Causal Mediation Analysis with High-Dimensional Surrogates: Identification, Estimation and Inference under Latent Factor Models

Introduction

Causal mediation analysis provides a principled framework for understanding the mechanisms through which a treatment affects an outcome [5, 22, 38] (Pearl 2022). It moves beyond testing whether an overall effect exists to ask how that effect operates by separating the total effect into a direct effect, representing the effect through other pathways, and an indirect effect, representing the effect through an intermediate variable (referred to as the mediator). The statistical literature on mediation analysis has developed from its early regression-based formulations. The product-of-coefficients method of Baron and Kenny (1986) offered an intuitive way to study mediated pathways. Later, the potential-outcomes framework of Rubin (1974) allowed mediation to be defined in formal causal terms through natural direct and indirect effects [32] (Pearl 2022), along with the conditions for their identification [22]. Subsequent work developed semiparametric, weighting, and multiply robust methods for their estimation and inference [35, 38, 37]. More recent work has addressed high-dimensional mediation problems, even when the number of mediators is much larger than the sample size through variable selection, sparse regression, and related inferential techniques [41, 21, 10, 42, 46, 30, 18, 19, 20, 31, 8]. However, in many real-world applications, the true mediators are not directly observed; instead, researchers may rely on a potentially large number of surrogate measurements that provide indirect information about the underlying mediator. Our motivating application uses DNA methylation profiles from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study to investigate a latent biological pathway between depressive symptoms and cognitive impairment. Oxidative stress has been implicated in both depression and Alzheimer’s disease (AD) [7, 9]. Direct quantification of reactive oxygen species (ROS) is difficult because of their short lifespan and high reactivity [23]. Researchers therefore rely on more stable molecular signatures that reflect oxidative activity. Among them is DNA methylation, which can be altered in response to oxidative stress and can be measured at scale [26, 28, 7]. While DNA methylation provides a useful surrogate for investigating the mediating role of oxidative stress, its measurement across hundreds of thousands of CpG sites using high-throughput assays produces a high-dimensional data structure in which individual sites capture only partial aspects of the underlying biological process. When the pre-treatment covariates needed for confounding adjustment are also measured indirectly, both the mediator and the covariates must be related to their observed surrogates. One way to describe this measurement structure is through a latent factor model, in which a small number of unobserved variables generate the common variation across many noisy measurements [3, 13].

Mediation analysis with indirectly measured variables has received attention in fixed-dimensional settings [1], with fewer developments for high-dimensional measurements [11]. Proxy-based methods establish identification of mediation effects when the mediator is hidden and suitable proxy measurements are available [16]. Related factor-based approaches account for dependence among observed high-dimensional exposures and mediators [44] or accommodate individualized and time-varying mediation effects [45]. Despite these developments, existing results do not directly characterize how the dimensions of noisy surrogate measurements determine the error in recovering the latent mediation functional when both the mediator and the pre-treatment covariates are unobserved. In particular, it remains to be established when mediation regressions based on low-dimensional projections of these surrogates approximate the latent target closely enough to permit inference with the same first-order precision as if the latent factors were observed. This question becomes more delicate when the working factor dimensions exceed the true dimensions, because some directions then contain only measurement noise. For example, if there are three true factors but we estimate five, two linear combinations of the estimated factors contain no signal from the true factors. Their variances shrink as the number of surrogates grows, which can make the regression matrices nearly singular and the estimated regression coefficients unstable.

To address these questions, we propose Factor-Adjusted Regression for Mediation Analysis (FARMA), a factor-based framework for identification, estimation, and inference with high-dimensional surrogates of latent mediators and pre-treatment covariates. Building on factor-model approaches for high-dimensional data [4, 3, 13, 14, 12], FARMA constructs low-dimensional projections of the surrogate covariates and mediators and incorporates these projected factors into cross-fitted mediation regressions. Under some standard assumptions, these projections retain information about the underlying latent factor spaces while reducing the contribution of measurement noise. Our analysis allows the fixed working dimensions to exceed these true factor dimensions and accounts for the additional directions that contain only measurement noise. The main contributions of this paper are threefold:

Identification. We establish asymptotic identification of the causal natural direct and indirect effects when both the mediator and the relevant covariate factors are latent. With a fixed number of surrogates, the target effects are generally not point identified from the observed data alone without strong restrictions on the data-generating process. We show that suitable projections of the high-dimensional surrogates yield an observed-data parameter that approaches the latent target as the number of measurements increases, leading to asymptotic point identification.

Estimation. Building on the identification result, we develop a cross-fitted estimator of the causal mediation effects using projected surrogate factors. The method does not require exact knowledge of the true factor dimensions and remains valid when fixed working dimensions are chosen as upper bounds on the true dimensions. Under suitable regularity and growth conditions on the surrogate dimensions, we show that the resulting estimator is $\sqrt{n}$-consistent and asymptotically normal for the parameter of interest. This result provides the basis for statistical inference on the natural direct and indirect effects even when mediators are not observed directly.

Application. We apply FARMA to investigate the effect of depression on Alzheimer’s disease (AD) mediated by oxidative stress, where the mediating process is not directly observed and is instead represented by high-dimensional DNA methylation measurements. We illustrate the approach using data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) Study.

The remainder of the paper is organized as follows. § 2 presents the problem setting and model. § 3 establishes identification of the target parameter. § 4 introduces the FARMA. § 5 develops the asymptotic theory. § 6 reports the simulation results. § 7 presents the application to the ADNI dataset. § 8 concludes with a discussion of future work.

Model Setting

We consider a binary treatment variable $A \in\{0,1\}$ (also called exposure or action variable), and a real-valued outcome $Y \in\mathbb{R}$. For each individual, we consider mediators $\mathbf{f}_M = [f_{M1}, \ldots, f_{M\bar{q}}] \in\mathbb{R}^{\bar{q}}$ through which treatment may influence the outcome, and pre-treatment covariates $\mathbf{f}_X = [f_{X1}, \ldots, f_{X\bar{p}}] \in\mathbb{R}^{\bar{p}}$, which capture baseline characteristics that may confound the relationship among treatment, mediators and outcome. We use lowercase notation, such as $f_{Mi}$ and $f_{Xj}$, to denote the individual coordinates within these factors, respectively. The treatment can influence the outcome in two ways, indirectly through the mediators, or directly without passing through them. Formally, the variables are assumed to be generated from the following structural equations:

where $\epsilon_0$ and $\epsilon_1$ are centered outcome errors on the control and treated arms, $\mathbf{u}_{X,M,0} \in\mathbb{R}^{\bar{q}}$ and $\mathbf{u}_{X,M,1} \in\mathbb{R}^{\bar{q}}$ are the corresponding centered mediator structural noises, $\mathbf{B}_0,\mathbf{B}_1 \in\mathbb{R}^{\bar{q}\times\bar{p}}$, $\beta_0,\beta_1 \in\mathbb{R}^{\bar{p}}$, and $\gamma_0,\gamma_1 \in\mathbb{R}^{\bar{q}}$ are unknown coefficients, and $\delta_0,\delta_1 \in\mathbb{R}^{\bar{q}}$ and $\alpha_0,\alpha_1 \in\mathbb{R}$ are intercepts. Note that we allow for an $A-f_X$ interaction in the generation of $\mathbf{f}_M$, and $A-f_X$ and $A-f_M$ interactions in the generation of the outcome $Y$.

In many real-world applications, $\mathbf{f}_X$ and $\mathbf{f}_M$ are latent (i.e., unobserved). Instead, we observe measurements $\mathbf{X} \in\mathbb{R}^{p}$ and $\mathbf{M} \in\mathbb{R}^{q}$ that serve as noisy surrogates for these latent variables. We focus on high-dimensional measurements with $p \gg\bar{p}$ and $q \gg\bar{q}$. We adopt the following factor model for the observed measurements:

where $\mathbf{\Gamma}_X \in\mathbb{R}^{p\times\bar{p}}$ and $\mathbf{\Gamma}_M \in\mathbb{R}^{q\times\bar{q}}$ are unknown loading matrices, and $\mathbf{u}_X$ and $\mathbf{u}_M$ are mean-zero errors. Henceforth, we refer to $\mathbf{f}_X$ and $\mathbf{f}_M$ as latent factors. Figure 1 summarizes the causal relationships among the latent factors, treatment, and outcome specified in (1) with the measurement relationships in (2). The standard mediation model with directly observed covariates and mediators is recovered as a special case by setting $p = \bar{p}$, $q = \bar{q}$, $\Gamma_X = I_p$, and $\Gamma_M = I_q$, with vanishing errors. In this case, $X = f_X$ and $M = f_M$, so the structural equations in (1) can be written directly in terms of the observed variables $X$ and $M$.

We first consider the setting in which $f_X$ and $f_M$ are directly observed, and define the causal estimands using the potential-outcomes framework ([33, 32]). For each individual, let $Y^{(a,f_M)}$ denote the potential outcome under treatment $A = a$ and mediator value $f_M = f_M$, and let $f_M^{(a)}$ denote the potential mediator value under treatment $A = a$, where $a \in\{0,1\}$. The potential outcome under treatment $a$ is then $Y^{(a)} = Y^{(a,f_M^{(a)})}$. The average treatment effect (ATE) admits the following decomposition:

The two terms on the right-hand side are the total indirect effect and the pure direct effect, respectively, in the terminology of Robins and Greenland [32]. Following Pearl (2022), we refer to them as the natural indirect effect (NIE) and the natural direct effect (NDE). In our setting, these effects are defined with respect to $f_M$. The NIE measures the effect of changing the mediator from its potential value under control to its potential value under treatment, while holding treatment fixed at $A = 1$. The NDE measures the effect of changing treatment from $A = 0$ to $A = 1$, while holding the mediator at its potential value under control.

The decomposition in (3) depends on three counterfactual means, $\mathbb{E}[Y^{(1)}]$ ($\equiv\mathbb{E}[Y^{(1,f_M^{(1)})}]$), $\mathbb{E}[Y^{(0)}]$ ($\equiv\mathbb{E}[Y^{(0,f_M^{(0)})}]$), and $\mathbb{E}[Y^{(1,f_M^{(0)})}]$. Under consistency, positivity and sequential ignorability, the first two can be identified using some standard covariate adjustment. $\mathbb{E}[Y^{(1,f_M^{(0)})}]$ is a cross-world mean that combines the outcome under treatment with the mediator value that would arise under control. Since its identification requires additional assumptions that links the outcome and mediator across treatment regimes, we consider this as our parameter of interest,

To identify $\theta_0$, we impose the following standard assumptions.