Causal mediation analysis with mediators subject to censoring by assay limits of detection
May 29, 2025
The ACTIV-2/ACTG A5401 clinical trial
Objective: Evaluate how monoclonal antibody (mAb) agents reduce risk of hospitalization due to COVID-19 via their role in viral clearance.
- ACTIV-2: phase 2/3 adaptive platform trial of investigational mAb agents to treat non-hospitalized COVID-19 patients (Evering et al. 2023; Giganti et al. 2025).
- The action mechanisms of mAb therapies is understood to revolve around their role in clearing viral (SARS-CoV-2, in this case) RNA.
- Viral RNA was measured in several ways: both anteronasal (AN; daily) and nasopharyngeal (NP; every 3 days) swabs, and blood plasma (twice)
- Viral RNA, as part of the putative action mechanism of mAbs, acts as a mediator of the effect of mAb agents.
- Viral RNA measurements subject to a significant degree of missingness.
- Clinical endpoint: Hospitalization or death through Day 28 (end of study).
Causal mediation analysis in the ACTIV-2 trial
Objective: Decompose the total effect of mAb agents on hospitalization or death into an indirect effect (thru viral RNA) and a direct effect (the remainder).
- Causal mediation analysis methods operate under the assumption that mediators are not subject to (severe) censoring.
- Problem: Mediator (viral RNA) is left-censored by assay limit of detection.
- Approximately 75% of plasma samples at baseline are reported as falling below the assay limits of detection (LoD).
- Mishandling censored mediators (e.g., “naive” imputation by LOD/2) can risk introducing (possibly severe) bias into the estimation process.
- Left-censoring of mediators reflects technical limitations and biologically meaningful clearance of virus. How to disentangle these?
- Recent work (e.g., Chernofsky, Bosch, and Lok 2024) proposes exptrapolation based on a parametric form or numerical integration of an observed-data likelihood.
Causal mediation analysis without censored data
- Data: \(n\) study units, sampled iid from \(\P \in \M\), with data on i^th unit as \(O_i = (L_i, A_i, M_i, Y_i)\), where
- \(L\) are baseline covariates, \(A \in \{0, 1\}\) is the treatment.
- \(M\) is a continuous mediator, \(Y\) is the clinical endpoint of interest.
- Total effect decomposition: \[\small{ \begin{aligned} \underbrace{\E\left[Y(A = a) - Y(A = a^{\prime})\right]}_{\text{Total effect (TE)}} &= \E\left[Y(A = a, M = M(a)) - Y(A = a^{\prime}, M = M(a^{\prime}))\right] \\ &= \underbrace{\E\left[Y(A = a, M = M(a)) - Y(A = a, M = M(a^{\prime}))\right]}_{\text{Natural indirect effect (NIE)}} \\ &\quad + \underbrace{\E\left[Y(A = a, M = M(a^{\prime})) - Y(A = a^{\prime}, M = M(a^{\prime}))\right]}_{\text{Natural direct effect (NDE)}} \end{aligned}} \]
Standard identification assumptions
- Consistency:
- Outcome: \((A, M) = (a, m) \implies Y = Y^{a,m}\)
- Mediator: \(A = a \implies M = M^a\)
- Positivity:
- Treatment: \(\Pr(A = a \mid L) > 0\) and \(\Pr(A = a^{\prime} \mid L) > 0\), with probability one
- Mediator: \(\Pr(M = m \mid A = a^{\prime}, L) > 0\) with probability one only whenever \(\Pr(M = m \mid A = a, L) > 0\)
- Exchangeability:
- Treatment: \(M(a) \indep A \mid L\)
- Treatment–mediator: \(Y(a,m) \indep (A, M) \mid L\)
- Cross-world counterfactuals: \(Y(a^{\prime}, m) \indep M^a \mid L\) for all \(a \neq a^{\prime}\) and \(m\)
- No unmeasured mediator–outcome confounder affected by exposure
Identification result: Statistical estimand
- Recall that the NDE and NIE contain terms that are trivially identified, i.e., \(Y(A = a, M = M(a))\) and \(Y(A = a, M = M(a))\)
- But the decomposition strategy hinges on \(\E[Y(A = a, M = M(a^{\prime}))]\), which is identified by the “mediation formula” as follows \[\small{ \begin{aligned} \mathbb{E}&\left[Y(A = a, M = M(a^{\prime}))\right] = \\ &\int_{\mathcal{L}} \int_{\mathcal{M}} \E[Y \mid A = a, M = m, L = l] f_{M \mid A, L}(m \mid A = a', L = l) f_{L}(l) \,\mathrm{d}m\,\mathrm{d}l \end{aligned}} \]
- From this identification result, we can construct plug-in or re-weighted estimators (Imai, Keele, and Tingley 2010; Imai, Keele, and Yamamoto 2010).
- Construct asymptotically efficient estimators (Tchetgen Tchetgen and Shpitser 2012; Zheng and van der Laan 2012) by using the efficient influence function (EIF).
- Can we extend the use of these results to the case with left-censored mediators?
Identification under informatively censored mediators
- The data unit is now \(O_i = (L_i, A_i, C_i M_i, C_i, Y_i)\), where the assay limit of detection \(\lambda_{\text{LOD}}\) results in censoring of \(M_i\), that is, \(C_i := \I(M_i > \lambda_{\text{LOD}})\).
- Censoring of the mediator complicates identification; key targets are
- \(\Pr(Y = y \mid A = a, M = m, L = l)\), and
- \(f(M = m \mid A = a, L = l)\)
- Under the assumption: \(C \perp Y \mid (M, A, L)\), conditional outcome mean can be estimated from complete cases only, that is, \[ \Pr(Y \mid M, A, L) = \Pr(Y \mid M, A, L, C = 1) \ . \]
- However, \(f(M \mid A, L)\) (or \(\Pr(Y, M \mid A, L)\)) is not identified non-parametrically due to informative censoring when \(M\) is continuous and \(Y\) is binary (Zuo et al. 2024).
- Identifiability in this setting requires a completeness assumption—that the “dimension” of \(Y\) be at least as large as that of \(M\).
Addressing identifiability: Parametric framework
- Non-parametric identification fails due to violation of the completeness assumption.
- To work around, impose parametric distributional constraints to achieve identification; for example, structural assumptions on \(f(M = m \mid A = a, L = l)\):
- exponential family distributions (Newey and Powell 2003), or
- location-scale families (Hu and Shiu 2018).
- Identifiability then relies on the uniqueness of the solution to score equations in standard maximum likelihood estimation.
- With this framing, the complete-data log-likelihood (conditional on \(A\) and \(L\)) is \[\small \begin{aligned} \ell_{\text{com}}(\theta) = \sum_{i=1}^n \Big[ \log \Pr(Y_i \mid M_i, A_i, L_i; \alpha) + \log f(M_i \mid A_i, L_i; \beta) + \log \Pr(C_i \mid M_i, A_i, L_i) \Big] \ , \end{aligned} \] for \(\theta = (\alpha, \beta)\), and where \(\alpha\) are parameters of the conditional outcome mean and \(\beta\) are those for the conditional mediator density.
From complete- to observed-data log-likelihoods
Given the limit of detection \(\lambda_{\text{LOD}}\) and known form of censoring, \(C_i = \I(M_i > \lambda_{\text{LOD}})\), the censoring mechanism is fully determined by \(M\).
Due to this, in the likelihood expressions, the censoring probability simplifies to \[ \Pr(C_i = c \mid M_i, A_i, L_i) = 1 \]
Then, we can express the observed-data log-likelihood as follows
Observed-data log-likelihood (under left-censoring by LOD ): \[ \begin{aligned} \ell_{\text{obs}}(\boldsymbol{\theta}) = \sum_{i : C_i = 1} &\left[\log \Pr(Y_i \mid M_i, A_i, L_i; \alpha) + \log f(M_i \mid A_i, L_i; \beta) \right] \\ + &\sum_{i : C_i = 0} \log \int_0^{\lambda_{\text{LOD}}} \Pr(Y_i \mid M_i, A_i, L_i; \alpha) \cdot f(M \mid A_i, L_i; \beta) \, dM \end{aligned} \]
Problem summary: Challenges and proposed solution
- Started with left-censored mediators and non-parametric definitions of causal mediation estimands.
- Reduced this to identification via parameter estimation in parametric (working) models with informatively censored mediators.
- Core challenge: Censoring mechanism depends on unobserved values; thus, falls outside of the missing-at-random (MAR) paradigm.
- Solution: Use a combination of fractional imputation (FI) and the expectation-maximization (EM) algorithm to handle left-censoring.
- FI: Generate multiple candidate values of the mediator for units subject to left-censoring, assigning importance weights to each.
- EM: Update importance weights iteratively to solve score equations.
Fractional imputation (FI, Kim (2011))
- For each study unit with censored mediator, replace missing mediator value with multiple plausible values, from a proposal distribution.
- Use importance sampling to assign fractional weights to each of the candidate imputed mediator values.
- Key steps:
- Generate \(S\) imputed values using rejection sampling—that is, \(m^{\star(j)}_{i}\) for \(j = 1, \ldots, S\) from a proposal distribution \(f(M \mid A, L)\).
- Compute importance-based fractional weights.
- Update model parameters iteratively.
Importance weighting
Target density: \(f(m_i^{\star(j)} \mid Y_i, C_i = 0, a_i, l_i)\)—direct estimation is expensive!
Importance weights: \[ \frac{\text{Target}}{\text{Proposal}} = \frac{ f(m_i^{\star(j)} \mid Y_i, C_i = 0, a_i, l_i)}{ f(m_i^{\star(j)} \mid a_i, l_i)} \]
Avoid explicit evaluation of the joint density via the identity: \[ \begin{aligned} &\frac{\Pr(Y_i,\ M_i = m_i^{\star(j)}, C_i = 0 \mid a_i, l_i) / f(m_i^{\star(j)} \mid a_i, l_i)}{ \sum_{k=1}^S \Pr(Y_i,\ M_i = m_i^{\star(k)},\ C_i = 0 \mid a_i, l_i) / f(m_i^{\star(k)} \mid a_i, l_i)} \\[1.2em] = &\frac{ f(m_i^{\star(j)} \mid Y_i,\ C_i = 0, a_i, l_i) / f(m_i^{\star(j)} \mid a_i, l_i)}{ \sum_{k=1}^S f(m_i^{\star(k)} \mid Y_i,\ C_i = 0,\ a_i, l_i) / f(m_i^{\star(k)} \mid a_i, l_i)} \end{aligned} \]
Importance weighting
Normalized Importance Weights
\[ w_{ij} = \frac{\tilde{w}_{ij}}{\sum_{k=1}^S \tilde{w}_{ik}}, \quad \tilde{w}_{ij} \propto \frac{ \Pr(Y_i, M_i = m_i^{\star(j)}, C_i = 0 \mid a_i, l_i)}{ f(m_i^{\star(j)} \mid a_i, l_i) } \]
Numerator term is the joint distribution of observed data. This can be factorized as
- Conditional mediator density: \(f(m_i \mid a_i, l_i)\)
- Conditional censoring probability: \(\Pr(C_i = 0 \mid M_i, a_i, l_i)\)
- Conditional outcome probability: \(\Pr(Y_i = 1 \mid M_i, a_i, l_i)\)
We will treat these as nuisance parameters, estimating each and updating the fractional weights assigned to imputation replicates via EM.
EM algorithm for informatively censored mediators
E-Step:
- Imputation step: Generate \(S\) values from a proposal distribution.
- Weighting step: Compute normalized importance weights: \[\small w_{ij} \propto \frac{ \Pr(Y_i,\ M_i = m^{\star(j)}_i,\ C_i = 0 \mid a_i, l_i) }{ f(m^{\star(j)}_i \mid a_i, l_i) }, \quad \text{subject to } \sum_{j=1}^S w_{ij} = 1 \]
M-Step:
- Solve weighted score equations to update parameters: \[\small S(\theta;\ Y_i,\ m^{\star(j)}_i, a_i, l_i) = \frac{\partial, \ell_{\text{com}}(M_i = m^{\star(j)}_i; \theta)} {\partial \theta} \]
EM-FI versus MCEM: Efficiency and convergence
Monte Carlo EM (MCEM)
- Resamples imputed values each iteration, leading to a higher computational cost
- Stochastic fluctuations may delay convergence
EM with Fractional Imputation
- Fixed imputations across iterations, only updating weights \(w_{ij}\) at each step \(t\)
- Computational savings: Importance sampling performed once
Theoretical Convergence
Under suitable regularity conditions, for a sufficiently large number of iterations \(t\) in the EM algorithm, the estimated parameter \(\hat{\theta}^{(t)}\) converges to its asymptotic limit \(\hat{\theta}^{\star}_{S}\), a stationary point of \(Q^{\star}\) for fixed \(S\); that is, \(\hat{\theta}^{(t)} \to \hat{\theta}^{\star}_{S}\,\, \text{as}\,\, t \to \infty\). Then, for a sufficiently large number of imputations \(S\), we have \(\hat{\theta}^{\star}_{S} \to \hat{\theta}_{\text{MLE}}\).
- Stability: Fixed imputation scheme reduces Monte Carlo variability.
- Reliability: Guaranteed convergence to MLE through imputation size \(S\).
- Efficiency: Much faster than MCEM in numerical experiments (and data analysis).
Inference: Adaptive \(m\)-out-of-\(n\) bootstrap
Key idea
Overcome failure of the standard bootstrap for non-smooth/irregular estimators (e.g., NDE/NIE functional with fractionally imputed mediator) by adapting the size of the resamples \(m\).
- Select \(m = \lfloor n^{f(p)} \rfloor\), where \(f(p)\):
- Monotone decreasing in degree of non-regularity wrt \(p\), with \(f(0) = 1\)
- Continuous, with bounded derivative
- This ensures \(m \to \infty\) and \(m = o(n)\) for consistency.
- Data-driven selection of \(m\) \[ m = \left\lfloor n^{c_i} \right\rfloor, \quad \text{where} \quad c_i = \frac{1 + \gamma_i \exp(-k\, p_{\text{cens}})}{1 + \gamma_i} \ , \] where \(p_{\text{cens}}\) is censoring rate and \(\gamma_i\) and \(k\) is hyperparameter. If \(p_{\text{cens}} = 0\), then \(m = n\) and the procedure reverts to the standard bootstrap.
Numerical experiment with censored mediator
Simulate data from an observational study with a mediator left-censored by an assay limit of detection (LoD): \[\small \begin{aligned} L_1 & \sim \text{Bern}(0.7), \quad L_2 \sim \text{Bern}(0.5), \quad L_3 \sim \text{Bern}(0.25) \\\\ A \mid \boldsymbol{L} & \sim \text{Bern}\left( \expit(-1 + 0.5 L_1 + 1.25 L_2 + 0.75 L_3 - 1.25 L_1 L_3) \right) \\\\ \log M \mid A, \boldsymbol{L} & \sim \mathcal{N}(-3 + 1.5 A + 1.75 L_1 + 1.5 L_2 - 0.25 L_3,\; 0.25^2) \\\\ Y \mid M, A, \boldsymbol{L} & \sim \text{Bern}\left(\expit(-1 + 2.5 A + 1.75 M + 0.5 A M - 2.25 L_1 - 1.75 L_2 - 1.5 L_3) \right) \end{aligned} \]
- Assay LoD defined by a fixed quantile (e.g., 30%) below which \(M\) is left-censored; used to generate several experimental scenarios.
- Mediator is left-censored: Observed \(M\) is reported as \(\text{LoD}\) when \(M < \text{LoD}\).
- Outcome: Binary endpoint (e.g., hospitalization, death—as in ACTIV-2).
Imputation strategies and estimation approach
Objective: Mitigate bias by applying the proposed imputation approaches with standard (e.g., plug-in) estimators of NDE/NIE statistical estimands.
Implemented 5 fractional imputation strategies, including a conventional method while keeping nuisance estimators correctly specified for NDE/NIE estimation.
- Compared performance across the five strategies and against an oracle EM strategy that used all correctly specified nuisances.
- Investigated performance in terms of
- standard metrics: bias, variance, MSE, CI coverage (based on both standard and m-out-of-n bootstrap); and
- severity of left-censoring of \(M\): 25%, 50%, 75% of observations.
- Our findings indicate that flexible, semi-parametric estimation strategies approach performance of the oracle strategy, even with heavy left-censoring (75%).
Simulation results: Natural direct effect
Details on imputation strategies
- EM: true nuisances (Best case!)
✅ Correct log-normal density for \(M\)
✅ Correct models for \(\E(M \mid A, L)\) and \(\E(Y \mid M, A, L)\) - LOD/2 imputation (Worst case!)
⚠️ Imputes censored \(M\) values by LoD/2 - EM: oracle dens., misspec. GLMs
✅ Correct log-normal density for \(M\)
❌ Misspecified mediator model \(\E(M \mid A, L)\) (omits covariates)
❌ Misspecified outcome model (omits covariates) - EM: misspec. dens., misspec. GLMs
❌ Incorrect log-normal density for \(M\)
❌ Misspecified mediator model \(\E(M \mid A, L)\) (omits covariates)
❌ Misspecified outcome model (omits covariates) - EM: heterosced. CDE, HAL
🔹 Semiparametric conditional density estimation (CDE) with heteroscedastic errors
🔹 Estimates conditional mean and variance of \(M\) using the highly adaptive lasso (HAL) - EM: homosced. CDE, HAL
🔹 Semiparametric conditional density estimation (CDE) with homoscedastic errors
🔹 Estimates conditional mean of \(M\) using the highly adaptive lasso (HAL)
Simulation results: Natural indirect effect
Work-in-progress
- Verify that fractional imputation-based approach is compatible with efficient estimators of the NDE/NIE functional.
- Apply methods to re-analysis of the ACTIV-2/ACTG A5401 trial data, informing prior evidence on the role of viral RNA clearance in COVID-19 disease (Li et al. 2022).
References
Appendix
Estimation of NDE and NIE
Natural direct effect (NDE) functional: \[ \Psi_{\text{NDE}}(\Pr) = \E \left[\E \left\{ \E(Y \mid A=1, M, L) - \E(Y \mid A=0, M, L) \mid A=0, L \right\} \right] \]
Plug-in (g-computation) estimator:
- Fit \(\hat{\E}(Y \mid A=1, M=m, L = l)\)
- Fit \(\hat{\E}(Y \mid A=0, M=m, L = l)\)
- Fit \(\hat{f}(M=m \mid A=0, L = l)\)
- Estimate \(\Pr(L = l)\) empirically (from the sample)
Evaluate plug-in estimator (g-computation): \[\small \Psi_{\text{NDE}}(\hat{\Pr}) = \hat{\E} \left[ \hat{\E} \left\{ \hat{\E}(Y \mid A=1, M, L) - \hat{\E}(Y \mid A=0, M, L) \mid A=0, L \right\} \right] \] The bootstrap may be used to obtain inference based on this plug-in estimator.
Double bootstrap algorithm
- Initialize a grid of \(\gamma\) values and specify \(k\) (e.g., \(k = 5\) for \(p_{\text{cens}} = 0.5\)); draw \(B_1\) samples of size \(n\).
- For each sample \(b_1\):
- Run \(m\)-out-of-\(n\) bootstrap (\(B_2\) iterations)
- Construct confidence interval (CI) based on the \(m\)-out-of-\(n\) samples
- Run \(m\)-out-of-\(n\) bootstrap (\(B_2\) iterations)
- Evaluate coverage:
- Calculate proportion of CIs that cover original estimate
- Adjust \(\gamma\) until target nominal coverage is achieved
- Calculate proportion of CIs that cover original estimate
- Final step: Compute final CIs using optimal \(\hat{m} = n^{\hat{c}}\)
Advantages
- Robust to non-regularity via adaptive \(m\)
- Seamless transition to standard bootstrap when \(p_{\text{cens}} = 0\)
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