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Efficient Targeted Maximum Likelihood Estimators for Two-Phase Design Problems

This paper reviews existing estimators for two-phase design problems and introduces a new class of asymptotically equivalent estimators within the Targeted Maximum Likelihood Estimation (TMLE) framework to improve efficiency under the coarsening at random assumption.

Original authors: Sky Qiu, Susan Gruber, Pamela A. Shaw, Brian D. Williamson, Mark J. van der Laan

Published 2026-03-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sky Qiu, Susan Gruber, Pamela A. Shaw, Brian D. Williamson, Mark J. van der Laan

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 a detective trying to solve a mystery: Does a new medicine actually prevent heart attacks?

To get the perfect answer, you would need to interview every single patient in your study about their age, their diet, their genetics, their lifestyle, and their medical history. You would also need to know exactly who took the medicine and who didn't, and then wait a year to see who had a heart attack.

But here's the problem: Interviewing everyone about their diet and genetics is incredibly expensive and time-consuming. You simply can't afford to ask everyone those deep questions.

The Two-Phase Detective Strategy (The Setup)

This paper is about a clever way to solve the mystery without breaking the bank. It uses a Two-Phase Design:

Phase 1 (The Wide Net): You interview everyone in your study, but only ask the easy, cheap questions (like age and gender). You also record who took the medicine and who didn't.
Phase 2 (The Deep Dive): You can't afford to ask the deep questions (like diet and genetics) for everyone. So, you pick a subsample—a smaller group of people—to interview in depth. Maybe you pick the people who look "high risk" based on the cheap info you already have.

The Challenge: Now you have a puzzle. You have deep data for only a few people, but you need to make a conclusion about everyone. If you just look at the deep-data group, you might get a biased answer because that group isn't perfectly representative of the whole crowd.

The Old Tools (The "Raking" and "IPCW" Methods)

Statisticians have had tools to fix this puzzle for a long time:

The "Weighted" Approach (IPCW-TMLE): Imagine you have a scale. You take the people you interviewed deeply and give them "extra weight" on the scale to represent the people you didn't interview deeply. If you picked 10 high-risk people to interview deeply, and they represent 100 people in the total crowd, you give them a weight of 10. This balances the scale.
The "Raking" Approach: This is like adjusting the weights on your scale so that the total number of "men" and "women" in your deep group matches the total number of men and women in the whole crowd. It forces the deep group to look like the whole crowd.

The Problem with the Old Tools:
The paper argues that while the "Raking" tool is popular, it has a hidden flaw. It tries to make the deep group look like the whole crowd based on a specific mathematical formula. If that formula is slightly wrong (which it often is in real life), the tool gives you the answer to the wrong question. It tells you about a "census" (a made-up version of reality) rather than the true causal effect of the medicine.

The New Tools (The "Targeted Maximum Likelihood Estimators")

The authors of this paper are like master mechanics who have built a new, more precise engine for solving this puzzle. They introduce a new class of tools called Targeted Maximum Likelihood Estimators (TMLE).

Think of TMLE as a Self-Correcting GPS:

The Initial Guess: It starts with a rough guess of the answer (like a GPS saying, "You are here").
The Targeting Step: It looks at the specific question you are asking ("How effective is the medicine?"). It then tweaks the initial guess just enough to make sure it answers that specific question perfectly, without messing up the rest of the data.
The Correction: It keeps adjusting the weights and the model until the "error" is zero.

The paper introduces three new variations of this GPS:

The "Efficient Estimating Equation" (EEE): A smart calculator that solves the math perfectly but doesn't quite fit the "plug-in" style of the old tools.
The "Quasi-TMLE": A version that takes the EEE and adds the "plug-in" feature, making it more stable and reliable in small groups (like having a spare tire).
The "Alternative TMLE": A completely different way of looking at the map that leads to the same destination but takes a slightly different route.

Why Does This Matter? (The "Double Robustness" Superpower)

The biggest superpower of these new tools is Double Robustness.

Imagine you are trying to guess the winner of a race. You have two sources of information:

Source A: The runners' training logs (Outcome Model).
Source B: The weather report (Sampling Model).

If you use the old "Raking" tool, you need both sources to be perfect. If the weather report is slightly wrong, your prediction fails.

But with the new TMLE tools, you only need one of them to be right.

If the training logs are perfect, the tool works, even if the weather report is wrong.
If the weather report is perfect, the tool works, even if the training logs are wrong.

This makes the new tools incredibly robust. In the messy real world, where data is never perfect, this means you are much less likely to get a wrong answer.

The Simulation Results (The Proof)

The authors ran thousands of computer simulations (like running the race 1,000 times in a video game) to test their new tools against the old ones.

When the data was tricky: The old "Raking" tool often failed, giving answers that were way off or had confidence intervals that didn't cover the truth.
The New Tools: They consistently hit the target. They were more accurate, had less error, and their "confidence intervals" (the range where the true answer likely lies) were reliable.

The Bottom Line

This paper is about upgrading the toolkit for researchers who have to work with "incomplete" data (where expensive info is only available for a few people).

Old Way: Try to force the small group to look like the big group using rigid rules. (Fragile; breaks if rules are slightly off).
New Way (TMLE): Use a flexible, self-correcting system that targets the specific question you are asking. It's like having a GPS that knows you only need to get to the destination, not follow every single rule of the road perfectly.

The authors show that these new methods are not only mathematically elegant but also practically superior, giving researchers more confidence in their conclusions about what actually works in medicine and public health.

1. Problem Statement

The paper addresses the statistical challenge of estimating full-data causal parameters (specifically the Average Treatment Effect, ATE) in two-phase sampling designs.

Design Structure:
- Phase 1: A random sample is drawn from the target population. A subset of variables ( $V$ , including baseline covariates $W_1$ , treatment $A$ , and outcome $Y$ ) is observed for all individuals.
- Phase 2: A subsample of the Phase 1 cohort is selected based on $V$ . Additional, often expensive, covariates ( $W_2$ ) are measured only for this subsample.
Data Structure: The observed data $O = (V, \Delta, \Delta W_2)$ exhibits a "coarsened" structure where $W_2$ is missing for Phase 1 individuals.
Assumptions: The authors assume Coarsening at Random (CAR), meaning the Phase 2 sampling indicator $\Delta$ depends only on the fully observed variables $V$ .
Goal: To estimate the ATE, $\Psi_F = E[Y_1 - Y_0]$ , efficiently and robustly, leveraging the full data structure while accounting for the missingness mechanism.

2. Methodology and Existing Approaches

The paper reviews and critiques existing methods before proposing new estimators within the Targeted Maximum Likelihood Estimation (TMLE) framework.

Inverse Probability of Censoring Weighted TMLE (IPCW-TMLE):
- Standard approach that weights Phase 2 observations by the inverse probability of selection ( $\Delta/\Pi(V)$ ).
- It targets the outcome regression ( $Q$ ) and treatment mechanism ( $g$ ) but typically treats the sampling mechanism ( $\Pi$ ) as a nuisance parameter to be estimated, not targeted.
IPCW-TMLE with Targeted $\Pi$ :
- An extension where the sampling mechanism $\Pi$ is also updated via a targeting step to solve the empirical mean of the $\Pi$ -component of the Efficient Influence Curve (EIC). This improves efficiency but requires iterative fitting.
Generalized Raking (GR):
- A calibration method that adjusts weights to satisfy calibration equations.
- Critical Limitation Identified: The authors demonstrate that standard GR targets a "census estimand" defined by a parametric working model. If the outcome model is misspecified, the GR estimator is inconsistent for the true non-parametric causal estimand, leading to poor confidence interval coverage even with large sample sizes.

3. Key Contributions: New Estimators

The authors propose a class of new estimators that are asymptotically equivalent to the optimal TMLE but differ in construction and finite-sample properties.

A. Estimators based on EIC Rearrangement (Sections 4.1 & 4.2)

By rearranging the A-IPCW representation of the EIC, the authors derive:

Efficient Estimating-Equation (EEE) Estimator:
- Solves the empirical mean of the EIC directly.
- It is not a plug-in estimator (i.e., it does not respect the bounds of the parameter space or the likelihood structure).
Quasi-TMLE:
- A modification of the EEE that enforces the plug-in property.
- It constructs a submodel for the conditional expectation of the EIC and solves a system of equations to ensure the estimator is a valid plug-in estimate.
- Significance: Retains the efficiency of EEE while gaining the stability and finite-sample benefits of plug-in estimators.

B. Alternative Representation TMLE (Section 5)

The authors propose a TMLE based on an alternative representation of the target parameter: $\Psi(P) = E_V [ E_{X|V} [Q(1, W) - Q(0, W) | \Delta=1, V] ]$ .
This leads to a canonical gradient with four components ( $D_Q, D_\Pi, D_\Gamma, D_{P_V}$ ).
Algorithm 3: A multi-step iterative procedure that targets $Q$ , $\Pi$ , and the conditional expectation of the outcome difference ( $\Gamma$ ) sequentially until convergence.

4. Theoretical Analysis

Exact Remainder Terms: The paper derives the exact remainder terms for these estimators.
Double Robustness: The estimators possess a double robustness structure. Consistency is achieved if either:
1. The sampling mechanism ( $\Pi$ ) and treatment mechanism ( $g$ ) are correctly specified, OR
2. The sampling mechanism ( $\Pi$ ) and outcome regression ( $Q$ ) are correctly specified.
Convergence Rates: The analysis shows that the product of the estimation errors for the nuisance parameters must converge at a rate faster than $n^{-1/2}$ . This allows for the use of flexible machine learning algorithms (e.g., Super Learner) provided they converge sufficiently fast.

5. Simulation Results

The authors conducted extensive simulations comparing the Raking estimator, A-IPCW, various IPCW-TMLE variants, EEE, Quasi-TMLE, and the new TMLE.

Double Robustness Verification:
- In scenarios with misspecified models, the Raking estimator failed significantly, showing large bias and coverage dropping to 0% as sample size increased (confirming it targets the wrong estimand).
- All TMLE-based estimators (including EEE, Quasi-TMLE, and the new TMLE) maintained nominal coverage (~95%) and low bias, demonstrating robustness.
Plug-in Property Advantage:
- Quasi-TMLE and other plug-in estimators consistently achieved lower Mean Squared Error (MSE) compared to their non-plug-in counterparts (A-IPCW and EEE).
- This suggests that enforcing the plug-in property provides tangible finite-sample benefits, particularly in reducing variance.
Efficiency Gains:
- Targeting the Phase 2 sampling mechanism ( $\Pi$ ) generally improved efficiency (lower MSE) compared to standard IPCW-TMLE, though gains were modest when nuisance parameters were estimated via Super Learner.
- The "Raking" approach to targeting $\Pi$ (replacing the logistic regression step in TMLE) yielded performance similar to the standard TMLE targeting of $\Pi$ .

6. Significance and Conclusion

Resolution of Raking Limitations: The paper clarifies that while Generalized Raking is popular, it is not a valid substitute for TMLE in causal inference unless the working model is perfectly specified. TMLE is necessary to target the non-parametric causal estimand.
New Estimator Class: The introduction of Quasi-TMLE and the Alternative Representation TMLE expands the toolkit for two-phase designs. These estimators offer the same asymptotic efficiency as standard TMLE but may offer better finite-sample performance due to the plug-in property.
Practical Guidance: The authors recommend using Super Learner (ensemble machine learning) for nuisance parameter estimation to ensure robustness against model misspecification. They also suggest that while targeting $\Pi$ improves efficiency, the primary driver of robustness is the double robustness structure of the TMLE framework itself.
Future Directions: The paper suggests exploring ensemble strategies to combine these asymptotically equivalent estimators and investigating imputation-based methods to better utilize Phase 1 data when Phase 2 sampling rates are low.

In summary, this work provides a rigorous theoretical and empirical foundation for using TMLE in two-phase designs, highlighting the superiority of plug-in estimators over traditional raking or non-plug-in methods when estimating causal effects under model uncertainty.