📊 statistics

Doubly robust Methods for Recurrent Event Outcomes: Causal Effects of Blood Pressure Medications on Acute Kidney Injuries

This paper proposes advanced doubly robust methods to estimate the causal effects of standard versus intensive blood pressure therapies on recurrent acute kidney injuries using Systolic Blood Pressure Intervention Trial data, while addressing time-varying confounding, model misspecification, medication adherence, and the semi-competing risk of death.

Original authors: Wenling Zhang, Cecilia Cotton, Lan Wen

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wenling Zhang, Cecilia Cotton, Lan Wen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out if a specific diet plan (let's call it the "Intensive Diet") causes more stomach aches than a "Standard Diet." But there's a catch: people don't always follow the diet perfectly, and some people might get so sick from other causes that they stop eating entirely (which we'll call "dropping out").

This paper is like a sophisticated detective story about how to measure the true effect of these diets on stomach aches, even when people cheat on the diet or drop out of the study. The researchers applied this detective work to a real medical study called SPRINT, which looked at two ways of treating high blood pressure: a Standard approach and an Intensive (stricter) approach.

Here is the breakdown of their work using simple analogies:

1. The Problem: It's Not Just a One-Time Decision

Usually, when scientists study a treatment, they ask, "Did you take the pill at the start?" But in real life, taking blood pressure medication is like a marathon, not a sprint. People might take their pills every day for a month, then forget for a week, then take them again.

The researchers wanted to know:

Intent-to-Treat (The "Assignment" View): If we assign someone to the Intensive Diet, do they get more stomach aches than if we assign them to the Standard Diet, regardless of whether they actually stick to it?
Per-Protocol (The "Perfect Adherence" View): If we imagine a world where everyone followed their assigned diet perfectly, would the Intensive Diet cause more stomach aches?

2. The Complication: The "Drop Out" Factor

In the SPRINT study, some patients died. In medical statistics, death is tricky. If a patient dies, they can't have another stomach ache (or in this case, another Acute Kidney Injury, or AKI).

The Trap: If you simply ignore people who died, you might get a wrong answer. It's like counting how many times people trip while running a race, but only counting the people who finished the race. If the "Intensive Diet" group had fewer people finish the race because they died earlier, you might falsely think they tripped less often.
The Solution: The researchers treated death as a "semi-competing risk." Think of it like a race where the track ends abruptly for some runners. They built a model that accounts for the fact that death stops the possibility of future kidney injuries, ensuring they didn't accidentally hide the true risk.

3. The Tool: The "Double-Robust" Safety Net

To solve this, the authors used a method called Doubly Robust Estimation (specifically using something called Targeted Maximum Likelihood Estimation, or TMLE).

The Analogy: Imagine you are trying to predict the weather. You have two different weather forecasters (models):

Forecaster A looks at the wind and clouds (Treatment/Adherence).
Forecaster B looks at the temperature and humidity (Patient History/Outcomes).

Usually, if one forecaster is wrong, your prediction is wrong. But with this "Double-Robust" method, you are safe as long as at least one of the forecasters is right. Even if the math for the wind is messy, if the temperature math is solid, you still get the correct answer. This makes the results very reliable even if the researchers' initial guesses about the data weren't perfect.

4. The Findings: What Did They Discover?

After running their complex math on the SPRINT data (which involved over 9,000 people), here is what they found:

The Intensive Approach Causes More Kidney Injuries: Whether they looked at the group as assigned (Intent-to-Treat) or imagined a world where everyone followed the rules perfectly (Per-Protocol), the Intensive Blood Pressure Treatment resulted in slightly more Acute Kidney Injury (AKI) episodes than the Standard treatment over the first four years.
The Difference is Small but Real: The increase was small (about 1 or 2 extra events per 100 people), but statistically significant.
Adherence Didn't Change Much: Interestingly, it didn't matter much if people actually took their pills perfectly or not. The results for "perfect adherence" were almost identical to the "assigned" results. This suggests that in this specific study, most people were already doing a pretty good job following their doctors' orders, so forcing them to be "perfect" didn't change the outcome much.
Death vs. Kidney Issues: The researchers noted that because the death rate in the study was low (under 5%), the "Total Effect" (which includes the impact of death) looked very similar to the "Direct Effect" (kidney issues only). However, they warned that in other studies with higher death rates, ignoring death would lead to very wrong conclusions.

5. The Bottom Line

The paper doesn't say "Stop taking intensive blood pressure meds." Instead, it provides a better way to measure the trade-offs.

It confirms that while intensive blood pressure control is great for the heart, it comes with a small, measurable cost to the kidneys. The researchers proved that their new mathematical "detective tools" can handle the messy reality of people changing their behavior over time and people dropping out of studies, giving doctors a clearer picture of the true risks and benefits.

Technical Summary: Doubly Robust Methods for Recurrent Event Outcomes

Problem Statement
Evaluating the average causal effects (ACE) of treatment strategies on recurrent event outcomes (e.g., acute kidney injury episodes) is critical in clinical research but methodologically challenging. Standard analyses often fail to adequately address the complexity of longitudinal settings involving time-varying treatments, time-varying confounding, and non-adherence. Furthermore, previous studies estimating ACE for recurrent events have predominantly focused on point exposure settings, neglecting the nuances of sustained treatment compliance. Additionally, the presence of "semi-competing risks," such as death, complicates the analysis of recurrent non-terminal events, as death truncates the risk set for future events. The Systolic Blood Pressure Intervention Trial (SPRINT) data presents a specific case where intensive versus standard blood pressure management may differentially affect acute kidney injury (AKI) recurrences and mortality, requiring robust causal inference methods to disentangle these effects.

Methodology
The authors propose a framework based on the counterfactual model and the longitudinal g-formula to estimate causal effects under time-varying treatments and adherence regimes.

Data Structure and Identification:
- The study defines a longitudinal data structure with baseline variables ( $L_0$ ), time-varying covariates ( $L_k$ ), treatment assignment ( $Z$ ), time-varying adherence ( $A_k$ ), and binary recurrent event indicators ( $Y_k$ ).
- Causal identification relies on three standard assumptions: Conditional Exchangeability, Positivity, and Consistency.
- The target estimand is the expected cumulative count of recurrent events ( $E[R_{K+1}^{z, \bar{a}_K}]$ ) under specific intervention regimes for treatment ( $z$ ) and adherence ( $\bar{a}_K$ ). This is expressed via an iteratively conditional expectation (ICE) form.
Doubly Robust Estimation via TMLE:
- The authors extend the Longitudinal Targeted Maximum Likelihood Estimation (TMLE) to handle recurrent event outcomes on a discrete time grid.
- The method utilizes the Efficient Influence Function (EIF) to construct estimators that possess double robustness: the estimator remains consistent if either the propensity score model (treatment/adherence mechanism) or the outcome regression model is correctly specified.
- The implementation incorporates machine learning algorithms (e.g., Extreme Gradient Boosting, Generalized Additive Models) for nuisance function estimation to mitigate model misspecification bias.
- The algorithm involves an iterative update process where outcome regression models are adjusted using a fluctuation parameter derived from the propensity scores to target the specific estimand.
Handling Semi-Competing Risks:
- To address the issue of death truncating recurrent event observation, death ( $D_k$ ) is modeled not as a censoring event to be eliminated, but as a semi-competing risk.
- Death is incorporated as a component of the time-varying covariates ( $L_k^*$ ), with a deterministic relationship where death implies the cessation of future AKI events ( $D_k=1 \implies Y_k=0$ ). This allows for the estimation of the total effect, which includes pathways mediated by death.

Key Contributions

Methodological Extension: The paper formalizes the target causal estimands and data structures for recurrent event outcomes under time-varying treatment and adherence, extending longitudinal TMLE to this specific context.
Practical Implementation: It provides a transparent, step-by-step implementation guide (Algorithm 1) for constructing doubly robust estimators using machine learning, facilitating application in real-world studies.
Handling Complexity: The framework explicitly addresses time-varying confounding, non-adherence (allowing for both Intent-to-Treat and Per-Protocol analyses), and the semi-competing risk of death without relying on hypothetical interventions that eliminate death.

Results (SPRINT Application)
The proposed methods were applied to the SPRINT dataset (9,322 subjects) to evaluate the effects of intensive ( $<120$ mm Hg) versus standard ( $<140$ mm Hg) blood pressure lowering therapy on AKI recurrences over a four-year period.

Total Intent-to-Treat (ITT) Effect: Assigning patients to intensive therapy resulted in a statistically significant increase in AKI events compared to standard therapy. The estimated difference was 0.019 (95% CI: [0.011, 0.026]).
Total Per-Protocol (PP) Effect: Under a hypothetical scenario of full adherence to assigned regimens, the intensive therapy group also showed a marginally higher expected number of AKI recurrences compared to the standard group. The estimated difference was 0.017 (95% CI: [0.007, 0.027]).
Adherence Effects: Supplementary analyses of the Population Intervention Effect (PIE) for adherence indicated that moving from observed adherence to full adherence had a negligible impact on AKI recurrence rates, suggesting that the similarity between ITT and PP estimates is due to high baseline adherence in the study population.
Robustness: Sensitivity analyses regarding temporal discretization (bi-yearly vs. annual) and the use of different machine learning algorithms (GLM, XGB, GAM) yielded consistent results, confirming the robustness of the causal conclusions.

Significance and Claims
The paper claims that its methodology offers a rigorous approach to estimating the total causal effect of treatment strategies on recurrent events in the presence of semi-competing risks. By treating death as a truncating event rather than a censoring mechanism to be hypothetically eliminated, the authors argue their approach reflects real-world public health implications more accurately than methods relying on controlled direct effects (which require assumptions about eliminating death that may be ill-defined).

The authors conclude that while intensive blood pressure lowering reduces mortality, it is associated with a marginal increase in the frequency of recurrent AKI episodes. They emphasize that their doubly robust, machine-learning-enhanced TMLE framework provides a reliable tool for clinicians and researchers to navigate the trade-offs between cardiovascular benefits and renal risks in longitudinal studies with complex adherence and competing risk structures. The paper modestly notes that the validity of the adherence definition relies on the assumption that interval-level adherence measurements accurately reflect true adherence, a limitation that may be more pronounced in populations with lower compliance rates.