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Generation of Multivariate Discrete Data with Generalized Poisson, Negative Binomial and Binomial Marginal Distributions

This paper proposes a new algorithm for generating multivariate discrete data with pre-specified correlations using generalized Poisson, negative binomial, and binomial marginal distributions, demonstrating its effectiveness through both simulated and real-world scenarios.

Original authors: Chak Kwong, Cheng, Hakan Demirtas

Published 2026-02-11

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CC BY 4.0

Original authors: Chak Kwong, Cheng, Hakan Demirtas

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 chef trying to create a "synthetic" version of a complex, multi-course meal. You want the meal to taste exactly like a real one—the steak should be juicy, the potatoes should be fluffy, and the wine should be crisp—but you want to be able to control exactly how much salt or pepper is in each dish to test how people react to different flavors.

In the world of statistics, researchers often need to do exactly this. They need to create "fake" (synthetic) data that looks and behaves exactly like "real" data. This is important because real data is often private (like medical records) or hard to get.

This paper, written by Tommy Cheng and Hakan Demirtas, introduces a new "recipe" for creating these synthetic datasets.

The Problem: The "Connectedness" Challenge

Imagine you are studying a group of people. You notice that people who exercise more tend to eat more vegetables and sleep better. These three things—exercise, diet, and sleep—are correlated. They are "connected."

If a scientist wants to create a fake dataset to test a new medical software, they can't just make up random numbers for exercise, random numbers for diet, and random numbers for sleep. If they do, the "fake" people will look impossible (e.g., someone who runs marathons every day but eats zero calories). The data wouldn't "make sense" because the connections between the variables would be broken.

The Solution: The "Collapsing" Trick

The researchers developed a clever way to build these connections. Instead of trying to build a massive, complicated structure all at once, they use a method that works like building with LEGO blocks.

Here is their step-by-step "recipe" in plain English:

The Simplification (The "Binary" Step): They take complex, multi-valued data (like "how many times did you visit the doctor this year: 0, 1, 2, 5, or 10?") and temporarily simplify it into a simple "Yes/No" question (e.g., "Did you visit the doctor? Yes or No").
The Blueprint (The Correlation): It is much easier to figure out how "Yes/No" answers connect to each other than it is to figure out how complex numbers connect. They create a blueprint that says, "If the answer to Question A is 'Yes,' there is a 70% chance the answer to Question B is also 'Yes'."
The Expansion (The "Reverse" Step): Once they have the "Yes/No" connections perfectly mapped out, they use a mathematical "expansion" trick to turn those simple "Yes/No" answers back into the original, complex numbers (0, 1, 2, 5, 10...) while keeping those connections intact.

What makes this special?

Before this paper, most methods were like a "one-trick pony." You could use them if your data followed one specific pattern, but if your data was a mix of different types, the method would break.

This new method is like a Swiss Army Knife. It can handle:

Generalized Poisson: Data that counts rare events (like how many times a lightbulb breaks).
Negative Binomial: Data where things tend to "clump" together (like how many people show up to a party).
Binomial: Data with a fixed limit (like how many heads you get in 10 coin flips).
The "Mixed" Bag: It can even mix all of these together in one single dataset!

Why does this matter to you?

While this sounds like heavy math, it has real-world impact:

Medicine: Researchers can create "fake patients" to test new drugs without violating anyone's privacy.
Crime Prevention: As shown in the paper, they can simulate crime patterns in cities to help police plan better.
Public Health: They can simulate how diseases spread through a population to prepare for future outbreaks.

In short: They have created a high-tech "digital twin" generator for data, allowing scientists to practice and experiment in a safe, controlled, and incredibly realistic virtual world.

Technical Summary: Generation of Multivariate Discrete Data with Generalized Poisson, Negative Binomial, and Binomial Marginal Distributions

1. Problem Statement

In fields such as epidemiology, genomics, and the social sciences, researchers frequently encounter multivariate discrete data (count data) where variables are correlated. While stochastic simulation is essential for assessing the performance of statistical models and evaluating estimator properties (bias, consistency, etc.), existing methods for simulating correlated count data often suffer from significant limitations:

Lack of Flexibility: Many algorithms (e.g., NORTA-based) are restricted to a single distribution type.
Correlation Constraints: Some methods are limited to positive correlations or low-correlation/low-mean scenarios.
Structural Limitations: Certain approaches can only handle exchangeable structures or require complex joint distribution models involving latent variables.

The authors aim to provide a robust, flexible framework capable of generating multivariate data with diverse marginal distributions (Generalized Poisson, Negative Binomial, and Binomial) and a wide range of correlation structures (both positive and negative).

2. Methodology

The proposed algorithm is an extension of the methodology developed by Demirtas (2006) for multivariate ordinal data. The core strategy involves a "collapse-and-reconstruct" approach using latent multivariate normal variables.

The Algorithm Steps:

Collapsing (Dichotomization): Each discrete marginal distribution is collapsed into a binary variable. To maximize flexibility, the authors use the median of the discrete variable as a threshold. If the median does not yield a binary expectation near 0.5, it is assigned to the category that brings the expectation closer to 0.5.
Iterative Correlation Matching: Because the correlation between discrete variables ( $\delta^*_{ij}$ ) is not identical to the correlation between their collapsed binary counterparts ( $\delta^b_{ij}$ ), the authors use an iterative simulation-based procedure. They start with the target correlation, generate binary data, convert it back to the discrete scale, and adjust the binary correlation until the resulting discrete correlation matches the target within a specified tolerance.
Positive Definiteness: Since the assembled binary correlation matrix ( $\Sigma$ ) may not be positive definite, the algorithm applies Higham’s (2002) method to find the nearest positive definite matrix.
Latent Variable Generation: Multivariate binary data is generated using the Emrich and Piedmonte (1991) approach, which utilizes a multivariate normal distribution and tetrachoric correlation.
Reconstruction (Reverse-Collapsing): The generated binary data is mapped back to the original discrete scale using the probability mass functions (PMFs) and the thresholds established in Step 1.

3. Key Contributions

Unified Framework: A single algorithm that accommodates three major families of discrete distributions (Generalized Poisson, Negative Binomial, and Binomial) and their mixtures.
Distributional Flexibility: Unlike previous methods, it allows for both over-dispersion (via Generalized Poisson/Negative Binomial) and under-dispersion.
Correlation Versatility: It supports various correlation structures, including negative correlations and non-exchangeable structures (e.g., Toeplitz).
Software Implementation: The authors released the MultiDiscreteRNG R package, providing validated functions for parameter checking, collapsing, and mixed-distribution generation.

4. Results

The methodology was evaluated through extensive simulation studies and real-world data applications:

Simulation Studies: Across four scenarios (Generalized Poisson, Negative Binomial, Binomial, and Mixed), the method demonstrated high accuracy and precision.
- Accuracy: Relative Bias (RB) was generally below the 5% threshold.
- Precision: Standardized Bias (SB) and RMSE decreased as sample size increased from $N=200$ to $N=2000$ .
- Reliability: Coverage Rates (CR) remained consistently near the nominal 95% level.
Real-Data Applications: The method successfully replicated parameters and correlations from three distinct datasets:
- NMES (Medical Expenditure): Recovered Generalized Poisson parameters for healthcare usage.
- AHS (Australian Health Survey): Recovered Negative Binomial parameters for healthcare utilization.
- LA Crime Data: Successfully handled a mixed-distribution scenario (combining Generalized Poisson and Negative Binomial) to model criminal offense frequencies.

5. Significance

This paper provides a critical tool for researchers needing to create synthetic datasets that mimic the complex, correlated, and dispersed nature of real-world discrete data. By decoupling the marginal distributions from the correlation structure, the method offers a level of flexibility that allows for more rigorous testing of statistical models in diverse scientific domains. While the authors note limitations regarding correlation bounds (imposed by marginal expectations), the framework represents a significant advancement in multivariate discrete data simulation.