Derivation of Passing-Bablok regression from Kendall's tau.

Florian Dufey

doi:10.1515/ijb-2019-0157

Abstract

It is shown how Passing's and Bablok's robust regression method may be derived from the condition that Kendall's correlation coefficient tau shall vanish upon a scaling and rotation of the data. If the ratio of the standard deviations of the regressands is known, a similar procedure leads to a robust alternative to Deming regression, which is known as the circular median of the doubled slope angle in the field of directional statistics. The derivation of the regression estimates from Kendall's correlation coefficient makes it possible to give analytical estimates of the variances of the slope, intercept, and of the bias at medical decision point, which have not been available to date. Furthermore, it is shown that using Knight's algorithm for the calculation of Kendall's tau makes it possible to calculate the Passing-Bablok estimator in quasi-linear time. This makes it possible to calculate this estimator rapidly even for very large data sets. Examples with data from clinical medicine are also provided.

Highlights

The comparison of measurements obtained with different instruments or test formats are a biometrical problem familiar to every scientist
In the field of clinical chemistry, Passing and Bablok [6] proposed a regression method, to be called classical Passing–Bablok regression in the following, which may be considered a variant of Theil–Sen regression (TSR) [7, 8], adapted to take into account the variation of the ordinate and of the abscissa
It can be seen that the least squares regression (LSR), TSR and the classical Passing–Bablok regression (cPBR) are sensitive to the choice of the dependent and independent variables, while the least product regression (LPR) and equivariant Passing–Bablok regression (PBR) (ePBR) yield results which are more centered than those of the other methods

Summary

Introduction

The comparison of measurements obtained with different instruments or test formats are a biometrical problem familiar to every scientist. While disciplines differ considerably with respect to the preferred methods, regression is of general importance [1] Since both measurements to be compared have some level of imprecision, least squares regression (LSR) is not the best choice, as it assumes the abscissa values to be error free. The results of these methods may be seriously biased by a small number of outlying measurements To overcome this problem, in the field of clinical chemistry, Passing and Bablok [6] proposed a regression method, to be called classical Passing–Bablok regression (cPBR) in the following, which may be considered a variant of Theil–Sen regression (TSR) [7, 8], adapted to take into account the variation of the ordinate and of the abscissa

Dufey: Derivation of Passing–Bablok regression

Synopsis of the relevant regression methods

Coordinate transformations

Passing–Bablok regression

Circular median of the doubled angle

Confidence interval of the slope estimate

Confidence intervals for the intercept and bias

Findings

Discussion