A statistical note on extending Christensen's limits of agreement with the mean

📅 2025-08-22
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🤖 AI Summary
This study addresses the neglect of subject–observer interaction effects in assessing inter-observer agreement for continuous measurements. We extend Christensen’s mean-based Limits of Agreement (LOAM) to a two-factor random-effects model incorporating interaction terms. By decomposing variance components, we rigorously distinguish between repeatability LOAM (within-observer) and reproducibility LOAM (between-observers), and derive their asymptotic confidence intervals, sample size formulas, and statistical tests for comparing LOAMs across measurement systems. The proposed framework unifies variance component estimation, LOAM inference, and hypothesis testing, thereby enhancing both the precision and interpretability of measurement system analysis. It provides a theoretically rigorous yet practically implementable tool for evaluating measurement consistency in clinical and biomedical research.

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📝 Abstract
Limits of agreement with the mean (LOAM) can be used for assessing agreement of continuous measurements made by different observers. Definitions of a LOAM for measuring reproducibility has been introduced under a two-way random effects model without interaction between subject and observer. Here we extend that model framework to include a subject-observer interaction, allowing the separation of residual measurement error and systematic variation in how individual observers measure specific objects. Further, our framework extends the LOAM concept to two metrics: one quantifying reproducibility and the other repeatability. We supply estimates and confidence intervals for the reproducibility and repeatibility LOAM and discuss sample size calculations and a test to compare LOAMs between two groups. To make the text self-contained for a complete agreement analysis, we additionally provide estimates and confidence intervals for the variance components.
Problem

Research questions and friction points this paper is trying to address.

Extending LOAM to include subject-observer interaction effects
Separating measurement error from systematic observer variation
Developing reproducibility and repeatability metrics with confidence intervals
Innovation

Methods, ideas, or system contributions that make the work stand out.

Extends LOAM with subject-observer interaction term
Introduces dual metrics for reproducibility and repeatability
Provides variance components estimates with confidence intervals
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H
Heidi Søgaard Christensen
Center for Clinical Data Science, Department of Clinical Medicine, Aalborg University, Denmark
M
Martin Bøgsted
Center for Clinical Data Science, Department of Clinical Medicine, Aalborg University, Denmark
J
Jens Borgbjerg
Department of Radiology, Akershus University Hospital, Norway