Partial Identification of Causal Effects Using Proxy Variables
 

Evaluation of the causal effect of a certain treatment variable on an outcome variable of interest from purely observational data is the main focus in many scientific endeavors. One of the most common identification conditions for causal inference from observational data is that of conditional exchangeability. The assumption essentially presumes that the researcher has collected a sufficiently rich set of pre-treatment covariates, such that within the covariate strata, it is as if the treatment were assigned randomly. Unfortunately, this assumption is violated in many real-world settings as it essentially requires that there should not exist any unobserved common causes of the treatment-outcome relation (i.e., no unobserved confounders).

To address the challenge of unobserved confounders, the proximal causal inference framework was recently introduced by Tchetgen Tchetgen and colleagues (Tchetgen Tchetgen et al.,, 2020; Miao et al., 2018a, ). This framework shows that identification of the causal effect of the treatment on outcome in the presence of unobserved confounders still is sometimes feasible provided that one has access to two types of proxy variables of the unobserved confounders, a treatment confounding proxy and an outcome confounding proxy that satisfy certain assumptions. The proximal causal inference framework was recently extended to several other challenging causal inference settings such longitudinal data (Ying et al.,, 2023), mediation analysis (Dukes et al.,, 2023; Ghassami et al., 2021a, ), outcome-dependent sampling (Li et al.,, 2023), network interference settings (Egami and Tchetgen Tchetgen,, 2023), graphical causal models (Shpitser et al.,, 2023), and causal data fusion (Ghassami et al.,, 2022). It has also recently been shown that under an alternative somewhat stronger set of assumptions with regards to proxy relevance, identification is sometimes possible using a single proxy variable (Tchetgen Tchetgen et al.,, 2023).

A key identification condition of proximal causal inference is an assumption that proxies are sufficiently relevant for hidden confounding factors so that there exist a so-called confounding bridge function, defined in terms of proxies, that matches the association between hidden factors and either the potential outcomes or the treatment variable; identification of which is an important step towards identification of causal effects via proxies. However, non-parametric identification of such a bridge function, when it exists, has to date involved a completeness condition which, roughly speaking, requires that variation in proxy variables reflects all sources of variation of unobserved confounders (Tchetgen Tchetgen et al.,, 2020). However, completeness is a strong condition which may significantly limit the researcher’s choice of proxy variables, and hence the feasibility of proximal causal inference in important practical settings where the assumption cannot reasonably be assumed to hold. For instance, in many settings, only one of the two required types of proxies may be available rendering point identification infeasible, even if the proxies are highly relevant. More broadly, point identification of causal effects using existing proximal causal inference methods crucially relies on identification of a confounding bridge function, and remains possible even if the latter is only set-identified (Zhang et al.,, 2023; Bennett et al.,, 2023). Notably, failure to identify such a confounding bridge function is generally inevitable without a completeness condition; in which case partial identification of causal effects may be the most one can hope for.

In this paper, we propose partial identification methods for causal parameters using proxy variables in settings where completeness assumptions do not necessarily hold. Therefore, our methods obviate the need for identification of bridge functions which are solutions to integral equations that involve unobserved variables. We provide methods in single proxy settings and demonstrate that certain conditional independence requirements in that setting can be relaxed in case that the researcher has access to a second proxy variable. For the causal parameter of interest, we initially focus on the average treatment effect (ATE) and the effect of the treatment on the treated (ETT) in Sections 2 and 3. Then we extend our results in Section 4 to partial identification in related settings where identification hinges upon hidden mediators for which proxies are available, however such proxies fail to satisfy a completeness condition which would in principle guarantee point identification of the causal effect of interest; such settings include causal mediation with a mis-measured mediator and hidden front-door models. Our bounds are non-smooth functionals of the observed data distribution. As a consequence, in the context of inference, in Section 5, we initially employ the LogSumExp approximation, which represents a smooth approximation of our bounds. Subsequently, we leverage bootstrap confidence intervals on the approximated bounds.

Consider a setting with a binary treatment variable $A\in\{0,1\}$, an outcome variable $Y\in\mathcal{Y}\subseteq[0,+\infty)$,¹¹1We use capital calligraphic letters to denote the alphabet of the corresponding random variable. and observed and unobserved confounders denoted by $X$ and $U$, respectively. We are interested in identifying the effect of the treatment on the treated (ETT), which is defined as

as well as the average treatment effect (ATE) of $A$ on $Y$ defined as

where $Y^{(A=a)}$ is the potential outcome variable representing the outcome had the treatment (possibly contrary to fact) been set to value $a$. Due to the presence of the latent confounder $U$ in the system, without any extra assumptions, ETT and ATE are not point identified. Therefore, we focus on partial identification of conditional potential outcome mean parameters, $\mathbb{E}[Y^{(A=a)}\mid A=1-a]$, and marginal potential outcome mean, $\mathbb{E}[Y^{(A=a)}]$, for $a\in\{0,1\}$.

In this section, we show that having access to two conditionally independent but invalid proxy variables of the unobserved confounder, one can relax the requirements of conditional independence of proxy variables from the treatment and the outcome variables while still providing valid bounds for the treatment effect. The proxy variables in our setting do not necessarily satisfy the strong exclusion restrictions of the original proximal causal inference. Therefore, the point identification approach in the original framework cannot be used here. We require the existence of two proxy variables $W$ and $Z$ that satisfy the following.

Figure 3 demonstrates an example of a graphical model consistent with our two-proxy setting. Note that neither proxy variable $W$ technically needs to be an outcome confounding proxy variable (defined in Assumption 1) nor proxy variable $Z$ needs to be a treatment confounding proxy variable (defined in Assumption 4). Specifically, there can exist a direct causal link between $A$ and $W$ and one between $Z$ and $Y$ (shown in the figure by the red edges).

We have the following partial identification result for the conditional potential outcome mean.

Note that in the absence of the unobserved confounder $U$, $\min_{w,z}p(w,z\mid a,x)=\max_{w,z}p(w,z\mid a,x)=p(w\mid a,x)p(z\mid a,x)$. Hence, again the upper and lower bounds match and will be equal to the g-formula and IPW formula in the conditional exchangeability framework given $X$.

In this Section we show that our partial identification ideas can also be used for dealing with unobserved mediators in the system. The results in this section are the counterparts of the results of Ghassami et al., 2021a where the authors obtained point identification results under completeness assumptions. We focus on two settings with hidden mediators: Mediation analysis and front-door model. We only show the extension of the method presented in Subsection 2.1 and do not repeat extensions for the methods of Sections 2.2 and 3, although they are easily deduced from our exposition.

Let $M$ be the mediator variable in the system which is unobserved. We denote the potential outcome variable of $Y$, had the treatment and mediator variables been set to value $A=a$ and $M=m$ (possibly contrary to the fact) by $Y^{(a,m)}$. Similarly, we define $M^{(a)}$ as the potential outcome variable of $M$ had the treatment variables been set to value $A=a$. Based on variables $Y^{(a,m)}$ and $M^{(a)}$, we define $Y^{(a)}=Y^{(a,M^{(a)})}$, and $Y^{(m)}=Y^{(A,m)}$. We posit the following standard assumptions on the model.

So far, we presented non-parametric partial identification formulae for causal parameters in the presence of unobserved confounders or mediators. In this section, we focus on the estimation aspect of the bounds. We restrict our analysis to the scenario where all variables involved are categorical in nature. However, it is worth noting that our findings possess the potential for extension to situations where the observed confounders $X$ and the outcome variable $Y$ are continuous. Nonetheless, such an extension necessitates careful consideration of selecting suitable statistical models to mitigate bias or ensure satisfactory convergence rates. For the results in this section, we only focus on the bounds for conditional potential outcome mean $\mathbb{E}[Y^{(A=a)}\mid A=1-a]$ in Section 2; the proposed approach can be applied to the bounds in Sections 3 and 4.

For the case of categorical variables, it is relatively straightforward to estimate the bounds for the conditional potential outcome mean by employing the formulae outlined in Theorems 1 and 2. However, it is important to acknowledge that these bounds represent non-smooth functionals of the underlying distribution. Consequently, a naive implementation of the bootstrap may result in confidence intervals that are invalid and unreliable. To circumvent this issue, we propose the use of smooth approximations of the aforementioned functionals, described as follows.

Consider a finite set of real numbers $\{x_{1},...,x_{n}\}$. Define the LogSumExp (LSE) function, $LSE(\cdot;\alpha)$ as

It can be shown that $LSE(\{x_{1},...,x_{n}\};\alpha)$ can be used to approximate the maximum and minimum of $\{x_{1},...,x_{n}\}$: for any choice of $\alpha>0$, we have

Similarly, for any choice of $\alpha<0$, we have

Therefore, $LSE(\{x_{1},...,x_{n}\};\alpha)$ approximates the maximum and the minimum of $\{x_{1},...,x_{n}\}$ as $\alpha$ tends to $\infty$ and $-\infty$, respectively. Applications of LSE approximation approach has appeared in the fields of optimization (Boyd and Vandenberghe,, 2004), machine learning (Murphy,, 2012; Goodfellow et al.,, 2016; Calafiore et al.,, 2019, 2020), and statistics (Wainwright,, 2019; Chernozhukov et al.,, 2012; Tchetgen Tchetgen and Wirth,, 2017; Levis et al.,, 2023). Notably, Tchetgen Tchetgen and Wirth, (2017) and Levis et al., (2023) employed this approximation technique to facilitate statistical inference pertaining to bounds within the instrumental variables model for missing data and causal effects, respectively.

Applying the LSE function to the bounds in Theorems 1 and 2, we have the following result.

Equipped with the bounds in Corollary 3, which are smooth functionals of the underlying distribution, the standard bootstrap technique can be used to acquire valid confidence intervals. Formally, for $\psi(\alpha)\in\{\psi_{LW}(\alpha),\psi_{UW}(\alpha),\psi_{LZ}(\alpha),\psi_{UZ}(\alpha)\}$, let $\hat{\psi}_{n}(\alpha)$ denote our estimator based on $n$ samples, $\hat{\psi}^{*}_{n,1}(\alpha),...,\hat{\psi}^{*}_{n,B}(\alpha)$ denote $B$ bootstrap replications of $\hat{\psi}_{n}(\alpha)$, and $\psi^{*}_{\beta}(\alpha)$ denote the $\beta$ sample quantile of the bootstrap replications. Using Part (a) of Corollary 3, we have the following 95% confidence interval for the conditional potential outcome mean

Using Part (b) of Corollary 3, we have the following 95% confidence interval for the conditional potential outcome mean

In this section, we provide simulation results to demonstrate the performance of our proposed estimators. We designed the data generating process for variables $(U,X,W,Z,A,Y)$ as follows.

• •

  $p(U,X)$: We chose $|\mathcal{U}|\times|\mathcal{X}|$ parameters for the joint distribution $p(U,X)$ uniformly from $Unif[0.1,1]$, and normalized them to sum up to one.

• •

  $p(W\mid U,X)$: For any $w,u,x$,

  $$p(W=w\mid U=u,X=x)=\frac{\exp(\beta_{w,0}+\beta_{w,U}u+\beta_{w,X}x)}{\sum_{w\in\mathcal{W}}\exp(\beta_{w,0}+\beta_{w,U}u+\beta_{w,X}x)}.$$

• •

  $p(Z\mid U,X)$: For any $z,u,x$,

  $$p(Z=z\mid U=u,X=x)=\frac{\exp(\beta_{z,0}+\beta_{z,U}u+\beta_{z,X}x)}{\sum_{z\in\mathcal{Z}}\exp(\beta_{z,0}+\beta_{z,U}u+\beta_{z,X}x)}.$$

• •

  $p(A\mid U,X,Z)$: For any $a,u,x,z$,

  $$p(A=a\mid U=u,X=x,Z=z)=\frac{\exp(\beta_{a,0}+\beta_{a,U}u+\beta_{a,X}x+\beta_{a,Z}z)}{\sum_{a\in\mathcal{A}}\exp(\beta_{a,0}+\beta_{a,U}u+\beta_{a,X}x+\beta_{a,Z}z)}.$$

• •

  $p(Y\mid U,X,W,A)$: For any $y,u,x,w,a$,

  $$p(Y=y\mid U=u,X=x,W=w,A=a)=\frac{\exp(\beta_{y,0}+\beta_{y,U}u+\beta_{y,X}x+\beta_{y,W}w+\beta_{y,A}a)}{\sum_{y\in\mathcal{Y}}\exp(\beta_{y,0}+\beta_{y,U}u+\beta_{y,X}x+\beta_{y,Z}z+\beta_{y,A}a)}.$$

In all the conditional distributions above, coefficients $\beta_{\cdot,\cdot}$ are chosen according to uniform distribution $Unif[-0.5,0.5]$. We require the coefficients connecting $U$ to $W$ and $Z$ to be non-zero to ensure that $W$ and $Z$ are relevant to the latent confounder $U$.