Hyperbolic confidence bands of errors-in-variables regression lines applied to method comparison studies

Francq, Bernard G.; Govaerts, Bernadette B.

Journal de la société française de statistique, Tome 155 (2014) no. 1, pp. 23-45.

Cet article se concentre sur la construction et l’évaluation de la qualité de bandes de confiance de droites de régression à erreurs sur les variables utilisées dans le contexte particulier de la comparaison de méthodes de mesure. La comparaison de méthodes de mesure vise à vérifier, sur base de données expérimentales, que deux méthodes de mesure fournissent des résultats équivalents. En l’absence de biais analytique, deux méthodes, entachées d’erreurs de mesure, doivent fournir des résultats en moyenne identiques, c’est-à-dire distribués autour de la droite $Y=X$ de pente $\left(\beta \right)$ égale à 1 et d’ordonnée à l’origine $\left(\alpha \right)$ égale à 0. Pour tester cette hypothèse, un intervalle de confiance (IC) joint est idéalement utilisé en tenant compte des erreurs de mesure sur les deux axes. Les régressions DR (Régression de Deming) et BLS (Bivariate Least Square regression) fournissent des droites de régression consistantes et confondues sous homoscédasticité. Leurs IC joints sous forme d’ellipses dans un plan $\left(\beta ,\alpha \right)$ sont présentés et cet article propose de transformer ces IC joints en des bandes de confiance hyperboliques pour la droite dans le repère $\left(X,Y\right)$ qui sont plus faciles à mettre en graphique et à interpréter par le praticien. Quatre méthodes pour les calculer sont proposées et leurs propriétés et avantages respectifs discutés. Une comparaison détaillée est fournie basée, entre autre, sur des simulations et des données réelles. Lorsque les variances des erreurs sont connues, les taux de couverture sont semblables, mais les IC joints calculés avec le maximum de vraisemblance (ML) ou la méthode des moments fournissent des taux de couverture légèrement meilleurs. Quand les variances des erreurs sont inconnues et hétéroscédastiques, les taux de couverture du ML chutent de façon spectaculaire tandis que le BLS donne des meilleurs taux de couverture.

This paper focuses on the confidence bands of errors-in-variables regression lines applied to method comparison studies. When comparing two measurement methods, the goal is to ’proof’ that they are equivalent. Without analytical bias, they must provide the same results on average notwithstanding the errors of measurement. The results should not be far from the identity line $Y=X$ (slope $\left(\beta \right)$ equal to 1 and intercept $\left(\alpha \right)$ equal to 0). A joint-CI is ideally used to test this joint hypothesis and the measurement errors in both axes must be taken into account. DR (Deming Regression) and BLS (Bivariate Least Square regression) regressions provide consistent estimated lines (confounded under homoscedasticity). Their joint-CI with a shape of ellipse in a $\left(\beta ,\alpha \right)$ plane already exist in the literature. However, this paper proposes to transform these joint-CI into hyperbolic confidence bands for the line in the $\left(X,Y\right)$ plane which are easier to display and interpret. Four methodologies are proposed based on previous papers and their properties and advantages are discussed. The proposed confidence bands are mathematically identical to the ellipses but a detailed comparison is provided with simulations and real data.When the error variances are known, the coverage probabilities are very close to each other but the joint-CI computed with the maximum likelihood (ML) or the method of moments provide slightly better coverage probabilities. Under unknown and heteroscedastic error variances, the ML coverage probabilities drop drastically while the BLS provide better coverage probabilities.

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