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A Law of Iterated Expectation Primer for Causal Inference

This paper provides a primer clarifying the relationship between the law of iterated expectation and the g-formula for causal inference, presenting both non-iterative and iterative forms of the formula and illustrating their application through progressively complex numerical examples.

Original authors: Ashley I. Naimi, Razieh Nabi, Lindsay J. Collin, Paul N. Zivich, Stephen R. Cole

Published 2026-06-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashley I. Naimi, Razieh Nabi, Lindsay J. Collin, Paul N. Zivich, Stephen R. Cole

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

The Big Picture: Why Do We Need This?

Imagine you want to know if a specific medicine (let's call it "Tamoxifen") actually prevents breast cancer from coming back. In a perfect world, you could give the medicine to one group of people and a sugar pill to another group, then compare the results. This is a randomized trial.

But in the real world, we often only have observational data. We can't force people to take medicine; we just watch what they choose to do. The problem is that people who choose the medicine might be different from those who don't (maybe they are sicker, or have different genetics). These differences are called confounders. If we don't account for them, we might blame the medicine for something that was actually caused by the patient's underlying health.

This paper is a "primer" (a beginner's guide) on how to fix this math problem. It explains a specific mathematical tool called the Law of Iterated Expectation and shows how it helps us turn messy, real-world data into a clear answer about cause and effect.

The Core Concept: The "Law of Iterated Expectation"

Think of this law as a way to calculate a weighted average.

Imagine you are a school principal trying to find the average test score for the whole school.

The Simple Way: You just add up every single student's score and divide by the number of students. This is the "marginal expectation."
The "Iterated" Way: You realize the school has different grades (1st grade, 2nd grade, etc.). You calculate the average score for 1st graders, then the average for 2nd graders, and so on. Then, you take those grade-level averages and combine them, but you weight them by how many students are in each grade.

The Law of Iterated Expectation simply says: You get the same final answer whether you average everyone at once, or if you average the groups first and then combine the groups.

In the paper, the authors explain that this mathematical identity is the engine behind the g-formula, a famous tool used to figure out causal effects.

Two Ways to Drive the Same Car: NICE and ICE

The paper introduces two different ways to use this math to solve the causal problem. They are mathematically identical (they give the exact same answer), but they look at the data differently. The authors call them NICE and ICE.

1. NICE (Non-Iterative Conditional Expectation)

The Analogy: The "Recipe Book" approach.
Imagine you want to know the average height of all people in a city, but you only have data on people who wear red hats and blue hats.

How NICE works: You look at the "Red Hat" group and calculate their average height. You look at the "Blue Hat" group and calculate their average height. Then, you look at the city census to see what percentage of people wear red hats vs. blue hats. Finally, you mix those two averages together using the census percentages as weights.
In the paper: The authors show this using a simple example of Tamoxifen and lymph nodes. They calculate the recurrence rate for different groups and then "plug in" the numbers to get a final weighted average.

2. ICE (Iterative Conditional Expectation)

The Analogy: The "Prediction Machine" approach.
Imagine you are a weather forecaster. Instead of just averaging past data, you build a model that predicts the weather for every single day based on the conditions of that day.

How ICE works: You take your data, run it through a model, and generate a "predicted outcome" for every single person in your dataset (as if they had all taken the medicine). Then, you just take the average of all those predictions.
In the paper: The authors show that you can do this by creating a list of "what-if" predictions for every person and then averaging them up.

The Key Takeaway: Whether you do the "Recipe Book" (NICE) or the "Prediction Machine" (ICE), you end up with the same number. The paper proves that these two methods are just two different ways of writing the same mathematical sentence.

Getting More Complex: Time and Moving Parts

The paper doesn't stop at simple examples. It shows how this works when things get complicated:

More Variables: What if you have age, income, race, and sex all mixed together? The "Recipe Book" (NICE) becomes very hard to write out because there are too many combinations. The "Prediction Machine" (ICE) is much easier because you just let the computer handle the math.
Time-Varying Confounders: This is the hardest part. Imagine a scenario where:
- You take a drug at Time 1.
- That drug changes your health (a confounder) at Time 2.
- That new health status affects whether you take a second dose of the drug at Time 2.
- Finally, you look at the outcome at Time 3.
In this scenario, standard statistics fail because the "confounder" (your health) was changed by the treatment itself. The paper shows that the g-formula (using the Law of Iterated Expectation) is the only way to untangle this knot. It does this by working backwards:
- First, predict the outcome at the very end.
- Then, work backward to predict what happened at Time 2.
- Then, work backward to Time 1.
- Finally, average it all out.
The paper calls this "backward recursion." It's like solving a maze by starting at the exit and walking backward to the entrance.

What the Authors Actually Claim (and What They Don't)

They DO claim: The Law of Iterated Expectation is the mathematical foundation that allows us to turn "what we observed" into "what would have happened" (causal effects).
They DO claim: The NICE and ICE methods are mathematically equivalent. They represent the same thing, just written differently.
They DO claim: In simple situations (time-fixed), both methods are easy. In complex situations (time-varying), the ICE method (working backward) is often easier to code and more robust against certain types of errors.
They DO NOT claim: This paper provides new medical results, new clinical guidelines, or specific advice for doctors on how to treat patients. It is purely a guide on the mathematics and logic of how to analyze data.
They DO NOT claim: One method is "better" than the other in a general sense; they are just different tools for the same job. However, they note that if your mathematical models are wrong, both methods can give wrong answers.

The Bottom Line

This paper is a translator. It takes a very dense, scary mathematical concept (the Law of Iterated Expectation) and explains how it acts as the bridge between raw data and causal truth. It shows researchers that whether they use a "weighted average" approach or a "step-by-step prediction" approach, they are using the same fundamental logic to answer the question: "What would have happened if we had done something different?"

Technical Summary: A Law of Iterated Expectation Primer for Causal Inference

Problem Statement
The g-formula is a foundational tool for identifying causal effects in observational data, relying heavily on the law of iterated expectation (LIE). However, the mathematical notation used to express the LIE and the g-formula—often involving integration with respect to probability measures (e.g., $dP(x)$)—can be opaque to researchers with limited backgrounds in mathematical statistics. This opacity creates a barrier to understanding the mechanics of widely used causal estimators, such as parametric g-computation. Furthermore, there is often confusion regarding the relationship between the statistical identity of the LIE and the causal identification provided by the g-formula, as well as the distinction between the two nonparametrically equivalent forms of the g-formula: the non-iterative conditional expectation (NICE) and the iterative conditional expectation (ICE).

Methodology
This paper serves as a primer designed to bridge the gap between abstract mathematical identities and practical causal inference. The authors employ a pedagogical approach using three progressively complex numerical examples to illustrate the application of the LIE and the g-formula:

Time-Fixed Example (Binary Confounder): Using data on tamoxifen use and breast cancer recurrence, the authors demonstrate the g-formula with a single binary confounder. They explicitly calculate the causal risk difference using both the NICE form (a weighted sum of conditional means) and the ICE form (nested expectations), showing they yield identical results.
Time-Fixed Example (Mixed Confounders): Utilizing NHANES data to estimate the effect of smoking cessation on weight change, the authors address a scenario with a multivariate vector of confounders (continuous, binary, and categorical). They contrast the computational labor of the NICE form (requiring explicit stratification and weighting) with the computational simplicity of the ICE form (averaging predictions over the empirical distribution).
Time-Varying Example: Using a simulated dataset with two timepoints, the authors address settings where standard regression fails due to time-varying confounding affected by prior exposure. They demonstrate how the LIE extends to sequential settings, deriving the g-formula for counterfactual means. They detail the implementation of both the NICE form (requiring modeling of the confounder distribution to generate weights) and the ICE form (using backward recursion/sequential regression).

Throughout these examples, the authors rely on standard causal identification assumptions: causal consistency, positivity (and sequential positivity), and conditional exchangeability (and sequential conditional exchangeability).

Key Contributions

Clarification of Notation: The paper demystifies the integration notation ( $\int \cdot dP(x)$ ) used in the LIE, explaining it as a weighted average where the weights are determined by the probability distribution of the covariates. It distinguishes between the statistical identity of the LIE and the causal g-formula, noting that the latter is derived by combining the LIE with causal identification assumptions that allow counterfactual quantities to be replaced by observed data quantities.
Dual Formulation of the G-Formula: The authors explicitly define and demonstrate two nonparametrically equivalent forms of the g-formula:
- NICE (Non-Iterative Conditional Expectation): A single weighted average of conditional outcome means. In time-varying settings, this requires modeling the distribution of time-varying confounders to generate weights.
- ICE (Iterative Conditional Expectation): A sequence of nested expectations. In time-varying settings, this is implemented via backward recursion (g-computation), regressing outcomes on covariates and exposures at each time step moving backward from the end of follow-up.
Computational vs. Mathematical Equivalence: The paper clarifies that while NICE and ICE are mathematically equivalent (representing the same causal estimand), they differ computationally. In time-varying settings, NICE requires correct specification of the confounder distribution models, whereas ICE does not, making ICE robust to misspecification of confounder models (though both require correct outcome models).

Results
Through the numerical examples, the authors demonstrate that:

Both NICE and ICE formulations produce identical estimates of causal effects in time-fixed settings (e.g., $\hat{\psi} = -0.03$ in the tamoxifen example and $\hat{\psi} = 3.1$ kg in the smoking cessation example).
In time-varying settings, the ICE approach (via backward recursion) avoids the need to explicitly model the joint distribution of time-varying confounders, whereas the NICE approach requires these models to construct the necessary weights.
The ICE approach, implemented as a sequence of regressions, naturally aligns with the mechanics of longitudinal g-computation and targeted maximum likelihood estimation (TMLE).

Significance and Claims
The authors position this work as a "primer" intended to build intuition for applied researchers. They claim that a firm grasp of the LIE and its two forms makes the mechanics of causal estimators transparent. The paper argues that understanding the LIE provides a "conceptual bridge" connecting causal identification to practical estimation.

The authors modestly state that while NICE and ICE are different computational implementations that may yield different numerical values in finite samples or under model misspecification, they are nonparametrically equivalent representations of the same observed-data functional. The paper concludes that familiarity with the structured, assumption-justified weighted average of conditional outcome means—whether in iterative or non-iterative form—equips researchers to understand, implement, and critically evaluate the full spectrum of g-methods in causal inference. The authors do not propose new algorithms or future applications beyond the scope of clarifying existing methodologies.