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

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.