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Set-valued data analysis for interlaboratory comparisons

Sébastien Petit, Sébastien Marmin, Nicolas Fischer

TL;DR

This work develops a Bayesian framework for set-valued data arising in interlaboratory comparisons, by modeling a central consensus set $A\in\mathcal{P}_n$ through Hamming-distance based distributions $P_{A,u}$ and extending to hierarchical structures to capture within-laboratory effects. It introduces two inference strategies for one-stage data (brute-force and MCMC) and a specialized two-stage approach for laboratory-level dependencies, with Fisher's noncentral hypergeometric and Binomial families as interpretable dispersion mechanisms. The methodology is demonstrated on an EMC Eurolab France dataset to identify a consensual $n=10$ subset among $M=55$ items and to assess deviations via posterior $p$-values and decision signals. A substantial within-laboratory effect is detected, supported by a Bayes factor vastly favoring the hierarchical model, and the analysis provides practical guidance for labeling atypical responses and quantifying lab-specific biases. The work offers a rigorous, scalable framework for set-valued analysis in ILC and lays groundwork for extensions to alternative dispersion families and Bayesian outlier detection in set-valued data.

Abstract

This article introduces tools to analyze set-valued data statistically. The tools were initially developed to analyze results from an interlaboratory comparison made by the Electromagnetic Compatibility Working Group of Eurolab France, where the goal was to select a consensual set of injection points on an electrical device. Families based on the Hamming-distance from a consensus set are introduced and Fisher's noncentral hypergeometric distribution is proposed to model the number of deviations. A Bayesian approach is used and two types of techniques are proposed for the inference. Hierarchical models are also considered to quantify a possible within-laboratory effect.

Set-valued data analysis for interlaboratory comparisons

TL;DR

This work develops a Bayesian framework for set-valued data arising in interlaboratory comparisons, by modeling a central consensus set through Hamming-distance based distributions and extending to hierarchical structures to capture within-laboratory effects. It introduces two inference strategies for one-stage data (brute-force and MCMC) and a specialized two-stage approach for laboratory-level dependencies, with Fisher's noncentral hypergeometric and Binomial families as interpretable dispersion mechanisms. The methodology is demonstrated on an EMC Eurolab France dataset to identify a consensual subset among items and to assess deviations via posterior -values and decision signals. A substantial within-laboratory effect is detected, supported by a Bayes factor vastly favoring the hierarchical model, and the analysis provides practical guidance for labeling atypical responses and quantifying lab-specific biases. The work offers a rigorous, scalable framework for set-valued analysis in ILC and lays groundwork for extensions to alternative dispersion families and Bayesian outlier detection in set-valued data.

Abstract

This article introduces tools to analyze set-valued data statistically. The tools were initially developed to analyze results from an interlaboratory comparison made by the Electromagnetic Compatibility Working Group of Eurolab France, where the goal was to select a consensual set of injection points on an electrical device. Families based on the Hamming-distance from a consensus set are introduced and Fisher's noncentral hypergeometric distribution is proposed to model the number of deviations. A Bayesian approach is used and two types of techniques are proposed for the inference. Hierarchical models are also considered to quantify a possible within-laboratory effect.
Paper Structure (20 sections, 1 theorem, 31 equations, 11 figures, 4 tables, 2 algorithms)

This paper contains 20 sections, 1 theorem, 31 equations, 11 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Notations and introducing example
  4. A statistical model for set-valued data
  5. Hamming-distance probability distributions over subsets
  6. Bayesian statistical model on $\mathcal{P}_n$
  7. Taking a possible within-laboratory effect into account
  8. Fisher's noncentral hypergeometric distribution
  9. Binomial distribution
  10. Inference
  11. Inference for a one-stage model
  12. Inference for a two-stage model
  13. Application
  14. Case study
  15. Statistical analysis, all operators pooled together
  16. ...and 5 more sections

Key Result

Proposition 1

Let $A \in \mathcal{P}_n$. If $u < N/(N + n)$, then $A$ is the unique mode of $P_{A, u}$. Furthermore, if $u \leq 1/2$ and $n \leq N / 2$, then $P_{A, u}$ is Hamming decreasing.

Figures (11)

  • Figure 1: Choices of $78$ operators from $26$ laboratories. The $x$-axis represents the elements of $\mathcal{O}$ and each row represents the choices made by one operator. Colors are used to identify laboratories.
  • Figure 2: The number of $X_i$s containing each element of $\mathcal{O}$.
  • Figure 3: Representation of model \ref{['eq:hierarchical']} as a directed acyclic graph.
  • Figure 4: Left: uniform prior on $\{ 0 < p_2 \leq p_1 < 1\}$. Middle: the resulting prior probability density on $u$. Right: kernel density estimate of the resulting prior probability distribution function on the mean $\sum_{k=0}^n k \cdot e_u(k)$ of $e_u$.
  • Figure 5: Kernel density estimates of the posterior probability distribution function of $u$ using $10^3$ posterior samples obtained using the brute-force scheme on the data from Table \ref{['tab:example']} with Fisher's noncentral hypergeometric distribution, a uniform prior for $A$, and the prior \ref{['eq:prior_omega']} for $u$. A uniform grid of size $10^4$ was used to invert the posterior cumulative distribution function of $u$. Left: posterior probability distribution function of $u$. Right: posterior probability distribution function of the mean of $e_u$.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Remark
  • Proposition 1