sum_{i=1}^n \mathbb{E}[Y_i(1, \boldsymbol{W}_{-i})] - \mathbb{E}[Y_i (0, \boldsymbol{W}_{-i})]. \end{align}` When potential outcomes are considered fixed, the inner average is over treatment assignment. If no interference, `\(\bar{\tau}_\text{DIR} = \text{ATE}\)` because `\(\mathbb{E}[Y_i(1, \boldsymbol{W}_{-i})] = \mathbb{E}[Y_i(1)] = Y_i(1)\)`. -- Also, in this scenario, we could consider multiple causal estimands. For example, the indirect effect of treating a larger or smaller part of the population. Will defer this question for another time. --- ## Estimand & Estimation As hinted at on the previous slide, we are interested in estimating what we will refer to as the *direct effect*: `\begin{equation} \bar{\tau}_\text{DIR} = \frac{1}{n} \sum_{i=1}^n \mathbb{E}[Y_i(1, \boldsymbol{W}_{-i}) - Y_i (0, \boldsymbol{W}_{-i})]. \end{equation}` -- Nice result: an unbiased estimator for `\(\bar{\tau}_\text{DIR}\)` is the well-known Horvitz-Thompson estimator: $$ `\begin{equation} \hat{\tau}_\text{DIR}^\text{HT} = \frac{1}{n} \sum_{i=1}^n \frac{Y_i W_i}{\pi} - \frac{Y_i(1-W_i)}{1-\pi} \end{equation}` $$ This is unbiased for the ATE if no interference present, and `\(\hat{\tau}_\text{DIR}^\text{HT}\)` if interference is present. --- ## Open Questions So, we have an unbiased estimator for our estimand at interest. What we do not have, is ... an unbiased variance estimator of the HT estimator -- ... asymptotic normality of the HT estimator in general <!-- --- --> <!-- layout: false --> <!-- # Previous Work on Variance Estimation and Asymptotic Results --> <!-- ### Partial and Stratified Interference --> <!-- In some of the earliest work on estimation of direct effects under interference, [Hudgens and Halloran (2008)](#references) present unbiased variance estimators for the estimators `\(\hat{Y}_1 = \frac{\sum_{i=1}^n Y_iW_i}{\sum_{i=1}^n W_i}\)` and `\(\hat{Y}_0 = \frac{\sum_{i=1}^n Y_i(1-W_i)}{\sum_{i=1}^n 1-W_i}\)` under the two assumptions --> <!-- 1. partial interference --> <!-- 2. stratified interference --> <!-- Note: `\(\hat{Y}_1 - \hat{Y}_0\)` is unbiased for `\(\bar{\tau}_\text{DIR}\)`. --> <!-- -- --> <!-- Also construct finite sample confidence intervals for estimands of interest, including `\(\bar{\tau}_\text{DIR}\)`. --> <!-- --- --> <!-- # Previous Work on Variance Estimation and Asymptotic Results --> <!-- ### Partial Interference with Groups from Superpopulation --> <!-- To avoid the stratified interference assumption, [Liu, Hudgens, and Becker-Dreps (2016)](#references) consider the disjoint groups as random draws from a superpopulation. --> <!-- -- --> <!-- Consider an Inverse Probability-Weighted Estimator and a Hájek-type estimator in settings where treatment assignment is done independently. --> <!-- Show both estimators are asymptotically normal. --> --- layout: false # Part I: Possible Answers [Hudgens and Halloran (2008)](#references): assume partial and stratified interference to get conservative variance estimators, and finite sample confidence intervals. -- [Liu, Hudgens, and Becker-Dreps (2016)](#references): assume partial interference with groups being random draws from superpopulation to get asymptotic normality of Inverse Probability-Weighted estimator and Hájek-type estimator. -- Both heavily restrict the interference structure. **Example**: power plant emission data. [Papadogeorgou, Mealli, and Zigler (2019)](#references) consider power plant emission reduction technology. Can imagine power plant A influences power plant B influences power plant C. --- layout: true # Part I: Random Interference Graph --- [Li and Wager (2021)](#referencescont) show how viewing the interference graph as a random draw from a graphon can help get asymptotic results. -- This is done under the following assumptions on the interference graph. **Assumption 1**: `\(E_{ij} = E_{ji}\)` **Assumption 2**: `\(\mathcal{G}\)` is randomly generated as follows: * each individual has a random latent position `\(X_i \overset{\text{iid}}{\sim} \text{Uniform}(0,1)\)` * `\(G_n: [0,1]^2 \mapsto [0,1]\)` is a symmetric function * probability an edge exists based on the latent positions: `\(P(E_{ij} = 1 | X_i, X_j) = G_n(X_i, X_j)\)`, `\(i < j\)` `\(G_n\)` is called a *graphon* **Assumption 3**: `\(G_n(X_i, X_j) = \min\{1, \rho_n G(X_i,X_j)\}\)` where * `\(G(\cdot, \cdot): [0,1]^2 \mapsto [0, \infty)\)` * `\(0 < \rho_n \le 1\)` such that either `\(\rho_n = 1\)` or `\(\rho_n \to 0\)` and `\(\rho_n n \to \infty\)`. --- Also need a few assumptions on the potential outcomes. **Assumption 4**: `\(Y_i(w, \boldsymbol{w}_{-i}) = f_i(w, X_i, M_i / N_i; \epsilon_i)\)` where `\(M_i = \sum_{j \neq i} E_{ij} W_j\)` and `\(N_i = \sum_{j\neq i}E_{ij}\)`. (This is similar to the stratified interference assumption made by [Hudgens and Halloran (2008)](#references)) **Assumption 5**: `\(f_i\)` is three-times differentiable, and `\(|f_i|, |f_i'|, |f_i''|, |f_i'''| \le B\)` where the derivative is taken with respect to `\(M_i / N_i\)`. -- **Central Limit Theorem**: under Assumptions 1-5 and some mild assumptions on the graphon, `\begin{equation} \sqrt{n}(\hat{\tau}_\text{DIR}^\text{HT} - \bar{\tau}_\text{DIR}) \to N(0, \pi(1-\pi)\mathbb{E}[(R_i + Q_i)^2]). \end{equation}` `\(R_i\)`: would show up if no interference is present; depends on `\(f_i\)` `\(Q_i\)`: additional term due to interference; depends on first derivatives of `\(f_i\)` -- Now, a natural question is: can we do better (be more efficient) than the somewhat simple Horvitz-Thompson estimator? --- layout: true # Part II: Improving/Building on Li and Wager (2021) ## More Efficient Estimator (?) --- [Li and Wager (2021)](#referencescont) considered the well-known and intuitive Horvitz-Thompson estimator. Could there be a more efficient estimator out there? -- The idea: lower the variance due to interference by "projecting out" the effect of the interference graph. Suggestion: fit a model `\(\hat{Y}\)` that predicts the potential outcomes from latent positions. Then estimate the direct effect `\(\bar{\tau}_\text{DIR}\)` using `\begin{equation} \hat{\tau}^\text{new} = \frac{1}{n} \sum_{i=1}^n \left(\frac{W_i(Y_i - \hat{Y}_i(1))}{\pi} + \hat{Y}_i(1)\right) - \frac{1}{n} \sum_{i=1}^n \left(\frac{(1-W_i)(Y_i - \hat{Y}_i(0))}{1-\pi} + \hat{Y}_i(0)\right) \end{equation}` where `\(\hat{Y}_i(w)\)` is the predicted potential outcome of individual `\(i\)` had they received treatment `\(w\)`. -- Latent positions unknown, so use some estimate of latent positions. --- ### Early Simulation Results Five different potential outcome models. Latent positions: `\(X_i \overset{\text{iid}}{\sim} \text{Uniform}(0,1)\)`. Graphon: `\(G_n(X_i, X_j) = (3/10 + 3/5\cdot 1[X_i > 0.5])\cdot(3/10 + 3/5\cdot 1[X_j > 0.5])\)`. Two predictive models for `\(\hat{Y}\)`: 1. use the actual potential outcome model with actual latent positions (oracle) 2. linear model with formula `Y ~ W*(E1 + ... + E10)` where `E1, ..., E10` are the first `\(10\)` eigenvectors of the adjacency matrix. --- <img src="data:image/png;base64,#figures/new-estimator-sims.png" width="4000" /> --- layout: true # Part II: Improving/Building on Li and Wager (2021) ## More inclusive model --- The model proposed by [Li and Wager (2021)](#referencescont) excludes models that might seem plausible. **Example**: power plant emission data. [Papadogeorgou, Mealli, and Zigler (2019)](#references) consider power plant emission reduction technology. -- Distance between power plants might influence spill over effect. -- Peer effects might not be symmetrical. -- Spill over effect might not be smooth enough for [Li and Wager (2021)](#referencescont). --- We propose a slightly tweaked model, and are working to show that the same `\(\sqrt{n}\)`-convergence result holds. .pull-left[ For the potential outcomes, assume `\begin{align} &Y_i(w, \boldsymbol{w}_{-i}) \\ &\quad = f_i(w, X_i, \pi; \epsilon_i) \\ &\qquad + \frac{1}{n\rho_n} \sum_{j \neq i} A_i(w, \boldsymbol{E}_i, X_i, X_j; \alpha_i) E_{ij} (w_j - \pi) \\ &\qquad + \left(\frac{1}{n\rho_n} \sum_{j \neq i} B_i(w, \boldsymbol{E}_i, X_i, X_j; \beta_i) E_{ij} (w_j - \pi)\right)^2 \\ &\qquad + r_i(w, \boldsymbol{E}_i, (\boldsymbol{EW})_i, X_i; \eta_i) \end{align}` ] -- .pull-right[ Compare to the Taylor expansion `\begin{align} &f_i(w, X_i, M_i/N_i; \epsilon_i) \\ &\quad = f_i(w, X_i, \pi; \epsilon_i) \\ &\qquad + f_i'(w, X_i, \pi; \epsilon_i)(M_i / N_i - \pi) \\ &\qquad + \frac{1}{2}f_i''(w, X_i, \pi; \epsilon_i)(M_i / N_i - \pi)^2 \\ &\qquad + \frac{1}{6}f_i'''(w, X_i, \pi_i^*; \epsilon_i)(M_i / N_i - \pi)^3 \end{align}` Remember: `\(M_i = \sum_{j\neq i} E_{ij}W_j\)`. ] --- <!-- Rewrite 0 --> We propose a slightly tweaked model, and are working to show that the same `\(\sqrt{n}\)`-convergence result holds. .pull-left[ For the potential outcomes, assume `\begin{align} &Y_i(w, \boldsymbol{w}_{-i}) \\ &\quad = f_i(w, X_i, \pi; \epsilon_i) \\ &\qquad + \frac{1}{n\rho_n} \sum_{j \neq i} A_i(w, \boldsymbol{E}_i, X_i, X_j; \alpha_i) E_{ij} (w_j - \pi) \\ &\qquad + \left(\frac{1}{n\rho_n} \sum_{j \neq i} B_i(w, \boldsymbol{E}_i, X_i, X_j; \beta_i) E_{ij} (w_j - \pi)\right)^2 \\ &\qquad + r_i(w, \boldsymbol{E}_i, (\boldsymbol{EW})_i, X_i; \eta_i) \end{align}` ] .pull-right[ Compare to this rewrite of the Taylor expansion `\begin{align} &f_i(w, X_i, M_i/N_i; \epsilon_i) \\ &\quad = f_i(w, X_i, \pi; \epsilon_i) \\ &\qquad + \frac{1}{n\rho_n} \sum_{j\neq i} \frac{f_i'(w, X_i, \pi; \epsilon_i)n\rho_n}{N_i} E_{ij}(W_j - \pi) \\ &\qquad + \left(\frac{1}{n\rho_n} \sum_{j\neq i} \frac{\sqrt{f_i''(w, X_i, \pi; \epsilon_i)}n\rho_n}{N_i}(W_j - \pi)\right)^2 \\ &\qquad + \frac{1}{6}f_i'''(w, X_i, \pi_i^*; \epsilon_i)(M_i / N_i - \pi)^3 \end{align}` Remember: `\(M_i = \sum_{j\neq i} E_{ij}W_j\)`. ] --- <!-- Rewrite 1 --> We propose a slightly tweaked model, and are working to show that the same `\(\sqrt{n}\)`-convergence result holds. .pull-left[ For the potential outcomes, assume `\begin{align} &Y_i(w, \boldsymbol{w}_{-i}) \\ &\quad = f_i(w, X_i, \pi; \epsilon_i) \\ &\qquad + \frac{1}{n\rho_n} \sum_{j \neq i} \color{#56B4E9}{A_i(w, \boldsymbol{E}_i, X_i, X_j; \alpha_i)} E_{ij} (w_j - \pi) \\ &\qquad + \left(\frac{1}{n\rho_n} \sum_{j \neq i} \color{#009E73}{B_i(w, \boldsymbol{E}_i, X_i, X_j; \beta_i)} E_{ij} (w_j - \pi)\right)^2 \\ &\qquad + \color{#E69F00}{r_i(w, \boldsymbol{E}_i, (\boldsymbol{EW})_i, X_i; \eta_i)} \end{align}` <!-- #eb4034 --> ] .pull-right[ Compare to this rewrite of the Taylor expansion `\begin{align} &f_i(w, X_i, M_i/N_i; \epsilon_i) \\ &\quad = f_i(w, X_i, \pi; \epsilon_i) \\ &\qquad + \frac{1}{n\rho_n} \sum_{j\neq i} \color{#56B4E9}{\frac{f_i'(w, X_i, \pi; \epsilon_i)n\rho_n}{N_i}}E_{ij}(W_j - \pi) \\ &\qquad + \left(\frac{1}{n\rho_n} \sum_{j\neq i} \color{#009E73}{\frac{\sqrt{f_i''(w, X_i, \pi; \epsilon_i)}n\rho_n}{N_i}}(W_j - \pi)\right)^2 \\ &\qquad + \color{#E69F00}{\frac{1}{6}f_i'''(w, X_i, \pi_i^*; \epsilon_i)(M_i / N_i - \pi)^3} \end{align}` Remember: `\(M_i = \sum_{j\neq i} E_{ij}W_j\)`. ] --- **Example**: power plant emission data. [Papadogeorgou, Mealli, and Zigler (2019)](#references) consider power plant emission reduction technology. Distance between power plants might influence spill over effect. Peer effects might not be symmetrical. Spill over effect might not be smooth enough for [Li and Wager (2021)](#referencescont). Our model could allow for all of this: `\begin{equation} Y_i(w, \boldsymbol{w}_{-i}) = f_i(w) + \frac{1}{n\rho_n} \sum_{j\neq i} (\max\{0, X_i - X_j\} + \alpha_i) E_{ij}(w_j - \pi) + \eta_i \end{equation}` --- We derive same asymptotic result when `\(f_i, A_i, B_i\)` are all bounded, and `\(r_i\)` is "well behaved". **Lemma**: Under Assumptions 1-3 and the tweaked model, `\begin{align} \hat{\tau}_\text{DIR}^\text{HT} - & \bar{\tau}_\text{DIR} = \frac{1}{n}\sum_{i=1}^n \left(\frac{f_i(1, X_i, \pi; \epsilon_i)}{\pi} + \frac{f_i(0, X_i, \pi; \epsilon_i)}{1-\pi}\right)(W_i - \pi) \\ &\quad + \frac{1}{n^2 \rho_n} \sum_{i=1}^n \sum_{j \neq i}\left(A_i(1, \boldsymbol{E}_i, X_i, X_j; \alpha_i) - A_i(0, \boldsymbol{E}_i, X_i, X_j; \alpha_i)\right)E_{ij}(W_j - \pi) \\ &\quad + \mathcal{O}_p\left(\delta \right). \end{align}` where `\begin{align} \delta = \frac{\sqrt{\max_i N_i}}{n^{3/2} \rho_n} + \frac{\sqrt{n \max_i N_i^2 + \max_i N_i \sum_{i\neq j} \gamma_{ij}}}{n^3 \rho_n^2} + \sqrt{\frac{\sum_{i\neq j} \gamma_{i,j}}{n^{5}\rho_n^{3}} + \frac{1}{n_4 \rho_n^3}} \end{align}` --- **Conjecture**: Under Assumptions 1-3, the tweaked model, and mild assumptions on the graphon, `\begin{equation} \sqrt{n}(\hat{\tau}_\text{DIR}^\text{HT} - \bar{\tau}_\text{DIR}) \to N(0, \pi(1-\pi)\mathbb{E}[(R_i + Q_i)^2]). \end{equation}` --- layout: false # Part II: Summary/Future Work We have discussed: * interference is challenging -- especially variance estimation. * strict assumptions on interference structure not always reasonable. * interpreting the interference graph as a realization from a graphon model can help us get asymptotic results without interference structure restrictions. * Li and Wager (2021) exclude non-smooth potential outcomes, and do not allow for asymmetrical or non-smooth interference. * We can include both by making minor tweaks * early indications of potential efficiency gains using more complex estimators. --- # Part II: Summary/Future Work In the near future, we hope to 1. complete the proof of asymptotic normality of the Horvitz-Thompson estimator under our slightly tweaked model 2. formalize our newly proposed estimator, and start work on asymptotic results * hopefully helps us determine when this is more/less efficient than Horvitz-Thompson estimator -- More long term projects include: 1. individualized propensity scores to maybe extend to observational data 2. heterogeneous Treatment Effects 3. consider indirect effect estimators --- layout: false name: references # References Manski, C. F. (1990). "Nonparametric Bounds on Treatment Effects". In: _The American Economic Review_ 80.2, pp. 319-323. ISSN: 0002-8282. JSTOR: [2006592](https://www.jstor.org/stable/2006592). Balke, A. and J. Pearl (1997). "Bounds on Treatment Effects from Studies with Imperfect Compliance". In: _Journal of the American Statistical Association_ 92.439, pp. 1171-1176. ISSN: 0162-1459. DOI: [10.1080/01621459.1997.10474074](https://doi.org/10.1080%2F01621459.1997.10474074). URL: [https://doi.org/10.1080/01621459.1997.10474074](https://doi.org/10.1080/01621459.1997.10474074) (visited on Feb. 05, 2020). Hudgens, M. G. and M. E. Halloran (2008). "Toward Causal Inference With Interference". In: _Journal of the American Statistical Association_ 103.482, pp. 832-842. ISSN: 0162-1459, 1537-274X. DOI: [10.1198/016214508000000292](https://doi.org/10.1198%2F016214508000000292). 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URL: [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6784535/](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6784535/) (visited on Mar. 03, 2022). --- name: referencescont # References (cont.) Li, S. and S. Wager (2021). "Random Graph Asymptotics for Treatment Effect Estimation under Network Interference". URL: [http://arxiv.org/abs/2007.13302](http://arxiv.org/abs/2007.13302) (visited on Mar. 01, 2022). Trane, R. M. and H. Kang (2022). "Nonparametric Bounds in Two-Sample Summary-Data Mendelian Randomization: Some Cautionary Tales for Practice". In: _Statistics in Medicine_ n/a.n/a. ISSN: 1097-0258. DOI: [10.1002/sim.9368](https://doi.org/10.1002%2Fsim.9368). URL: [https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.9368](https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.9368) (visited on May. 07, 2022).