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Inverse Probability Weighting of Count Exposures in the Presence of Missing Data: A Simulation Study

This simulation study evaluates five inverse probability weighting methods for count exposures under missing data conditions, finding that multinomial binning, CBPS, GBM, and energy balancing generally yield low bias and nominal coverage, while highlighting the critical need for imputation models that accommodate right-truncated overdispersed counts.

Original authors: Martin N. Danka, Jessica K. Bone, George B. Ploubidis, Richard J. Silverwood

Published 2026-03-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Martin N. Danka, Jessica K. Bone, George B. Ploubidis, Richard J. Silverwood

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: Trying to Find the "Real" Cause

Imagine you are a detective trying to solve a mystery: Does having a lot of psychological stress (like worrying or feeling down) actually cause people to get sick later in life?

You have a huge notebook of data from thousands of people born in 1970. You can see who was stressed at age 34 and who got sick at age 42. But there's a problem: Life isn't a controlled experiment.

People who are stressed might also be poorer, smoke more, or have had a harder childhood. These other factors (confounders) also cause sickness. If you just look at the raw numbers, you might think stress causes sickness, but really, it might just be that "hard childhoods" cause both stress and sickness. You need a way to isolate the stress from the other messy life factors.

The Tool: Inverse Probability Weighting (IPTW)

To fix this, statisticians use a trick called Inverse Probability Weighting (IPTW).

Think of your dataset as a crowded party.

Some people at the party are very "stressed" (the exposure).
Some are "not stressed."
The "stressed" people happen to be mostly wearing red hats (a confounder, like low income), while the "not stressed" people are wearing blue hats.

If you want to know if the stress causes the sickness, you can't just look at the crowd; the red hats are skewing the results.

IPTW is like a magical scale.
It gives a "weight" to every person at the party.

If a stressed person is wearing a red hat (which is common among stressed people), the scale makes them lighter (down-weights them) because they are over-represented.
If a stressed person is wearing a blue hat (which is rare for them), the scale makes them heavier (up-weights them) because they are under-represented.

The goal is to create a "fake universe" (a pseudopopulation) where the distribution of red and blue hats is exactly the same for both the stressed and non-stressed groups. In this fake universe, the only difference between the groups is the stress itself. Now, if the stressed group gets sicker, you can be much more confident it's the stress causing it, not the hats.

The New Problem: Counting Symptoms

Usually, this "weighting" trick works great for simple "Yes/No" exposures (like: Did you smoke? Yes/No).

But in this study, the exposure is Count Data.
Instead of just "Stressed" or "Not Stressed," the data is a score: "How many symptoms did you have?" (0, 1, 2, 3... up to 9).

This is like trying to balance a scale where people aren't just "Red Hat" or "Blue Hat," but are wearing hats with 0 to 9 feathers on them.
The feathers are unevenly distributed. Most people have 0 or 1 feather. Very few have 8 or 9. Some people have missing data (we don't know how many feathers they have).

The researchers asked: "Does our magical scale still work when the exposure is a messy count of symptoms, especially when some data is missing?"

The Five Methods Tested

The researchers tested five different ways to build this "magical scale" to see which one worked best for counting symptoms:

Multinomial Binning: Imagine grouping the feathers into buckets (0, 1-2, 3-4, 5+). It turns the messy count into simple categories. Verdict: Works well, easy to understand.
CBPS (Covariate Balancing Propensity Score): A smart algorithm that tries to force the red and blue hats to balance perfectly, even if the math is a bit rigid. Verdict: Works very well, fast.
npCBPS (Non-Parametric): A "free-for-all" version that tries to balance without assuming any specific math rules. Verdict: Disaster. It created "super-heavy" weights for a few people, making the results unstable and unreliable.
GBM (Generalised Boosted Models): A machine learning approach that learns the patterns of the feathers by building many small decision trees. Verdict: Works well, but takes a long time to compute.
Energy Balancing: A fancy method that measures the "distance" between groups to ensure they are identical. Verdict: Works well, but is incredibly slow and computationally expensive.

The Missing Data Challenge

Real-world data is messy. People forget to answer questions.

MCAR (Missing Completely at Random): People forgot to answer just by bad luck.
MAR (Missing at Random): People forgot to answer because of something else (e.g., people with high stress were too depressed to fill out the form).

The researchers combined their "magical scale" with a technique called Multiple Imputation (filling in the missing blanks with educated guesses based on other data).

The Findings on Missing Data:

When data was missing randomly, the good methods (Multinomial, CBPS, GBM) still worked great.
When data was missing because of the stress itself (MAR), all methods (even the standard ones) got a little bit biased.
The Twist: The bias wasn't because the "scale" was broken. It was because the guessing game (imputation) for the missing symptom counts was hard. Trying to guess a missing number of symptoms (which are often skewed and have weird spikes) is tricky. The researchers found that the problem was in how they filled in the missing numbers, not in the weighting method itself.

The Real-World Test

They took these methods and applied them to the actual 1970 British Cohort Study data.

Result: They found that higher psychological distress at age 34 did increase the risk of having a long-term illness at age 42.
The Lesson: The "standard" way of looking at the data (without fixing the confounders) made the risk look huge (almost double). After using the "magical scale" (IPTW), the risk was still significant, but more moderate (about 1.5 times higher).
The Winner: The CBPS and Multinomial methods were the stars. They were accurate, fast, and didn't break the scale. The npCBPS method was a flop and should be avoided for this type of data.

The Takeaway

If you are a researcher trying to figure out the effect of a "count" variable (like number of hospital visits, number of symptoms, or number of cigarettes) on an outcome:

Don't use the "non-parametric" (npCBPS) method. It creates unstable weights and gives bad answers.
Use Multinomial Binning or CBPS. They are reliable, fast, and handle the data well.
Be careful with missing data. If you have to guess missing counts, make sure your guessing model is good, or your results will be off, regardless of how good your weighting method is.

In short: The "magical scale" works for counting symptoms, but you have to pick the right type of scale, and you have to be very careful when filling in the missing blanks.

1. Problem Statement

Inverse Probability of Treatment Weighting (IPTW) is a standard method for estimating causal effects in observational studies by creating a pseudo-population where exposure is independent of confounders. While well-established for binary exposures, its application to count exposures (discrete, non-negative integers like symptom counts or hospital visits) is under-researched. Count data often exhibit specific distributional challenges, including overdispersion, zero-inflation, and truncation, which complicate the specification of the exposure model required for weight estimation.

Furthermore, real-world datasets frequently contain missing data. While Multiple Imputation (MI) combined with IPTW (using the "within-imputation" approach) is a common strategy for binary exposures, its performance and the behavior of flexible weighting methods under missingness mechanisms (Missing Completely at Random - MCAR, and Missing at Random - MAR) for count exposures remain unclear.

2. Methodology

Study Design

The authors conducted a simulation study adhering to the ADEMP framework, informed by a motivating example from the 1970 British Cohort Study (BCS70). The goal was to estimate the causal effect of psychological distress (measured as a symptom count at age 34) on longstanding illness (binary outcome at age 42).

Data-Generating Mechanisms (DGMs):

Sample Size: 2,000 datasets with 5,000 observations each.
Exposures: Two types were simulated:
1. Truncated Negative Binomial: Mimicking overdispersed count data (mean $\approx$ 1.8, dispersion $k \approx$ 1.3).
2. Truncated Poisson: Mimicking equidispersed count data (mean $\approx$ 1.9).
Missingness: Introduced in confounders, exposure, and outcome at levels of 20%, 40%, and 60% under both MCAR and MAR mechanisms.

Methods Evaluated

Five IPTW approaches were compared for weight estimation:

Multinomial Binning: Grouping the count exposure into categories and using multinomial regression to estimate weights.
Covariate Balancing Propensity Scores (CBPS): Parametric approach optimizing weights to balance covariates.
Non-Parametric CBPS (npCBPS): Directly optimizing weights without specifying the exposure distribution.
Generalised Boosted Models (GBM): Using machine learning (gradient boosting) to flexibly model exposure-covariate relationships.
Energy Balancing: Minimizing energy distance metrics to ensure independence between exposure and covariates.

Analysis Strategy:

Weights were applied to a modified Poisson regression model to estimate the Risk Ratio (RR).
Missing data was handled via Multiple Imputation by Chained Equations (MICE) using the "within-imputation" approach, pooling results via Rubin's rules.
Performance was assessed using bias, coverage, effective sample size (ESS), covariate balance, and computational speed.

3. Key Contributions

First Systematic Evaluation of IPTW for Counts: This study fills a critical gap by specifically testing IPTW methods on count exposures with realistic distributional features (overdispersion, truncation).
Integration with Missing Data: It provides empirical evidence on how these weighting methods perform when combined with MI under varying missingness mechanisms, a scenario rarely tested for count data.
Comparative Analysis of Flexible Methods: It benchmarks modern, flexible weighting techniques (GBM, Energy Balancing, npCBPS) against traditional binning and parametric approaches in a count-data context.
Practical Guidance: The study offers concrete recommendations on method selection based on trade-offs between bias, precision, and computational cost.

4. Key Results

Performance Under Complete Data

Top Performers: Multinomial, CBPS, GBM, and Energy Balancing produced low bias (<3%) and near-nominal coverage (95%) for both Negative Binomial and Poisson distributions.
Poor Performer: npCBPS failed significantly, producing large negative bias (-15% to -26%) and poor coverage. This was attributed to the generation of extreme weights and low Effective Sample Size (ESS).
Winsorisation: Trimming weights at the 99th percentile had a modest impact. It slightly increased bias for top performers but significantly improved the performance of npCBPS (though it remained inferior to other methods).

Performance Under Missing Data

MCAR: All top-performing methods (Multinomial, CBPS, GBM, Energy) maintained low bias even with 60% missing data.
MAR: Bias increased as missingness increased for all methods (including standard covariate-adjusted regression).
- Critical Finding: The increased bias under MAR was not specific to IPTW. It was also observed in regression adjustment.
- Sensitivity Analysis: When the exposure was kept fully observed (missingness only in covariates/outcome), bias disappeared. This suggests the bias under MAR arises from the imputation model's inability to handle right-truncated, overdispersed counts (specifically the limitations of Predictive Mean Matching for sparse tail values), rather than a failure of the IPTW framework itself.

Computational Efficiency

Fastest: CBPS (approx. 0.05s).
Slowest: Energy Balancing (approx. 1,285x slower than Multinomial).
Trade-off: GBM and Energy Balancing offer flexibility but at a high computational cost.

5. Significance and Conclusions

Recommendation for Practice: Researchers analyzing count exposures should use Multinomial Binning, CBPS, GBM, or Energy Balancing. These methods provide robust causal estimates with good balance and precision.
Avoid npCBPS: The non-parametric CBPS approach is not recommended for count exposures due to its tendency to generate extreme weights and unstable estimates in this context.
Handling Missing Data: IPTW can be successfully combined with MI for count data under MCAR. Under MAR, the primary source of bias is the imputation model, not the weighting method. The study highlights a need for better imputation models capable of handling right-truncated, overdispersed counts (e.g., models that generate values from a distribution rather than relying on donor matching).
Real-World Application: In the motivating BCS70 example, all robust weighting methods yielded consistent risk ratios (approx. 1.50–1.53) for the effect of psychological distress on longstanding illness, confirming the simulation findings.

In summary, the paper validates several IPTW methods for count data while cautioning against the use of npCBPS and highlighting the critical role of appropriate imputation models when dealing with missing count data.