Bland–Altman Limits of Agreement from a Bayesian and Frequentist Perspective

Oke Gerke,Sören Möller

doi:10.3390/stats4040062

Abstract

Bland–Altman agreement analysis has gained widespread application across disciplines, last but not least in health sciences, since its inception in the 1980s. Bayesian analysis has been on the rise due to increased computational power over time, and Alari, Kim, and Wand have put Bland–Altman Limits of Agreement in a Bayesian framework (Meas. Phys. Educ. Exerc. Sci. 2021, 25, 137–148). We contrasted the prediction of a single future observation and the estimation of the Limits of Agreement from the frequentist and a Bayesian perspective by analyzing interrater data of two sequentially conducted, preclinical studies. The estimation of the Limits of Agreement θ1 and θ2 has wider applicability than the prediction of single future differences. While a frequentist confidence interval represents a range of nonrejectable values for null hypothesis significance testing of H0: θ1 ≤ −δ or θ2 ≥ δ against H1: θ1 > −δ and θ2 < δ, with a predefined benchmark value δ, Bayesian analysis allows for direct interpretation of both the posterior probability of the alternative hypothesis and the likelihood of parameter values. We discuss group-sequential testing and nonparametric alternatives briefly. Frequentist simplicity does not beat Bayesian interpretability due to improved computational resources, but the elicitation and implementation of prior information demand caution. Accounting for clustered data (e.g., repeated measurements per subject) is well-established in frequentist, but not yet in Bayesian Bland–Altman analysis.

Highlights

The Bland–Altman plot for method comparison on continuous outcomes has its roots in Tukey’s mean-difference plot in which means and differences of paired measurements are shown in a scatterplot [1]
The prediction interval for a future observation follows the description of Carstensen [20]
Vock [26]: estimated mean q difference ± c PI times the standard deviation of the difn +1 α ferences with c PI = t1− α ;n−1 n, n observed differences, and the (1 − 2 ) quantile of Student’s t distribution with (n − 1) degrees of freedom, t1− α ;n−1

Summary

Introduction

The Bland–Altman plot for method comparison on continuous outcomes has its roots in Tukey’s mean-difference plot in which means and differences of paired measurements (from, for instance, two measurement devices) are shown in a scatterplot [1]. Altman and Bland defined the two parameters θ1 = μ − 1.96σ and θ2 = μ + 1.96σ for normally distributed differences, with mean μ and standard deviation σ, and called these lower and upper Limits of Agreement (LoA), respectively [2]. They extended Tukey’s mean-difference plot by adding empirical estimates for θ1 and θ2 as θ1 and θ2 are the boundaries within which, under the assumption of normally distributed differences, 95% of all population differences are supposed to fall [2,3]. Further extensions have been proposed [6,7,8,9,10,11], as well as reporting standards for [12,13] and alternative modes of analysis to the LoA [14,15,16,17,18]

Objectives

Methods

Conclusion